Group Theoretical Methods in Solid State Physics

Transcription

1 Group Theoretical Methods in Solid State Physics

2 Preface This lecture introduces group theoretical concepts and methods with the aim of showing how to use them for solving problems in atomic, molecular and solid state physics. It consists of a three-hours per week lecture and three-hours per week exercises and extends over 13 weeks. The definitive list of chapters is the following: 1. Abstract Group Theory and representation theory of groups 2. Group theory and eigenvalue problems 3. Continuos groups and symmetry breaking interactions 4. Space groups and their representations 5. Kronecker (direct) products 6. Applications of direct products: Spin 1 2 particles. The relevant literature for the topics presented in this lectures is: L.D. Landau, E.M. Lifshitz, Lehrbuch der Theor. Pyhsik, Band III, Quantenmechanik, Akademie-Verlag Berlin, 1979, Kap. XII; Ibidem, Band V, Statistische Physik, Teil 1, Akademie-Verlag 1987, Kap. XIII and XIV. Further relevant papers and literature is distributed as part of the manuscript. D. Zanin, Th. Gumbsch and Y. Forrer provided most of the solutions to the exercises, H. Pescia rendered the manuscript in LaTex. The student in charge of exercises this year is Y. Forrer ( yforrer@stud.phys.ethz.ch) Zürich, February 2016 D. Pescia pescia@solid.phys.ethz.ch ii

3 Contents Preface ii 1 Abstract Group Theory and representation theory of groups Transformation groups Point and space groups Examples of point groups Elementary properties of groups Representation theory of groups Matrix representations of a group (H. Weyl, ) Reducible and irreducible representations Properties of irreducible representations Characters of a representation and theorems involving them Symmetry adapted vectors A practical example Group theory and eigenvalue problems 43 3 Continuos groups and symmetry breaking interactions Continuos groups Representations of continuous Lie groups Example: the circle group The full rotation group The translation group Cristal field splitting Appendix I: Proof of The Peter-Weyl theorem Appendix II: The Harisch-Chandra character iii

4 Contents 3.6 Appendix III: The Harisch-Chandra character for the non-compact group Space groups and their representations Elements of crystallography An example of space groups The Translation group and its irreducible representations Irreducible representations of the group of k Non-symmorphic space groups Kronecker (direct) products The Kronecker product of matrices The Kronecker product of groups Applications of direct products An introduction to the universal covering group The universal covering group of SO SU(2) Symmetry adapted functions in atomic physics: the Clebsch-Gordan coefficients Double point groups The Clebsch-Gordan coefficients for point groups The Wigner-Eckart-Koster theorem Appendix Some linear algebra iv

5 1 Abstract Group Theory and representation theory of groups 1.1 Transformation groups We consider a set M containing, as elements, objects x, y,.. such as e.g. vectors, numbers, atoms, etc. A transformation g is a one-to one map from M to M, i.e g : x g(x) M, x M Consider now a physical body and a transformation that transform a physical body into itself. The transformation is called a symmetry transformation. Theorem: The set G of all symmetry transformations forms a group. The group is called the symmetry group of the system. Proof. First, we define a composition law consisting of performing two symmetry transformations of the system successively: by convention, b a means that a is performed first, followed by b. The result of performing two symmetry transformations successively is to transform the system into itself: i.e. Figure 1.1: If the atoms above and below the plane are identical, the plane is a mirror plane 1

6 1 Abstract Group Theory and representation theory of groups 1. if a and b are symmetry operations, b a is also a symmetry operation This means that the set G is closed under the law of successive transformations. 2. We can also define an identity transformation e which leave the system invariant: e belongs obviously to G 3. Given a symmetry transformation a we see that there exists an inverse transformation a 1 which transform back the system into itself: a 1 also belongs to G 4. Finally, the set G and the product are endowed with a rule of computation, i.e. obey the associative laws: a (b c) = (a b) c These four properties define the set G to be a group. In general, a set which has the above four properties is called a group. examples of groups are the following. 1. The additive group of real numbers. The set of real numbers becomes a group under the usual addition. The identity is the number zero and the inverse of x is x. This group will be denoted by. 2. The additive group of integers: The set of integers under the usual addition is a group. It is denoted by. 3. The multiplicative group of non-zero real numbers: The set of the real numbers without the number zero is a group under the usual multiplication. The identity is the number 1 and the inverse element of x is 1/x. This group is denoted by. 4. The Symmetrical Group: Assume M is a set consisting of N elements. A bijective map π : M M is called a permutation of M. The permutations of M form a group under the usual composition of functions. This group is called the symmetrical group and it is denoted by N. 5. The n-dimensional space: The set of all n-dimensional vectors with real components is a group under the normal addition of vectors. This group will be denoted by n. 6. the set consisting of (1, 1) under the standard rule of multiplication 2

7 1 Abstract Group Theory and representation theory of groups 7. the set complex numbers (1, i, 1, i) under the standard rule of multiplication 8. the set of all matrices of order n n under addition 9. Complex Numbers: The complex numbers under addition and the non-zero complex numbers under multiplication form groups. These groups are denoted by and 0, respectively. 10. The general linear group: The set of complex matrices of size n n with nonzero determinant is a group under ordinary matrix multiplication. It is denoted by GL(n, ). In the same way we define GL(n, ) as the set of real n n matrices with non-zero determinant. 11. The special linear groups: The sets of real or complex matrices of size n n with determinant 1 are groups. These groups are denoted by SL(n, ) and S L(n, ) respectively. 12. The orthogonal groups: An orthogonal matrix R is an element of GL(n, ) that satisfies the condition R t R = E where R t is the transpose of R and E is the identity matrix. The set of all orthogonal matrices defines a group under matrix multiplication. It is denoted by O(n). 13. The special orthogonal groups: The set of all orthogonal matrices R O(n) with detr = 1 is a group, the special orthogonal group SO(n). 14. The unitary groups: a complex matrix U is unitary if U U = E, where U is the transposed, conjugate complex of U. The set of all n n unitary matrices forms a group under matrix multiplication. This group is denoted by U(n). 15. The special unitary groups: The set of all unitary matrices of size n n with detu = 1 is also a group, the special unitary group. These groups are denoted by SU(n). 1.2 Point and space groups Symmetry transformations can be divided into three types: 3

8 1 Abstract Group Theory and representation theory of groups 1. Rotations by a certain angle about some axis and reflections at a certain plane 2. translations by some vector and 3. combination of the type 1. and type 2. transformations. Type 2 and 3 exist only in an infinite body: a physical system with finite dimensions (a molecule, for instance) can only have symmetry transformations which leave at least one point of the body undisplaced in position. Type 1. transformations are symmetry transformations of a finite body provided they all have a point of intersection: rotating one body around two non-intersecting axes or reflecting at two non-intersecting planes leads to a translation of the finite body, which is not transformed into itself. When a group of transformations of a certain system consists of operations which leaves one point of the system undisplaced, it is called a point group. Symmetry groups containing translations are space groups. Notation In the following, we introduce the Schönflies notation to label the symmetry transformations. When a body is transformed into itself by a rotation about a given axis by an angle 2π n (n = 1, 2, 3,...), the axis is an n-fold rotation axis. The value n = 1 corresponds to a rotation by 2π (or equivalently, 0): the corresponding operation is the identity transformation. The rotation with n 0 are denoted by C n. Its integral power will be denoted by C k n : this operation represents k successive operations of C n on the system or a rotation by 2πk n. Evidently, C k n = C n/k, if n is a multiple of k and C n n = E. When a body transforms into itself by reflection at a plane, the plane is said to be a symmetry plane. Reflections are denoted by σ (or m). An important property is that σ 2 = e. A reflection at a plane perpendicular to a certain rotational axis will be denoted by σ h (the subscript h means horizontal). Reflections at planes containing the rotational axis are indicated by σ v (v stands for vertical ). σ d (or σ v ) is a diagonal or dihedral reflection in a plane through the origin and the axis with the highest symmetry, but also bisecting the angle between the twofold axes perpendicular to the symmetry axis. A body may transform into itself by the 4

9 1 Abstract Group Theory and representation theory of groups Figure 1.2: Illustration of symmetry transformations combination of a rotation and a reflection: this operation is a so called improper rotation and is denoted by S n. S n fulfills the equation S n = C n σ h = σ h C n. The special case S 2, corresponding to a rotation by π and a reflection to a plane perpendicular to the rotation axis, is an inversion, denoted by I: each point P is mapped into a point P t (the index t means transformed ) lying on the straight line passing through P and O (O being the intersection of the rotation axis with the symmetry plane). If a body is symmetric with respect to S 2 = C 2 σ h = I, we speak of O as its inversion center. Notice that σ can always be written as IC 2, i.e. the combination of an inversion and a two-fold rotation around a suitable axis. A rotation followed by a reflection or the inversion is called an improper rotation, to be compared with a proper rotation, which does not contain reflections. Example 1: the symmetry group of a square. Suppose to have identical atoms arranged on the corners of a square, lying in the x y-plane. Provided the atoms above and below the plane are different, the x y- plane is not a mirror plane and the symmetry operations move the atoms from 5

10 1 Abstract Group Theory and representation theory of groups Figure 1.3: Illustration of C 4v and O h Type Operation Coordinate Permutation e e xyz 1234 C 4 C 4z y xz 2341 C 2 4 C 2 4z = C 2z x ȳz 3412 C 3 4 C 3 4z ȳxz σ v IC 2y (m y ) xȳz 4321 IC 2x (m x ) xyz σ v (2σ d) IC 2x y (m x y ) ȳ xz 3214 IC 2x ȳ (m x y ) yxz 1432 one corner to another without changing the position of the corners (and of the boundaries): It can be readily verified that the set of eight transformations listed in the table is a group and that it is the complete group of symmetry of a square. This symmetry group of order 8 is denoted by C 4v (Schönflies) or 4mm (Int.notation) and is one of the point groups. In the table, we list all operations and their result on x, y. Example 2: The symmetry group of a cube This group is called the cubic group O h ( m m, Int.) and is a very important example in solid state physics. With the aid of the figure, we can illustrate all 48 symmetry elements of O h. the identity e 3 rotations by π about the axes x, y, z (3C 2 4 ) 6 rotations by ±π/2 about the axes x, y, z (6C 4 ) 6

11 1 Abstract Group Theory and representation theory of groups 6 rotations by π about the bisectrices in the planes x y, yz, xz (6C 2 ) 8 rotations by ±2π/3 about the diagonals of the cube (8C 3 ) the combination of the inversion I with the listed 24 proper rotations The corresponding Table indicating the transformation of the coordinates x, y, z is given: 7

12 1 Abstract Group Theory and representation theory of groups Type Operation Coordinate Type Operation Coordinate C 1 C 1 x yz I I x ȳ z C 2 4 C 2z x ȳz IC 2 4 IC 2z x y z C 2x x ȳ z IC 2x x yz C 2 y x y z IC 2y x ȳz C 4 C 1 4z ȳ xz IC 4 IC 1 4z y x z C 4z y xz IC 4z ȳ x z C 1 4x x z y IC 1 4x xz ȳ C 4x xz ȳ IC 4x x z y C 1 4y z y x IC 1 4 y z ȳ x C 4 y z y x IC 4y z ȳ x C 2 C 2x y y x z IC 2 IC 2x y ȳ xz C 2xz z ȳ x IC 2xz z y x C 2yz xz y IC 2 yz x z ȳ C 2x ȳ ȳ x z IC 2x ȳ y xz C 2 xz z ȳ x IC 2 xz z y x C 2y z x z ȳ IC 2 y z xz y C 3 C 1 3x yz zx y IC 3 IC 1 3x yz z x ȳ C 3x yz yzx IC 3x yz z x ȳ C 1 3x ȳz z x ȳ IC 1 3x ȳz zx y C 3x ȳz ȳ zx IC 3x ȳz yz x C 1 3x ȳ z z x y IC 1 3x ȳ z zx ȳ C 3x ȳ z ȳz x IC 3x ȳ z y zx C 1 3x y z zx ȳ IC 1 3x y z z x y C 3x y z y z x IC 3x y z ȳzx Table 1.1: Coordinate transformations under the action of the elements of the group O h 8

13 1 Abstract Group Theory and representation theory of groups Figure 1.4: Illustration of the point group C 2. Top left: stereogram. Top right: a 3d object with C 2 -symmetry. Bottom: a molecule with C 2 -symmetry. 1.3 Examples of point groups The 32 point groups in 3d can be roughly divided into types. We start with the simplest ones and construct the next ones by adding symmetry elements. Cyclic symmetries This class includes C n, C nh, C nv, and S n, which have only one proper or improper rotation axis. I. The first type are the Cyclic groups C n, with n = 1, 2, 3, 4, 6, which describe the group of rotations aroung a n-fold axis. The number of elements is n. The symmetry elements are E, C 1 n, C 2 n 1 n,..., Cn. For instance, C 2 H 4 Cl 2 has symmetry group C 2. II. The pyramidal groups C nv. The symmetry elements are E, C k, n (k = 1,..., n 1) and nσ v. The order of C nv is 2n. For example, N H 3 has a symmetry C 3v. III. Reflection group C nh. The symmetry operations are E, C k n, (k = 1,..., n 1), σ h, σ h C m, n m = 1,..., n 1. The order of the group is 2n. For instance C 2H 2 F 2 has the symmetry group C 2h. IV. Improper rotation group S n. S 1 = C 1h, S 3 = C 3h. S 2 = E, I. S n (n=4,6): The symmetry operations are E, S k n, k = 1,..n 1. 9

14 1 Abstract Group Theory and representation theory of groups Figure 1.5: Top: Illustration of the group C 4. Bottom: Illustration of C 4v. Figure 1.6: Top left: stereogranm of C 2v. Top right: a 3d body. Bottom: The molecule H 2 0 with symmetry elements. 10

15 1 Abstract Group Theory and representation theory of groups Figure 1.7: Top left: stereogranm of C 3v. Top right: a 3d body. Bottom: The molecule N H 3. Figure 1.8: Top left: stereogranm of C 4v. Top right: a 3d body. Bottom: molecules with C 4v -symmetry. 11

16 1 Abstract Group Theory and representation theory of groups Figure 1.9: Top left: stereogranm of S 2. Top right: a 3d body. Bottom: a molecule with S 2 -symmetry. Dihedral symmetries This class includes D n, D nh, and D nd, which have one proper rotation C n axis and n C 2 axis perpendicular to C n axis. I.Dihedral group D n (order 2n). The symmetry elements are E, C k n (k = 1,..., n 1), and nc 2. II.Prismatic group D nh (order 4n). The symmetry operations are E, C k n (k = 1,..., n 1), nc 2, nσ s v, σ h, S m n (m = 1,..., n 1). For example, the point group of benzene is D 6h. III. Antiprismatic group D nd (order 4n). The symmetry operations are E, C k n (k = 1,..., n 1), nc 2, nσ d, S m 2n (m = 1,..., 2n 1). For example, the point group of C 2 H 6 is D3d. Polyhedral symmetries This class includes T, T h, T d, O, O h, I and I h. T, T h, T d (tetrahedral symmetries), O, O h (octahedral symmetries) are cubic groups. The O h group e.g. has the full 48 symmetry elements of a cube: E, 8C 3, 6C 2, 6C 4, 3C 2,I, 6S 4, 8S 6, 3σ h, 6σ d. For example, the point group of SF 6 is O h. I and I h are the Icosahedral groups. These 12

17 1 Abstract Group Theory and representation theory of groups Figure 1.10: Top left: stereogranm of D 6h. Top right: a 3d body. Bottom: a molecule with D 6h -symmetry. Figure 1.11: Top left: stereogranm of T d. Top right: a 3d body. Bottom: a molecule with T d -symmetry. 13

18 1 Abstract Group Theory and representation theory of groups Figure 1.12: Top left: stereogranm of O h. Top right: a 3d body. Bottom: a molecule with O h -symmetry. groups contain the symmetry elements of icosahedral. They are forbidden in 3d crystals by the requirement of translation symmetry but appear in quasicrystals and molecuales. The order of I h is 120. For example, the point group of C 60 is I h. Linear groups Other symmetry groups are useful, e.g. for describing linear molecules: the linear groups. These groups include C v and D h, which are the symmetry of linear molecules. C v has symmetry operations E, C and σ v. Examples are CO, HCN, NO, HCl. The D h group has symmetry operations E, C, C v, σ h, I, C 2. Examples are CO 2, O 2, N 2. 14

19 1 Abstract Group Theory and representation theory of groups Figure 1.13: Top left: A C 60 molecule with I h -symmetry. Top right: a molecule with C -symmetry. Bottom: a molecule with D h -symmetry. 15

20 1 Abstract Group Theory and representation theory of groups 1.4 Elementary properties of groups A group G is called commutative (or Abelian) if a b = b a a, b G. For transformation groups there are simple rules that allow to establish the commutativity of two operations: two consecutive rotations about the same axis. This rule implies that C n is commutative Two reflections at orthogonal planes (they are equivalent to a rotation by π at the axis forming the intersection of the two planes) two rotations by π about two orthogonal axes (equivalent to a rotation by the same angle about a third axis, orthogonal to both) A rotation followed by a reflection at a plane orthogonal to the rotation axis The rearrangement theorem. If {e, g 1,..., g n } are the elements of a group G and g k is an arbitrary element, than g k G. = {g k e, g k g 1,..., g k g n } and Gg k. = {egk, g 1 g k,..., g n g k } contain each group element once and only once. Proof: g k g j is certainly an element of the group, by definition of the composition law. Suppose g i g k = g j g k with g i g j Then, multiplying both sides of the equation by g 1 from the right we obtain g k i = g j which is contrary to the auumption. The rearrangement theorem has an important application for finite groups: the representation of the composition law as a multiplication table. Suppose to order the elements of a finite group at the top of a square table and on its left hand column. The bulk of the table contains the result of multiplying the top element with one element of the left hand column, whereby the top operation (by convention) is performed first. Notice that, in virtue of the rearrangement theorem, each column contains all elements only once. The ordering of the top and left hand 16

21 1 Abstract Group Theory and representation theory of groups e C 4 C 2 4 C 3 4 m y m x m x y m x y e e C 4 C 2 4 C 3 4 y x x y x y C 3 4 C 3 4 e C 4 C 2 4 x y x y x y C 2 4 C 2 4 C 3 4 e C 4 x y x y x y C 4 C 4 C 2 4 C 3 4 e x y x y y x m y y x y x x y e C 2 4 C 4 C 3 4 m x x x y y x y C 2 4 e C 3 4 C 4 m x y x y x x y y C 3 4 C 4 e C 2 4 m x y x y y x y x C 4 C 3 4 C 2 4 e Table 1.2: Multiplication Table for C 4v column elements is immaterial: the one chosen in the figure in such that along the principal diagonal one finds the operation e. A subset H of elements of a group G is a subgroup of G if it is closed with respet to the group operations. For instance {e} is a subgroup of G and G is a subgroup of itself. These are trivial subgroups. Non-trivial (proper) subgroup are very important in physics: they arise upon introduction of symmetry breaking interactions or at the transition point of second order phase transitions. Conjugacy classes. Two elements a and b of a group G are conjugate elements if there exists a group element x such that a = x bx 1 (or equivalently, a = x 1 bx). Applying the conjugation operation to one element a using every possible element x of the group produces classes of mutually conjugate elements. All element of one class are conjugated to each other but no two element belonging to different classes are conjugate to each other (this is because if a is conjugate to b and to c, then b and c are also conjugate: for from a = x 1 bx and a = y 1 c y follows b = (y x 1 ) 1 c(y x 1 ). The identity element e constitutes a class by itself. Each class is therefore completely determined by giving one single element of the class, which might be taken as the representative of the class. Each element of the group can only belong to one single class. In the case we are dealing with groups of transformations consisting of rotations, reflections and inversion of a physical system, there are some simple rules which allow the determination of the classes of a non-abelian group without having to perform explicit calculations for 17

22 all elements: 1 Abstract Group Theory and representation theory of groups Rotations through angles of different magnitudes must belong to different classes. Thus, e.g. C 4 and C 2 4 belong to different classes. Rotations by ± the same angle about an axis belong to the same class only if there is an element in the group that change the handiness of the coordinate system. Thus, C 4 and C 3 4 belong to the same class because of the existence of a vertical reflection plane Rotations through the same angle about different axes or reflections with respect two different planes belong to the same class only if the two different axis of the two different planes can be brought into each other by an element of the group. Thus, m x and m y belong to the same class as they can be brought into each other by C 4. Example: the classes of C 4v are {e}, {C 4, C 3 4 }, {C 2 4 }, {m x, m y }, {m x y, m x y } Each element of a commutative group builds a class on its own. In fact, from x a x 1 = a (because of commutativity) it follows that the conjugacy operation applied to an element a leads to the same element a. 1.5 Representation theory of groups Definition. If G has a finite number of elements then it is a finite group, otherwise G is infinite. The number of elements in the group is called the order of the group and it is often denoted by G. Finite groups are realized e.g. when the set of symmetry operations is a discrete set. Infinite groups are often continuous, i.e. some of the group elements are labeled by one or more continuous parameters a 1, a 2,..., a r. r is called the order of the continuos group. Example. The circle group C consists of all rotations by an angle ϕ [0, 2π] around a fixed axis in three dimensional Euclidean space (the two-dimensional or axial rotation group). The transformation operations of this continuous group can 18

23 1 Abstract Group Theory and representation theory of groups be regarded as mapping one point residing on a circle into another point residing on the same circle. The most natural way of describing this transformation group is to establish two mutual orthogonal vectors e i in a plane and write down the transformation of these two vectors under a rotation aroung a third vector pointing perpendicular to the plane, using an orthogonal matrix with determinant 1: e t 1 e t 2 = cos ϕ sin ϕ e 1 e 2 sin ϕ cos ϕ This establishes a matrix group which is isomorphic to the axial rotation group C and is called the two-dimensional special orthogonal group SO(2). This matrix group does not have a finite number of elements but there are a non-denumerable infinite number of them. In the framework of these lectures we deal with both types of groups, provided they are endowed with a group average, Mittelwert (H. Weyl) or Haar measure. Definition. Let x be an element of a group G (discrete or continuous) and f (x) be a real valued function defined on G. If a functional exists with the following properties: 1. linearity: x G f (x) x G [α f (x) + β g(x)] = α x G f (x) + β x G g(x) 2. positivity: if f (x) 0 and real, then x G 0, being 0 only if f (x) 0 3. normalizability: if f (x) = 1, then x G f (x) can be chosen to be left and right invariance with respect to group operations: y x G f (y x) = x G f (x y) = x G f (x) the group G is said to be a group with average (in german: eine Gruppe mit Mittelwert) and is the Haar measure of the group. 19

24 1 Abstract Group Theory and representation theory of groups Example: For a finite group a G f (a) =. 1 f (a) G a is an average (h: order of the group). For continuos groups the average takes, in general, the form of an integral over an appropriate element dτ G, dτ G being determined bei some differential form within the parameter space. According to the previous definition of average, one chooses dτ g such that the group volume dτ g assumes the value of 1. Example: The avergage for the group C is 1 2π 2π 0 f (ϕ)dϕ where f (ϕ) is a periodic function of ϕ with period 2π and dϕ is the length of the differential element on the unit circle in the plane Matrix representations of a group (H. Weyl, ) For groups with average there exists a precise representation theory due to H. Weyl. The advantage of having an average is that many theorems exist and/or can be easily proved that simplify the use of groups for solving concrete problems. Of course, some definition apply to more general groups, but we always have groups with avergae in mind. An homomorphism is a map φ : g i φ(g i ) that associates to each element g i of a group one element and only one element φ(g i ) of another group such that φ(g l g k ) = φ(g l ) φ(g k ) The element φ(g i ) is called the image or map of the element g i under the homomorphism. Although each element has only one image, several elements of G may be mapped onto the same image. Thus it might happen that φ(g i ) = φ(g j ) although g i g j. 20

25 1 Abstract Group Theory and representation theory of groups An isomorphism between two groups G = g 1, g 2,.. and F = f 1, f 2,... is a one-to-one homomorphism between the elements of G and F. If two finite groups are isomorphic, then they have e.g. the same multiplication table, i.e. they are essentially the same, although the concrete significance of their elements may be different. Definition. Let G be any group, L a complex vector space and denote by GL(L) the group of invertible, linear mappings that carry L into itself. We define a representation of the group as a homomorphism : G GL(L); a (a) i.e. has the property (a) (b) = (ab) L is called the representation space, (g) is called a representation operator (for simplicity, we assume that the mapping is continuos but we drop the word continuous when we speak of representations ). If the vector space is finitedimensional the representation is finite or finite-dimensional. Otherwise it is called infinite or infinite-dimensional. The dimension of L is called the dimension of the representation. Theorem: A finite-dimensional representation (a) can be described by a matrix, whereby the law of composition in GL(L) is the the matrix multiplication. One speaks in this case of matrix representation of a group. Proof: choose, within L a e 1,..., e n. Find the i th column of the matrix (a)e i = T ji (a)e j j [T ji (a)] = [(e j, T(a)e i )]. = T(a) 21

26 1 Abstract Group Theory and representation theory of groups contains the coefficients used for representing the transformed basis vector e t i linear combination of the basis vectors e j. e t 1 T 11 (a) T 12 (a)... T 1n (a).... T 21 (a) T 22 (a)... et = n e e n T n1 (a).... T nn (a) as Now let us test for homorphism of these matrices. (a) (b)e i = (a) k T ki (b)e k = T lk (a)t ki (b)e l = ( T kl (a)t li (b))e k = (a b)e i = T ki (ab)e k By comparison we deduce that k l kl T ki (ab) = T kl (a)t li (b) l k or T(ab) = T(a) T(b) being the operation of matrix multiplication. From now on, when we speak of a finite representation of a group we always mean a matrix representation under the operation of matrix multiplication. In addition, if we do not say otherwise, the representations we deal with in this lecture are assumed to be finite-dimensional. The coordinates (x 1,..., x n ) of any vector r in L transform according to x t 1 T 11 (a) T 12 (a)... T 1n (a) x 1 x t 2. = T 21 (a) T 22 (a)... x x t n T n1 (a).... T nn (a) As the correspondence is merely an homomorphism, several elements of G may be represented by the same matrix. In the special case of an isomorphism, the representation is said to be faithful. x n 22

27 1 Abstract Group Theory and representation theory of groups Example: The simplest representation of a group is obtained when we associate the number 1 with every element of the group. This representation is one-dimensional and unfaithful and is called the identical representation. Example: C. The circle group C consists of all rotations by an angle ϕ [0, 2π] around a fixed axis in three dimensional Euclidean space (the twodimensional or axial rotation group). The transformation operations of this continuous group can be regarded as mapping one point residing on a circle into another point residing on the same circle. The most natural way of representing this transformation group is to establish two mutual orthogonal vectors e i in a plane and write down the transformation of these two vectors under a rotation aroung a third vector pointing perpendicular to the plane, using an orthogonal matrix with determinant 1: e t 1 e t 2 = cos ϕ sin ϕ e 1 e 2 sin ϕ cos ϕ This establishes a two-dimensional matrix representation of C. The so constructed matrix group is isomorphic to the axial rotation group C and is called the twodimensional special orthogonal group SO(2). Example: C 4v. The rotational elements are represented by the suitable SO 2 -matrices. The reflections are given by 1 0 m x = m y = m x y = m x y = 1 0 Matrix representations are useful because the algebra of matrices is generally simpler to carry through than abstract symmetry operations. Furthermore, quantum mechanical operators are often written as matrices so that matrix representations associates elements of an abstract group with linear operators acting within some Hilbert space. 23

28 1 Abstract Group Theory and representation theory of groups Theorem. Suppose we have a representation T 1 (a) of dimension l. Then the set of matrices T 2 (a) S 1 T 1 (a)s also form a representation of dimension l with S being any non-singular matrix. Proof: T 2 (a)t 2 (b) = S 1 T 1 (a)ss 1 T 1 (b)s = S 1 T 1 (ab)s = T 2 (ab) The two representations T 2 and T 1 are said to be equivalent and the transformation between the two is similarity transformation. Clearly, by similarity transformation, one can produce an infinite number of equivalent representations. Maschke s Theorem. Any representation (finite or infinite) of a group with average is equivalent to a unitary representation. For matrix representations this means that the matrices are unitary matrices (their inverse coincides with the complex conjugate transpose. Proof: having a representation (a), we must find an operator S such that (S (a)s 1 ) = (S (a)s 1 ) 1. We now show that the operator S for which S 2 = M b (b) (b) is the sought for matrix. The first step is to write S is hermitic, i.e. S = S (S 1 ) = S 1 (a)s 2 (a) = M b (a) (b) (b) (a) = M b (ba) (ba) = M c (c) (c) = S 2 S 1 (a) S 2 (a) (a) 1 S }{{ 1 = S } S 2 (a) 1 S }{{ 1 } } {{ S } (S (a)s 1 ) 1 S 1 (a) S The left hand side of this equation amounts to S 1 (a) S = S T(a)(S 1 ) = S (a)s 1 24

29 1 Abstract Group Theory and representation theory of groups and this is the proof of unitarity. Owing to this theorem, we need to consider representations by unitary matrices only. This will produce a great simplification: for instance, the scalar product in L remains invariant if two vectors are transformed by T(a). Notice that unitary representations are not unique, for if T 1 is a unitary representation and S is a unitary matrix, then T 2 is also an equivalent unitary representation. Theorem. The basis set carrying unitary representations is an orthonormal set. Proof The following equation (e i, e j ) = (T(a)e i, T(a)e j ) = ( l,m has the solution (e i, e j ) = δ i j. T li (a)t m j(a)(e l, e m ) = l T li (a)e l, l,m m T mj (a)e m ) = T 1 il (a)t mj (a)(e l, e m ) Reducible and irreducible representations The direct sum of two or more Hilbert spaces. Consider a vector space L n with a coordinate system e 1,..., e n and a vector space L m with basis vectors i 1,..., i m. Provided that the two spaces have no common vector except the null vector, the direct sum space L t L n L m is the vector space defined by the t = n + m basis vectors (e 1,..., e n, i 1,...i m ). If L n and L m are complete spaces, so is L t, i.e. any vector in L t can be expanded as a linear combination of the basis vectors. As a simple example we consider a plane with basis vectors e x, e y and a line with basis vector e z which does not lie in the x, y plane. The direct sum of the two spaces is the three dimensional vector space with basis vectors e x, e y, e z. In conjunction with the direct sum of spaces one can introduce the direct sum of two square matrices [A i j ] of the order n and [B i j ] of the order m. The resulting matrix C is of the order 25

30 1 Abstract Group Theory and representation theory of groups n + m and is defined as A 0 C = A B = 0 B = A A 1n A A 2n A n1... A nn B B 1m B B 2m B m1... B mm The direct sum of matrices can be easily extended to more than two matrices. Such a matrix, which has non-vanishing elements in square blocks along the main diagonal and zeros elsewhere, is said to be in a block-diagonalized form. It has the important properties that detc = deta detb and t rc = t ra + t rb (det = determinant and tr = trace). Also if A 1 and A 2 are matrices with the same order n and B 1 and B 2 are matrices with the same order m, then (A 1 B 1 )(A 2 B 2 ) = (A 1 A 2 ) (B 1 B 2 ) Suppose that T 1 and T 2 are two matrix representations of a group of dimensions l 1 and l 2 respectively, and define for each element of the group a (l 1 + l 2 )-dimensional matrix T by the direct sum T(a). = T 1 (a) T 2 (a) = T1 (a) 0 0 T 2 (a) T(a) is a partitioned matrix in which the elements of the top-right-hand block and bottom left-hand block are all zero, i.e. T(a) = T 1 (a) T 2 (a). As T(a)T(b) = T1 (a)t 1 (b) 0 0 T 2 (a)t 2 (b) = T1 (ab) 0 0 T 2 (ab) = T(ab) 26

31 1 Abstract Group Theory and representation theory of groups T is also a representation of the group. The representation T constructed by direct sum of representations can be partitioned in block form, by construction. Definition. Given a representation T, if the same similarity transformation brings all the matrices of the representation T into the same block-form, then T is said to be reducible. Notice that a similarity transformation could obscure the block form by making the zero matrix elements disappear, but the representation would be still considered to be reducible, that is a representation is said to be reducible if it can be put into block form by an appropriate similarity transformation. It might be possible that the representations T i appearing in the reduction of T are further reducible, i.e. that a similarity transformation exists, which transforms all matrices T i (a) into block forms themselves. Clearly, this process of reduction can be carried out until we can find no unitary transformation which reduces all matrices of the representation further. Thus, the final form of the matrices of a representation T may look like T 1 (a) T 2 (a)... 0 T(a) = T s (a) with all matrices of T having the same reduced structure and all the blocks of T being no further reducible. When such a complete reduction of a representation is achieved, the component representations T 1, T 2,... are called irreducible representations of the group and the representation T is said to be fully reduced. Definition A representation is called irreducible if it cannot be partinioned into blocks by any similarity transformation. Notice that an irreducible representation may occur more than once in the reduction of a reducible representation: in general T = i n i T i Clearly, the irreducible representations are the basic representations from which all others can be constructed Properties of irreducible representations It is the main aim of (a useful) group theory 27

32 1 Abstract Group Theory and representation theory of groups to construct all irreducible representations to test any representation for its irreducibility To this aim there is a set of key theorem which we are going to illustrate (and partly prove). These theorems and tools are based on two Lemma by Schur. Schur s Lemma 1: If T α is an irreducible representation of a group G and a matrix A commutes with all matrices of T α, then A = λ E. Schur s Lemma 2: If T α and T β are two irreducible representations of dimensions l α and l β of a group G and A is a matrix of order l α l β satisfying the relation T α (a)a = AT β (a), a then A = 0 (the null matrix) or the two representations are equivalent. The proof of both Lemmas can be found in the numerous standard books of group theory. here we use the Lemmas to prove the Great Orthogonality Theorem for irreducible representations of a group with average. Theorem (Great Orthogonality Theorem, GOT) Suppose that T α and T β are two unitary irreducible representations of a group, and that they are not equivalent if α β and identical if α = β. Then a T (α) (a)t β ik lm (a) = 1 δ αβ δ il δ km l α where l α the dimension of T α. Thus, knowing the matrix elements of the representations allows, in principle, to find out whether the representation is irreducible without having to search for the similarity transformation that puts the representation in block form. Proof: Let X be a non-singular l α l β matrix. We first show that the matrix defined by b T α (b)x T β (b 1 ) 28

33 1 Abstract Group Theory and representation theory of groups has the property of the A matrix in Schur s lemmas. In fact, T α (a)a = b T α (a)t α (b)x T β (b 1 ) = b T α (ab)x T β (b 1 )T β (a 1 )(T β (a) = b T α (ab)x T β ((ab) 1 T β (a) = AT β (a) by virtue of the definition of M. If T α = T β then A = λe, otherwise is A = 0, i.e. the matrix A = λ δ αβ E. Thus, in term of matrix elements, we have just proved the equation λδ αβ δ i j = a T α (a) ik X km T β (a 1 ) mj k,m in which the rectangular matrix is quite arbitrary but λ will be determined by the choice of X. In particular, chosing α = β and taking the trace on both sides of we obtain the general relation λ E = b T α (b)x T β (b 1 ) λ l α = Tr(X ) The GOT follows by making use of the freedom of choosing the appropriate matrix X. Let us, e.g., take a matrix X with zero elements everywhere except for the element in the row k and columns m which is taken to be 1. In this special case we have Considering that λ = δ km l α T β (a 1 ) mj = T β (a) jm we obtain the sought for orthogonality relation. We will now use this theorem to prove a set of further theorems that allow to test more easily for irreducibility and to construct irreducible representations. 29

34 1 Abstract Group Theory and representation theory of groups Characters of a representation and theorems involving them Because representation related by a similarity transformation are equivalent, there is a considerable degree of arbitrariness in the actual form of the matrices. However, there are quantities which do not change under similarity transformations, and which therefore provides a unique way of characterizing a representation. These quantities are appropriately called the characters of a representation. Definition. Let T be a representation (reducible or irreducible) of a group G. Then χ(a) =. T kk (a) is defined as the character of the group element a in this representation (that is the character is the sum of the diagonal elements). The set of characters corresponding to a representation is called the character system of the representation. If the representations are one dimensional, their character is the same as the representation. Theorem: The character is invariant under similarity transformation. Proof: (T(a)T(b)) ik = t r(t(a)t(b)) = = = k T(a) i j T(b) jk j T(a) i j T(b) ji i, j T(b) ji T(a) i j i j T(b) i j T(a) ji i j = t r T(b)T(a) = t r(s 1 T(a)S) = t r(t(a)ss 1 ) = t r T(a) Thus, equivalent representations have the same characters. The dimension of a representation is the character of the matrix representing the identity element of the group. 30

35 1 Abstract Group Theory and representation theory of groups Conjugate elements have the same character, i.e. the character is identical within the same class of conjugate elements (in other words: the character is a map associating to every class a complex number, i.e. it is a class function). Each class is therefore completely determined as far as its character is concerned by giving one single element of the class, which might be taken as the representative of the class. Each element of the group can only belong to one single class. There is a number of important theorems involving characters, which can be used to give very useful results from the knowledge of the characters alone, without requiring the explicit knowledge of the matrices. Theorem 1: orthogonality theorems for characters We can immediately transform the orthogonality relation a T (α) (a)t β ik lm (a) = (1/l α) δ αβ δ il δ km into an orthogonality relation for the characters of a group. In fact, setting i = k and l = m and summing both sides over the indices i and l we obtain a χ α (a)χ β (a) = δ αβ This theorem allows for an immediate test for irreducibility because the average of the square of the absolute value of the characters of an irreducible representation of a group with average is just one: a χ α (a) 2 = 1 if T α is irreducible. As each group admits the 1-representation, a further consequence of this orthogonality theorem is the equation a χ α (a) = 0 31

36 1 Abstract Group Theory and representation theory of groups provided T α is not the 1-representation. An alternative form of the orthogonality relation is obtained by starting again from the relation (see proof of GOT) and setting X = T α (b). Then we have T α (a)x (T α (a)) 1 = Tr(X ) E l α T α (a)t α (b)(t α (a)) 1 = χα (b) E l α We multiply both sides with a matrix T α (c): T α (c)t α (a)t α (b)(t α (a)) 1 = χα (b) T α (c) l α Taking the trace of both sides we obtain the multiplication theorem for characters: χ α (b) χ α (c) = l α a χ α (caba 1 ) Theorem 2: decomposition theorem (reduction formula) It very often happens that we have a representation of the group which is a reducible one. Such a representation T may be written as the direct sum of various irreducible representations, according to T = α n α T α where T α are irreducible representations of the group. n α is the number of times the irreducible representation T α is contained in the direct sum. To find n α, we take the traces of both sides in the expression for the direct sum. If χ(a) is the trace of the representation T and χ (α) (a) is the character of the element a in the irreducible representation T α, then χ(a) = n α χ α (a) for all a G. Multiplying both sides by χ γ (a) and averaging over a we obtain α a χ γ (a)χ(a) = n γ n γ = a χ γ (a)χ(a) 32

37 1 Abstract Group Theory and representation theory of groups This last equation gives a method for obtaining the coefficients n γ solely knowing the characters of the representation to be reduced and those of the irreducible component, without needing explicitly the matrix elements. A consequence of this formula is that any finite representation of a group with average is completely reducible into a number of irreducible representations. A formula for finite groups For finite groups, there is an important formula which is very helpful in constructing the character table of finite groups. This formula states that the sum of the squares of the dimensions of the inequivalent irreducible representations is equal to the order of the group. To prove this formula, we introduce the regular representation T r (h) of the element h of a finite group. The (right) regular representation is a linear map that associates to every element h of the group the result of right traslations g h for every element g of G: T r (h) : h G T r (h) = g h Alternatively one can defines the left regular representation of G as the left translation T l : h G T l (h) = h 1 g The matrix representation of T r or T l are produced in two steps: As a first step, one associates by a one-to-one map to every element of the group a basis vector in a space R G. In a second step, one associates to each element h taken as the top entry in the multiplication table a G G matrix whose columns consist of the basis vectors of the elements in the corresponding column of the multiplication table. This representation has some important properties. Each element appears only once in a column of the representation table and all diagonal elements are 0, except for the identity operation, which has exactly 1 on the diagonal. In other words: the regular representation of e is the unit matrix in R G and has character χ r (e) = G. The character for the representation of all other elements is zero, as their matrices have all zero on the diagonal. We shall now find the coefficients n i in T r = n i T i 33

38 1 Abstract Group Theory and representation theory of groups Using the suitable formula we obtain n i = 1 G a χ i (a)χ r (a) = 1 G χ i (e) G = l i with l i being the dimension of the i th irreducible representation. We thus have the equation T r = i l i T i. This equation means that the decomposition of the regular representation contains all irreducible representations with a multiplicity corresponding to their dimension. Taking the traces on both side of this equation for the matrix represenating e leads to l 2 i = G which is the important formula we have anticipated. i Theorem 3: the number of irreducible representations of a finite group For a finite group, the orthogonality relation between characters writes C n k χ α (k) χ β (k) = G δ αβ k=1 C being the number of classes and n k the number of elements within each class. This orthogonality theorem can be used to deduce a relationship between the number classes of a group and the number of irreducible representations. We rewrite the orthogonality relation as We then introduce the vectors k n k G χα (k) n k G χβ (k) = δ αβ χ α. = 1 G n1 χ α (1), n 2 χ α (2),..., n C χ α (C) 34

39 1 Abstract Group Theory and representation theory of groups to write the orthogonality relation as χ α χ β = δ αβ The χ α are mutually orthogonal vectors in a space whose dimension is the number of classes C in the group. There can be a maximum number of C of such mutually orthogonal vectors. But these vectors are labelled by an index α corresponding to the irreducible representations of the group. Hence, the number of irreducible representations must be less than or equal to the number of classes. This means that the rows of the character matrix G C n1 Γ α1 G χ α 1 (1) n 2 G χ nc α 1 (2)... G χ α 1 (C) n1 Γ αn G χ α n (1) n 2 G χ nc α n (2)... G χ α n (C) are mutually orthogonaly and that n C. On the other side, using the multiplication theorem for characters χ α (b) χ α (c) = l α a χ α (c 1 aba 1 ) one can show that the columns of the character matrix are also mutually unitary orthogonal, and this means that n C. In fact, χ α (b) χ α (c) = l α a χ α (c 1 aba 1 ) α α = a χ reg (c 1 aba 1 ) If we assume that c 1 is not in the conjugacy class of b than aba 1 c 1 and c 1 aba 1 e. As all characters of the regular representation are zero (except for the character of the element e), we conclude that χ α (b) χ α (c) = 0 α 35

40 1 Abstract Group Theory and representation theory of groups for b and c in different classes. This means that the columns of the character matrix are indeed mutually orthogonal. This leads to an important result for finite groups, namely that the number of irreducible representations of a group is exactly equal to the number of conjugacy classes. Technically speaking, the characters of the irreducible representations of a finite group are conveniently displayed in the form of character tables. The classes of the group (at least one representative) are usually listed along the top row of the table, and the irreducible representations down the left-hand side. The bulk contains the charatcers themselves. As a consequence of this theorem, this table is always square and the corresponding rows are normalized to the order of the finite group and orthogonal to each other. We summarize now some simple rules for constructing the character table of a finite group. 1. It is convenient to display in table form the characters of the irreducible representations. Such a table gives less information than a complete set of matrices, but it is sufficient for classifying the electronic states 2. The number of irreducible representations is equal to the number of classes in the group 3. The sum of the squares of the dimensions l α of the irreducible representations is equal to the number of elements of the group l 2 α = G α 4. The characters of the irreducible representation must be mutually orthogonal and normalized to the order of the group: χ α (a) χ β (a) = G δ αβ a 5. Every group admits the one-dimensional identical representation in which each element of the group is represented by the number 1. The orthogonality re- 36

41 1 Abstract Group Theory and representation theory of groups C 4v e 2C 4 C 2 2m 2m lation between characters then shows that for any irreducible representation other than the identity representation χ(a) = 0 a As a first example we construct the character table of C 4v. There will be 5 irreducible representations, one of them being the trivial one. Specializing SO 2 to the rotations of C 4v and computing the trace of the matrices for the m-operations we arrive at the conclusion that there is an irreducible representation of dimension 2 (called the representatioon 5 ). The equation l l l l l 2 5 = 8 has then a unique solution given in the column on the table corresponding to the element e. The remaining entries of the character table can be constructed by trial and error, taking into account the orthogonality relations of both rows and columns (and also taking care that the multiplication table is fullfilled). Further informations can be found in the review paper by G.F. Koster, Space groups and their representations, in Solid State Physics Vol.5, ed. by F. Seiitz and D. Turnbull (Academic Press, New York, 1957), pp , which contains all relevant informations about the 32 point groups of crystallography and their character tables, using a notation which is common in Solid State Physics. 37

42 1 Abstract Group Theory and representation theory of groups 1.6 Symmetry adapted vectors Consider a n-dimensional vector space carrying a n-dimensional representation T of a group G. Either is T irreducible or it can be brought into a block diagonalized form: T = i n i T i The reduction of T into a block-diagonalized from produces a number of subspaces which, by construction, are invariant with respect to all operations of G, i.e. the operations transform a vector lying within this subspace into another vector in the same subspace. The invariant subspaces are pairwise orthogonal, i.e. the original vector space L n is the orthogonal sum of invariant subspaces, each carrying an irreducible representation: L n = i n i L i Definition: The basis vectors {e i 1,..., ei l i } in the subspace L i carrying the l i - dimensional irreducible representation T i are said to be symmetry adapted vectors, transforming according to the columns of the irr. representation T i. The following theorem is fundamental for constructing symmetry adapted vectors: Theorem: consider a vector e in the space L n. The vector P (i) j e =. l i a T,i j j (a)t(a) e transforms according to the j th column of the irreducible representation T i. Proof: we think of the expansion of e as a linear combination of {e (1) 1,..., e(1),..., e (p),..., e(p) l 1 1 l 1 } i.e. and we compute e = q,m C q m eq m P (i) j e = l i a T,i j j (a)t(a) = l i q,m C q m at,i j j (a) k q,m C q m eq m T q km eq k 38

43 1 Abstract Group Theory and representation theory of groups = l i q,m,k C q m [ at,i j j (a)t q km }{{} ] 1 l δ i iq δ jk δ jm = C (i) j e i j The projector P (i) j projects out of any vector in L n those parts transforming according to the various columns j of the i-th irreducible representation contained in the reduction of T. Thus, this projector can be used to create a set of symmetry adapted vectors. Of course, it might be that the sought for symmetry adapted wave functions are not contained in the trial vector e, so that the result of applying the projector is zero. In that case, in order to generate symmetry adapted wave functions, one must repeat the the procedure with another trial vector. This projector technique requires, of course, an explicit knowledge of the matrix element of the various representations, and not merely a knowledge of the characters alone, which is the usual information given in the published literature. For one-dimensional representations the characters are the matrix elements them selves, so that this projector technique provide the complete answer for the basis functions of one-dimensional representations. For representations with larger dimension, there is a procedure to obtaining the l i -partners of a basis set for an irreducible representation from the characters themselves. This procedure involves the projector P (i). = n P (i) n e q k = l i a χ (i) (a)t(a) which projects out of any normalizable vector the sum of all basis functions transforming according to the columns of the i-th irreducible representation. This last projector requires only the knowledge of the characters of the irreducible representation of which one would like to construct symmetry adapted wave functions. Of course, the projector dos not produce all partners within the irreducible subspace. However, having determined one e (i), we can then apply to e (i) the operations of the symmetry group to obtain exactly l i -linearly independent vectors which can be made orthonormal and thus can be used as a basis set for the irreducible representation T i (remember that the application of the operations of the group to a vector belonging to an invariant subspace carrying the irreducible representation T i does not carry us outside this subspace). Notice that, after having generated the basis 39

44 1 Abstract Group Theory and representation theory of groups functions transforming according to the irreducible representation T i, it is possible to construct the matrix representation of T i (in practice, this is the procedure most adopted to construct matrix representations of the irreducible representations), using T(a)e i j = T i k j (a)ei k This procedure requires only the initial knowledge of the characters. In summary: One starts with a trial vector e P i applied to e possibly produces one symmetry adapted vector e i T(a)e i produces all partners vectors transforming according to T i (take, for this purpose, a sufficiently large number of T(a)). If P i e = 0 use a different trial function. Remark: the projection technique can be used as alternative to the decomposition theorem for finding simultaneously the irreducible components of a representation and the invariant subspaces. k A practical example A way of illustrating the technical aspects entering the use of the projector technique we study the tower (or permutation) representation of C 4v. One obtain this representation by associating to each corner of a square a weight or scalar, thus producing unit vectors in a 4d space, and representing the operations of C 4v as 4 4 matrices along the column of which one can find the map of the four basis vectors under the action of the symmetry operation. As an example, take C 4z, which is represented, in the permutation representation, by the matrix All other matrices can be constructed accordingly. In order to find the irreducible components of the tower representation T we can use the decomposition theorem: 40

45 1 Abstract Group Theory and representation theory of groups C 4v e 2C 4 C 2 2m 2m T C 4v e 2C 4 C 2 2m 2m Basis (1, 1, 1, 1) (1, 1, 1, 1) (1, 0, 1, 0) (0, 1, 0, 1) T = We find the symmetry adapted vectors using the projector technique. Remark: Character tables appear often with an extra column hosting the (some of the) symmetry adapted vectors, transforming according to the irreducible representation listed in the same row. 41

46 1 Abstract Group Theory and representation theory of groups Figure 1.14: Construction of symmetry adapted vectors using the projector technique. 42

47 2 Group theory and eigenvalue problems Within a Hilbert space L (finite or infinite dimensional) we consider a linear operator H typically the Hamilton operator of quantum mechanics or any linear operator appearing in well defined problems of classical physics and the corresponding eigenvalue problem Hφ = Eφ We have an eigenvalue problem in mind where the operators are Hermitic in the sense of J. von Neumann, Math. Ann. 102, (1929) and the sought for eigenvalues E 1,.., E i,... each carry an eigenspace {φ E i 1,..., φe i} l i of non-zero eigenfunctions with possible some degeneracy l i. For a wide category of Hermitic operators there are so called Spectra Theorems that specify the characteristic of the spectrum σ(h), i.e. the set containing the eigenvalues. We summarize here some of the most important definitions and results. Hilbert spaces To define the concept of Hilbert space we start with a set of function ϕ of some variables q running within the configuration space Ω of a system. We define a sum of two elements of the set and the multiplication with a complex scalar such that the set becomes a complex vector space. Within the vector space we define a scalar product ϕ, ψ. = Ω dqϕ (q)ψ(q) 43

48 2 Group theory and eigenvalue problems with the usual properties ϕ, aψ = a ϕ, ψ aϕ, ψ = a ϕ, ψ ϕ 1 + ϕ 2, ψ = ϕ 1, ψ + ϕ 2, ψ We further require that the squared norm ϕ 2 = ϕ, ϕ = Ω ϕ(q) 2 dq is finite, calling a zero function all those functions with ϕ 2 = 0, and considering two functions ϕ and ψ as being equal if their distance (which defines a metric within the space) is zero: ϕ ψ = 0 In view of these definitions, an element of a Hilbert space is defined as a class of functions ϕ that differ is such a way that their distance is zero. Finally we require that the vector space has a complete orthonormal system, (COS) i.e. a countable set of orthonormal vectors ϕ k such that any function ϕ can be expressed as the sum of ϕ k, in the sense that n lim ϕ c k ϕ k = 0 n k=0 with c k = ϕ k, ϕ Remark: These definitions can be applied to finite dimensional vector spaces, provided the scalar product is defined suitably. The number of vectors building a complete orthonormal system is given by the dimension of the vector space. 44

49 2 Group theory and eigenvalue problems Linear Operators An operator A transforming vectors in L into vectors in L is called a linear operator or linear transformation, if A(a 1 ϕ 1 + a 2 ϕ 2 ) = a 1 Aϕ 1 + a 2 Aϕ 2 Notice that it is not necessary that Aϕ is defined for all vectors ϕ. For instance, the momentum operator i d d x is only defined for differentiable functions. However, it is required that Aϕ is defined for a set of functions ϕ which is dense in the Hilbert space, i.e. if ψ is any function in L then there exists a function ϕ within the domain of definition of A so that ψ ϕ can be made arbitrarily small. Notice also that the physical situation typically produces a true eigenvalue problem only if the equation Aϕ = λϕ is accompanied by some boundary conditions on the wave functions ϕ. These boundary conditions typically contribute in defining the domain of definition D(A) of the operator A. In other words: two operators might be formally identical, but different boundary conditions produce a different domain of definition D(A) and, accordingly, a different spectrum might appear. Remark. If the vector space with scalar product is finite-dimensional, linear operators ca be represented by matrices. To construct a matrix representation, take a set of orthonormal vectors e 1,..., e n and align the matrix elements e i, Ae j j = 1,..., n along the i-th column, i = 1,..., n. Symmetric (or Hermitic) operators. Let A be a linear operator and D(A) its domain. A is called symmetric (or Hermitic, in the sense of Von Neumann) if ϕ, Aψ = Aϕ, ψ 45

50 2 Group theory and eigenvalue problems for any pair of function within D(A). Examples of symmetric operators are the operators q and p defined as qϕ(x). = x ϕ(x) pϕ(x). = iħh d d x ϕ(x) The proof of this is given by partial integration. The boundary integrals occurring upon partial integration vanishes if the wave functions are required to vanish at infinity. The vanishing of the wave functions at infinity is necessary in order to ensure that the norm of the functions is finite. Similarly, the Hamilton operator can be shown to by symmetric, provided it is applied to such functions for which all derivatives have finite norm. A further example of Hermitic operator is the Hilbert-Schmidt integral operator b Fψ(x). = a d yg(x, y)ψ(y) provided the kernel obeys the relation G(x, y) = G(y, x). Proof: = b b a a ϕ(x), Fψ(x) = b b d xd y[g(y, x)ϕ(x)] ψ(y) = a a d xd yϕ(x) G(x, y)ψ(y) b b = Fϕ(x), ψ(x) a a d xd y[g(x, y)ϕ(y)] ψ(x) For Hermitic (symmetric operators there are two important theorems: Theorem 1. The eigenvalues of an Hermitic operator are real. Proof: Let ψ be an eigenfunction of A, i.e. Aψ = λψ. Than, from ψ, Aψ = λ ψ, ψ = Aψ, ψ = λ ψ, ψ we obtain λ = λ. Theorem 2. Eigenfunctions of an Hermitic operator belonging to different eigenvalues are orthogonal. 46

51 2 Group theory and eigenvalue problems Proof: Let ψ 1 and ψ 2 be eigenfunctions of A belonging to different eigenvalues λ 1, λ 2 : Aψ 1,2 = λ 1,2 ψ 1,2 By the requirement of Hermiticity we have ψ 1, Aψ 2 = Aψ 1, ψ 2 (λ 1 λ 2 ) ψ 1, ψ 2 = 0 ψ 1, ψ 2 = 0 as λ 1 λ 2. Remark. Consider l i linearly independent vectors within the eigenspace of a l i -time degenerate eigenvalue λ i. The Gram Schmidt ortho-normalization algorithm allows to construct an orthonormal set within each finite eigenspace itself. Remarks on the eigenvalue problem for Hermitic operators. One must distinguish between the finite and infinite dimensional case. The finite dimensional case L n, n <. For a finite dimensional vector space L n Hermitic operators are represented by Hermitic matrices, i.e. matrices for which A = A where the operation on a matrix A takes its transpose and is complex conjugate. The condition of Hermiticity means that for all vectors x, y L n we have Ax, y = x, Ay. The spectral theorem for Hermitic operators in the finite dimensional case reads Theorem: If A is an Hermitic operator in L n, then there is an orthogonal basis in L n consisting of eigenvectors of A. Accordingly, there are exactly n, real eigenvalues. Proof: the proof of the spectral theorem is typically performed by induction. First we show that the theorem holds for n = 1. In n = 1 A is the multiplication by a scalar Ax = λx x L 1, λ being a real number. Thus, the theorem is true in this particular case. We now assume that the theorem holds for dimension n 1. Because of the fundamental theorem of algebra, which asserts that the determinantal equation of 47

52 2 Group theory and eigenvalue problems A has at least one non-zero solution, we know that at least one non-zero eigenvalue λ 1 exists, and we call the corresponding eigenfunction e 1. We now perform the Gram Schmidt orthogonalization procedure to construct a set of vectors e 2,..., e 1 orthogonal to e 1. This means that the matrix element A 1 j. = e1, Ae j = Ae 1, e j = λ 1 e 1, e j = 0 = A j1 and in the orthogonal basis e 1, e 2,...e n the matrix representation of A reads λ1 0 0 A n 1 For the (n 1) (n 1) Hermitic matrix A n 1 the theorem has been assumed to hold, so that the theorem is proved by induction on n: A can be diagonalized within L n. In virtue of this theorem, in the basis set e 1, e 2,..., e n in which A is diagonal, we have, for any x L n n x = e k, x e k and Ax = k=1 n λ k e k, x e k k=1 The most straightforward generalization of the spectral theorem to infinite dimensional Hilbert foresees that the the n k=1... is replaced by k=1..., meaning that an Hermitian operator in a infinite dimensional Hilbert space has an infinite but countable set of eigenvalues, the eigenspaces of which providing a complete orthonormal set e 1, e 2,... Unfortunately, this generalization is not in general true. As a counterexample, there are Hermitian operators that have no eigenvalue, like the q and p operators defined as qϕ(x) = x ϕ(x) pϕ(x) = i d d x ϕ(x) 48

53 2 Group theory and eigenvalue problems Both are, within the Hilbert space of square integrable functions of the variable x (the square integrability representing the boundary condition) Hermitic but the functions solving the eigenvalue equations x ϕ(x) = λϕ(x) i d ϕ(x) = λϕ(x) d x are not square integrable and therefore the equations have no solution. On the other side, if one is able to discover at least one eigenvalue, one could proceed to construct a family of eigenvectors. In oder to be able to assume the existence of at least one eigenvalue, either one computes it explicitly or one must make some mild assumption about the Hermitic operator that goes beyond its Hermiticity. Such mild assumption allow to prove the existence of at least one eigenvalue (notice that for the infinite dimensional case one cannot use the fundamental theorem of Algebra, as in the case of Hilbert spaces with finite dimensions, as the determinantal equation if one can define such an equation at all is not a polynomial of finite power). Once one eigenvalue λ 1 has been found, than the proof of the generalization of the spectral theorem proceeds as follows. Let e 1 L H 1 be the normalized. eigenvector associated with λ 1. Construct H 2 = {x H1 : x e 1 }. H 2 is a subset of H 1, Ax H 2 as Ax, e 1 = x, Ae 1 = λ 1 x, e 1 = 0 and A is again Hermitic. So, by searching for the eigenvalues of A within the space H 2 we might get a new eigenvalue λ 2 and a (possibly) l 2 -degenerate eigenspace e 1 2,..el 2 2, which can be considered orthogonal to themselves and to e 1. We can continue this construction to generate a family of mutually orthogonalized eigenspaces H n. = {x H1 : x e 1, x e 2,..., x e n 1 } for each n. The operator A restricted to e 1,..., e n (where possibly some of the eigenvectors belong to the same eigenvalue) carries a countable set of eigenvalues along the diagonal of its matrix representation. Accordingly, this operation generates a set of countable eigenvalues and countable orthogonalized eigenvectors. The sets might have an infinite number of elements, with the eigenvalues accumulating 49

54 2 Group theory and eigenvalue problems toward a certain value (0, as in the spectrum of the Hydrogen atom) or with no accumulation point (the spectrum of the quantum mechanical harmonic oscillator). The set of eigenfunctions generated in this way might form COS in L. In this case we have the sought for exact generalization of the Spectral Theorem for finite dimensional Hilbert spaces to infinite dimensional ones. The spectrum of some selected Hermitic operators We consider the operator i d d x. = p and define it within within the Hilbert space 2 of square integrable functions f (x) in the interval [0, L] with f 2. The operator is not Hermitic: f, pg = L 0 d x f (x) [ i d P.I. L d x g(x)] {}}{ = i f (x) g(x) L d x( i d d x f (x)) g(x) = i f (x) g(x) L 0 + p f, g as the constant i f (x) g(x) L 0 arising from the integration by parts is, in general not vanishing. However, the operator can be rendered Hermitic by imposing some boundary conditions. A possible set of boundary conditions is to allow only functions that vanish at x = 0 and x = L, so that the constant i f (x) g(x) L 0 arising from the integration by parts is made to be vanishing. Under this boundary condition the eigenvalue equation i d f (x) = λ f (x) d x has no solution, as the only functions solving the differential equation and fulfilling the boundary condition is the function f (x) 0, which cannot be considered as an eigenfunction: the operator, with the boundary condition introduced, has neither eigenfunctions nor eigenvalues. A different set of boundary conditions that eliminates the constant i f (x) g(x) L 0 are periodic boundary conditions f (L) = f (0) 50

55 2 Group theory and eigenvalue problems Enforcing this specific boundary condition onto the general solution of the equation i d f (x) = λ f (x) d x produces an algebraic equation for the sought for eigenvalues: e iλl = 1 2π within solved by the set of eigenvalues λ n L n n and the set of orthogonalized eigenfunctions 1 e iλ n x L We know from the theory of Fourier series that these functions build a COS set onto the interval [0, L]. The relation between groups and eigenvalue problem In general, eigenvalues and eigenfunction are the result of a great computational effort, except in in some simple exactly solvable cases. Group theory helps in (a) simplifying the eigenvalues problem and (b) classifying the various eigenfunctions of an operator by the irreducible representations of the symmetry group of the some physically relevant operators. The significance of G for solving problems in quantum mechanics is contained in three important theorems. Consider a l-fold degenerate energy eigenvalue E i of H and let {φ E i n }, n = 1, 2,..., l i the set of linearly independent eigenfunctions corresponding to this eigenvalue, so that Hφ E i n = E iφ E i n n. Suppose to have a set G of transformations O a which leave the H-operator invariant, i.e. O a HO 1 a = H [O a, H] = 0 Theorem A: the set O a builds a group and is called the symmetry group of the operator H. Proof: We must prove that from [O a, H] = 0 and [O b, H] = 0 also [O b O a, H] = 0. 51

56 Indeed 2 Group theory and eigenvalue problems O b O a H HO }{{} b O }{{} a = O b HO a O b HO a = 0 HO a O b H Theorem B: Each subspace {φ E i n } is invariant with respect to the operations of the symmetry group of the operator H. Proof: HO a φ E i = O a Hφ E i = O a E i φ E i = E i O a φ E i This means that O a φ E i is also an eigenfunctions of H with the same eigenvalue E i. As E i was assumed to have only l i linear independent eigenfunctions, O a φ E i must be some linear combination of the eigenfunctions {φ E i }, so that the subspace to n E i is indeed invariant with respect to O a. Theorem C: The subspace {φ n } belonging to a given eigenvalue E i of H carries a l i -dimensional matrix representation of the symmetry group of H. Proof: Writing O a φ n = T(a) mn φ m establishes a l i l i matrix [T nm (a)] for each operator O a. Furthermore m O b φ n = T(b) mn φ m m O a φ q = T(a) pq φ p = p O b O a φ q = O b T(a) pq φ p = T pq (a) T mp (b)φ m p p m = T mp (b)t(a) pq φ m m p i.e. the consecutive application of two operators O a first and O b second can be computed by the matrix multiplication [T nm (b)][t nm (a)] 52

57 2 Group theory and eigenvalue problems We have therefore demonstrated that a set of energy eigenfunctions of the operator H corresponding to an eigenvalue builds a basis for a matrix representation of the symmetry group of H. The function φ n is said to transform according to the n-the column of the representation (or to belong to the n-th column of the representation). This means that the coefficients of the linear combination determining the transformed n-the basis function are to be read out from the n-the column of the matrix. Notice that the representation can be irreducible one speaks of essential degeneracy, because small perturbations of H which do not break the symmetry cannot lift the degeneracy of the eigenspace. If the representation is reducible, instead, it has irreducible components that block-diagonalize further a putative perturbed operator H + V, even if the perturbation V has the same symmetry as the original operator H. One speaks of accidental degeneracy because the original degeneracy l i of the eigenvalue E i is removed by a small, symmetry preserving perturbation of the original operator. This last theorem provides the base for using the theory of group representations for solving eigenvalue problems. Typically, the eigenvalue problem is cast as a matrix equation and E i are the solutions of some more or less cumbersome determinantal equation. Such equation can be solved by brute force but the use of symmetry can reduce the numerical complexity. The key idea is that symmetry adapted functions provide a good basis for the true eigenfunctions and thus computing the eigenvalues with symmtry adapted funcrtions might give good results. In contrast to the true eigenfunctions, symmetry adapted functions can be derived by symmetry arguments alone using the projector technique onto a suitable trial function. Clearly, it is a perturbative solution of the eigenvalue problem, which depends on how good the trial function is. However, strictly speaking, using symmetry adapted functions, one can achieve an exact block diagonalization of the H-operator, each block containing only those functions that transform according to the same irreducible representation of G. There might be many functions transforming according to the same irr.rep., which are distinguished by some characteristics other than symmetry. All these functions provide non-vanishing matrix elements for H within the same symmetry block (also non-diagonal ones), and the further diagonalization of the block of same 53

58 2 Group theory and eigenvalue problems symmetry (which might be even infinite dimensional) can only be done by explicit computation. Yet, by reducing the H-operator to blocks with the same symmetry, symmetry arguments have effectively achieved a reduction of the size of the blocks, to blocks carrying the same symmetry. Within a perturbative approach, one might even, within a symmetry block, go further and speculate that the matrix elements between some of the functions of the same symmetry are small and can be neglected, and consider the eigenvalue problem within even smaller blocks, so that, within a perturbative approach, one really deals with the diagonalization of comparatively much smaller matrices. Example 1 As a way of illustration we consider the eigenvalue problem of a vibrating square membrane, clamped along the boundary. The displacement u(x, y) perpendicular to the plane of the membrane satisfies the Helmholtz equation u(x, y) = ω2 u(x, y) c2 where represents the Laplace operator and ω are the frequencies of oscillation. The particular boundary condition is that the amplitude along the boundary of the membrane vanishes. The above equation, together with the boundary conditions, is a well defined eigenvalue problem for the negative of the Laplace operator. In order for numerically solve a partial differential equation, one generally establishes onto the domain of definition a fine lattice mesh. The precision and the number of eigenvalues obtained increases with the number of lattice points, but so does the size of the numerical problem, so that we limit ourselves for the sake of keeping the matrices small enough to be manageable to just approximating the square membrane with four points placed onto a square. The amplitude u(x, y) becomes, on a lattice, a lattice function u 1,..., u 4 associating to every point of the lattice a weight. In order to use group theory, we need to define the Laplace operator onto 54

59 2 Group theory and eigenvalue problems a lattice. This is done by noting that, with (x ±1 = x 0 ± a), u(x ±1, y) = u(x 0, y) + u x x 0 (x ±1 x 0 ) u 2 x 2 x 0 (x ±1 x 0 ) 2 and accordingly 2 u x 2 x 0 = u(x 1, y) 2 u(x 0, y) + u(x 1, y) a 2 A similar equation holds true along the y-coordinate, so that the lattice version of the Laplace operator 2 u x 2 x 0,y u y 2 x 0,y 0 writes 4 u(x 0, y 0 ) + [u(x 1, y 0 ) + u(x 1, y 0 ) + u(x 0, y 1 ) + u(x 0, y 1 ] a 2 This last equation is just the sought for Laplace operator on a square lattice and can be visualized as a cross operator carrying a 4 in the middle and a 1 at each corner (let us take a = 1 for simplicity). The matrix elements of M the discrete (negative of the) Laplace operator in the four dimensional space are obtained by placing the cross operator onto the i-lattice point and determining which are the lattice points touched by the edges of the cross. Those lattice points contribute a matrix element 1, while the central one gives 4. Those lattice points which are outside the lattice contribute zero, in virtue of the boundary conditions. The matrix representation M of the cross operator reads M = Finding the eigenvalues of this matrix requires, in principle, solving a quartic determinantal equation and can be performed only numerically. However, we can use group theory to find the eigenvalues by noting that the matrix M commute 55

60 2 Group theory and eigenvalue problems Figure 2.1: The negative of the Laplace operator on the membrane (left) defined as operator on a square lattice (right). with all elements of the group C 4v and thus the eigenspaces of M are the invariant subspaces of the four dimensional tower representation of C 4v. As found in the previous chapter, T = Using the projector technique we find the invariant subspaces corresponding to the irreducible components of the four dimensional tower representation. 1 = 1, 1, 1, 1 2 = 1, 1, 1, 1 1, 0, 1, 0 5 = 0, 1, 0, 1 These are also, by virtue of the fundamental theorem of group theory, the sought for eigenvectors of the operator M which commutes with all elements of the symmetry group. Therefore we can use them to compute the sought for eigenvalues of M, without solving any determinantal equation!! The sought for eigenvalues are 2 (the lowest one, corresponding to a uniform excitation of the membrane), 4 (twice degenerate, with nodes) and 6. These last two eigenvalues correspond to higher lying excitation modes that break the symmetry of the original operator (spontaneous symmetry breaking). 56

61 2 Group theory and eigenvalue problems Figure 2.2: Left: the eigenvectors and eigenvalues for fixed boundaries. Right: eigenvectors and eigenvalues with periodic boundary conditions. Example 2 Suppose now the membrane is subject to periodic boundary conditions. This would change the matrix M to M = The eigenvalues can now be computed onto the known eigenvectors: Molecular vibrations An illuminating application of group theory is the analysis of molecular vibrations. They can be treated within the framework of classical mechanics and provide a clear example on how numerics is simplified by using symmetry arguments. Within Newton mechanics, the motion of N atoms, building a molecule, around their equilibrium position is described, within the harmonic approximation, by a set 57

62 of Newton equations m k ü kl = 2 Group theory and eigenvalue problems h,m f 0 kl,hm u hm h, k = 1,, N; l, m = 1, 2, 3. h, k = 1,..., N counts the atoms and l, m = 1, 2, 3 label the coordinates of the various atoms (for a 3d space). m k is the mass of the k-th atom, the double dots stands for the second derivative with respect to time. u kl are the components of the displacement from equilibrium of the k th particle and f 0 are the set of force kl,hm constants, whose arbitrariness is restricted only by the symmetry of the system and the fact that actio reactio implies f 0 hm,kl = f 0 kl,hm Such a system of coupled differential equations is usually solved by searching fro the 3N eigenmodes through the eigenmode Ansatz: u kl = a 0 kl e iωt k = 1,, N; l = 1, 2, 3. Inserting this Ansatz, which contains the sought for eigenfrequency ω, produces a system of 3N coupled algebraic equations for the eigenmode coefficients a kl : ω 2 m k a 0 kl = h,m f 0 kl,hm a0 hm which can be written in matrix form m 1 a 0 f 0 1x 1x,1x f 0 1x,1 y.... f 0 m 1 a 0 1y ω 2.. = x.Nz a 0 1x a 0 1y... m N a 0 Nz f 0 Nz,1x..... f 0 Nz,Nz a 0 Nz The symmetric matrix of force constants f 0 is called the force matrix and denoted by F 0. One usually manipulate further the mathematical problem to obtain kl,hm a 58

63 2 Group theory and eigenvalue problems simplified, particularly appealing version of it: Define f kl,hm. = 1 mk m h f 0 kl,hm a kl. = mk a 0 kl and one obtains the eigenvalue problem of the matrix F: ω 2 a = F a with a 1x a 1 y a =... and F being the 3N 3N Matrix [f kl,hm ], representing the effective force matrix. a Nz f 1x,1x f 1x,1 y.... f 1x,Nz f Nz,1x..... f Nz,Nz The sought for eigenfrequencies are the solutions of the determinantal equation det F ω 2 E = 0 The solution of this determinantal equation provides the frequencies of the 3N eigenmodes. Notice that, included in these solutions, are the trivial cases of pure displacement and pure rotations of the molecule as a whole. These particular kinds of motion are characterized by ω = 0, as they are not influenced by restoring forces. In general, there will be 3N 6 non-zero solutions for ω. Each non-zero 59

64 2 Group theory and eigenvalue problems Figure 2.3: The water molecule with vectors describing the planar displacement of the atoms. solution gives a specific normal mode of vibration. Now the question: how can group theory help solving the problem of finding normal modes and eigenfrequencies? We consider as an example the vibrational modes of a H 2 O-molecule, which has C 2v s ymmet r y In order to generate a representation of C 2v within the space of atomic displacements we attach to each atom a local coordinate system, see the figure. Referring to the figure, we write: m O ẍ 1 = f 0 (x 1 + y 2 ) m O ÿ 1 = f 0 (y 1 + x 3 ) m H ẍ 2 = g0 (x 2 + y 3 ) m H ÿ 2 = f 0 (y 2 + x 1 ) m H ẍ 3 = f 0 (x 3 + y 1 ) m H ÿ 3 = g0 (y 3 + x 2 ) Using the transformation x1 y 1 x2,3 y 2,3 = x 1 m O y 1 = m O x 2,3 y 2,3 60

65 2 Group theory and eigenvalue problems we reduce the Newton equations as follow: ẍ 1 = f x1 + y 2 mo mh using now the Ansatzt: it follows: ω 2 x 0 1 = ẍ 1 = f f x 1 y 2 m O mo m H ÿ 1 = f f y 1 x 3 m O mo m H ẍ 2 = g m H x 2 g m H y 3 ÿ 2 = f f y 2 x 1 m H mo m H ẍ 3 = f f x 3 y 1 m H mo m H ÿ 3 = g m H y 3 xi y i = g m H x 2 x 0 i y 0 e iωt i f x 0 1 m + f y 0 2 O mo m = f 1x f 2 y 0 2 H ω 2 y 0 1 = f 1 y f 2x 0 3 ω 2 x 0 2 = g x 0 2 m + g y 0 3 H m = g 1x g 1 y 0 3 H ω 2 y 0 2 = f y 0 2 m + f x 1 = f 3 y 0 2 H mo m + f 2x 0 1 H ω 2 x 0 3 = f 3x f 2 y 0 1 ω 2 y 0 3 = g 1 y g 1x

66 2 Group theory and eigenvalue problems We can now read out the effective force matrix: F = f f f f g g 1 f f f f g g 1 = F The vector space carrying the matrix F is built by the 6 vectors: x 0 1 = x 0 2 = x 0 3 = x 0 4 = x 0 5 = x 0 6 = This space sustains a 6 dimensional Vectorial Tower (permutation) Representation of C 2v The name vectorial is used to distinguish it from the scalar tower representation, where each component of the system is allocated a scalar amplitude. In a vectorial representation each atom is associated with a coordinate system, which hosts displacements with vectorial character. The symmetry group C 2v has symmetry elements: E, C 2, σ u (reflection at the molecule plane) and σ v. The vectorial matrix representation reads: E = = σ u χ E = χ σu = 6 62

67 2 Group theory and eigenvalue problems C 2 = = σ v χ C2 = χ σv = 0 By looking at the characters table of C 2v and knowing the characters of the Σ V T representation we can decompose it in sum of irreducible representation of C 2v : C 2v E C 2 σ u σ v Σ Σ Σ Σ Σ V T Σ 1 3Σ 4 This represents a first, important result of symmetry considerations toward solving an eigenvalue problem: the original F-matrix (6 6) can be certainly reduced to two 3 3 blocks containing each the representations Σ 1, Σ 1, Σ 1 and Σ 4, Σ 4, Σ 4. This is the exact block-diagonalization which can be achieved using group theory. In the present case we are still left with the need of explicitly diagonalizing two 3 3 matrices, but this is a considerable improvement with respect to the original 6 6 matrix. The process of block diagonalization requires the use of the projection operator to construct the invariant subspaces transforming according to Σ 1 and Σ 4. For instance, we find λ 1 λ 1 λ 2 λ 1 + λ 2 λ 2 λ 2 λ 1 λ 2 + λ 1 P λ 3 Σ1 λ = 2 λ 3 4 λ + 2 λ 6 4 λ = λ 3 + λ 6 5 λ 4 + λ 5 λ 5 + λ 4 λ 5 λ 5 λ 4 λ 6 λ 6 λ 3 λ 6 + λ 3 63

68 2 Group theory and eigenvalue problems i.e the symmetry adapted vectors transforming according to Σ 1 are of the type λ 1 λ 1 λ 3 λ 2 λ 2 λ 3 Applying F to λ 1 λ 1 λ 3 λ 2 λ 2 λ 3 we obtain f f f f g g 1 f f f f 3 0 λ 1 λ 1 λ 3 λ 2 λ 2 f 1 λ 1 + f 2 λ 2 f 1 λ 1 + f 2 λ 2 = 2g 1 λ 3 f 2 λ 1 + f 3 λ 2 f 2 λ 1 + f 3 λ g g 1 and the eigenvalue problem reads λ 3 2g 1 λ 3 f 1 λ 1 + f 2 λ 2 = ω 2 λ 1 f 1 λ 1 + f 2 λ 2 = ω 2 λ 1 2g 1 λ 3 = ω 2 λ 3 f 2 λ 1 + f 3 λ 2 = ω 2 λ 2 f 2 λ 1 + f 3 λ 2 = ω 2 λ 2 2g 1 λ 3 = ω 2 λ 3 Using symmetry adapted vectors of the Σ 1 type we have reduced the eigenvalue 64

69 2 Group theory and eigenvalue problems problem of size 6 6 to two identical eigenvalue problems with reduced size 3 3 f 1 λ 1 + f 2 λ 2 = ω 2 λ 1 2g 1 λ 3 = ω 2 λ 3 f 2 λ 1 + f 3 λ 2 = ω 2 λ 2 This last problem must be solved numerically, but group theory can give us some hints. The determinantal equation writes i.e. f 1 ω 2 f 2 0 f 2 f 3 ω 2 0 = g 1 ω 2 (2g 1 ω 2 ) (f 1 ω 2 )(f 3 ω 2 ) f 2 2 = 0 Considering that f 1 f 3 = f 2 2 we have three solutions with Σ 1 symmetry: Let us now comment on these solutions. To ω 2 = 0 we find the eigenvector ω 2 = 2g 1 ω 2 = f 1 + f 3 ω 2 = 0 f 1 λ 1 + f 2 λ 2 = 0 2g 1 λ 3 = 0 f 1 λ 1 + f 3 λ 2 = 0 λ 1 = 1 2g 1 λ 3 = 0 λ 2 = f 1 f 2 = m h m O λ 3 = 0 65

70 2 Group theory and eigenvalue problems This eigenvector f 1 f 2 f 1 f 2 0 describes a translation: it does not include restoring forces and therefore has ω = 0. Notice the manifest Σ 1 symmetry of the eigenvector. ω 2 = 2g 1. The eigenvector reads ω 2 = f 1 + f 3. The eigenvector reads mo m H mo m H 0. We now proceed to investigate the Σ 4 sector. First, the application of the projector 66

71 2 Group theory and eigenvalue problems algorithm produces Σ 4 symmetry adapted vectors of the type λ 1 λ 1 λ 3 λ 2 λ 2 λ 3 The insertion of this vector into the matrix equation F a = ω 2 a produces the Σ 4 block, which must be diagonalized manually : f 1 λ 1 + f 2 λ 2 = ω 2 λ 1 0λ 3 = ω 2 λ 3 f 2 λ 1 + f 3 λ 2 = ω 2 λ 2 This gives a double degenerate eigenvalue ω 2 = 0 Within the invariant subspace sustained by this eigenvalue we can choose the following eigenvectors: m H mh m O m mh O m O m H m O 1 +1 The one on the left hand describes a translational motion, the one on the right hand a rotational one. It remains the non-trivial eigenvalue (vibrational) ω 2 = f 1 + f 3 67

72 2 Group theory and eigenvalue problems Figure 2.4: Left: the 3 Σ 1 eigenvectors. Right: the 3 Σ 4 eigenvectors. to which we can assign the eigenvector mo m H m O m H 0 68

73 3 Continuos groups and symmetry breaking interactions Atomic orbitals represent the starting point of any Ansatz toward solving the eigenvalues problem of more complex systems, arising when the electron is embedded in a potential with symmetry reduced with respect to the atom. This could be the case of electrons in a molecule, a crystal field or a solid. We thus start with the representation theory of SO 3, which is the symmetry group of an electron in a spherically symmetric potential. This is a continuous group (with average) and therefore all theorems worked out in the past chapters are valid for SO 3 as well, although for continuous groups some more concepts and procedures have been developed, in order to work out their irreducible representations and to facilitate their practical use. One of these tools is, e.g., the Peter-Weyl theorem, see F. Peter, H. Weyl: Über die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe. Mathematische Annalen, Band 97, 1927, S We will review in this chapter first some of these new elements which are special to continuos groups. We then illustrate the use of symmetry when symmetry breaking interactions here in the specific form of a crystal field are introduced. 3.1 Continuos groups We consider groups whose elements x can be put in one to one correspondence with the points of a subset of an r-dimensional space S r, which we assume to be a vector space with scalar product. We shall refer to this subset as the parameter space, and designate the r-parameters as a 1,..., a r. r is called the order of the continuos group. 69

74 3 Continuos groups and symmetry breaking interactions The set of all translations in a three dimensional real vector space of the form x = x + a y = y + b z = z + c is a three parameter, continuos group. If we denote the translation operator by T(a, b, c), the identity element is T(0, 0, 0) and the inverse is T( a, b, c). The set of all continuos rotations around a fixed axis is a continuos group of order 1, the axial rotation group. The parameter can be chosen to be the angle of rotation, lying in the the interval [0, 2π]. The group is denoted by C and is isomorphic to SO 2.. The set of continuos rotations about all axes passing though a fixed point is the 3d space is a group, whose elements can be characterized by a unit vector and a rotation angle about it. It is isomorphic to the matrix group SO 3. Topological groups One require usually of a continuos group that it has a topology. Let P(x) the image of the group element in the parameter space. If this parameter space is provided with a scalar product, we can define a neighborhood of the point P(x) in S r, i.e. a set of point P such that P P(x) < ε The element x such that P(x ) = P will lay in a neighborhood of the element x is G, so that by this procedure it is possible to define a neighborhood of an element x of a continuos group. By using the concept of neighborhood one can define the continuity of the laws of composition and inversion of group elements. Let us consider the law of composition x 1 x 2 = x 3 and assume to change slightly x 2 to a point in its neighborhood. Than, if the law of composition is continuos, the result of the composition will be in the neighborhood 70

75 3 Continuos groups and symmetry breaking interactions of x 3. More precisely speaking: Given a neighborhood δ of P(x 2 ), for all x with P(x) P(x 2 ) < δ the elements P(x 1 x) belongs to a neighborhood ε of x 3 is x 1 x 2 = x 3, i.e. P(x 1 x) P(x 3 ) < ε One can define the continuity of the law of inversion by stating that for any x such that P(x ) P(x 1 ) < δ the elements P(x x) belong to a neighborhood of e, i.e. P(x x) P(e) < ε If a group is endowed with law of composition and a law of inversion which are continuos in the group elements than it is said to be a topological group. For topological groups one can define further properties, like connectedness and compactness. Consider any two elements x 1 and x 2 of a topological group G with images P(x 1 ) and P(x 2 ) in S r. It is possible to connect P(x 1 ) and P(x 2 ) by one or more paths lying entirely within the parameter space the parameter space and the group are said to be connected, otherwise it is disconnected. A continuos connected group may further be simply connected or multiply connected, depending on the topology of S r. A subset of a Euclidean space S r is said to be k-fold connected if there are precisely k distinct paths connecting any two points which cannot be brought into each other by continuos deformation without going outside S r. As an example, the surface of a sphere is a simply connected region of E 3. A group is said to be compact if the parameter space is bounded and closed (a set is closed if every Cauchy sequence of elements of the set has a limit element which also belongs to the set). Example: let us consider the axial group C. This is a one parameter, continuos compact connected group with a parameter space which can be visualized as a 71

76 3 Continuos groups and symmetry breaking interactions unit circle in a plane. In addition, it has Weyl average. It is connected because any two elements of the group (think of two point on a circle) can be connected by a path lying entirely within the circle. Actually, it is at least twice connected, as two points A and B can be connected by a path going clockwise and one going anticlockwise. Going further, we can state that any two elements can be connected by an infinite number of paths which wind around the circle a number of times n = 0, ±1, ±2,... Accordingly, the group is infinitely manifold connected because paths with different winding numbers cannot be brought into each other by a continuos deformation without going outside the circle. Lie Groups Consider a topological group G, i.e. a group where the the law of composition and the inverse law are continuos. Define P(x). = a = (a 1, a 2,..., a r ) as the image of the element x in the parameter space. Take the image of e to be 0, for convenience. Assume that we have a topological group, i.e the laws of composition and the law of inverse are continuos. This means that if x(s) x(t) = x(u) then the function is continuos and if the function u = f (s, t) x(s) 1 = x(v) v = g(s) is also a continuos function. If one requires that the functions f and g are analytical, than the topological group is said to be a Lie group (the name originates from Sophous Lie, a Norwegian mathematician). Example: it can be shown (Gleason, Montgomery and Zippin) that in a Lie group 72

77 3 Continuos groups and symmetry breaking interactions of order one it is possible to write the laws of composition as u = s + t so that we can assume that v = s x(s)x(t) = x(s + t) x(s) 1 = x( s) x(0) = e for a Lie group of order 1. It is easily verified that the axial rotation group is a Lie group, as well as the group of continuos translations. In contrast to the axial rotation group, the group of continuos translation, however, is neither a compact group nor it does have an average (a finite group volume, e.g., cannot be defined). Remark: One can show within an exact theorem that compact Lie groups have an average, so that when we speak of the representation of continuos groups with average we have actually compact Lie groups in mind. 3.2 Representations of continuous Lie groups We first illustrate the ideas underlying the finding of the matrix representations of Lie groups at the example of a lie group of order one, such as the group of translations along the t-axis or the (isomorphic) group of real numbers under the + -operation. By virtue of the Gleason, Montgomery Zippin theorem the results on this group can be used for any one-parameter group. Starting from x(s)x(t) = x(s + t) x(0) = e 73

78 3 Continuos groups and symmetry breaking interactions and differentiating with respect to s and putting s = 0 we obtain d d t x(t) = d x(s) ds s=0 x(t) which, for convenience, we rewrite as d d t x(t) = i 1 d x(s) s=0 x(t) } i ds {{}. The matrix I =. 1 d x(s) s=0 i ds is called the infinitesimal generator of the matrix representation of the onedimensional Lie group in the space L n. The name originates from the fact that, once I is specified, the solution of the differential equation leads to a matrix representation of the group elements via the so called exponential map: =I x(t) = e i I t The exponential function on the right hand side is to be understood in terms of its power-law expansion: e i I t ( i I t) n = n! n=0 We have just proven rigorously following theorem: "Every continuos finite dimensional representation of a one-dimensional Lie group is generated by an infinitesimal generator I via the exponential map e i I t This Theorem was generalized to Hilbert spaces by Stone: Suppose the unitary transformation x(t) of a Hilbert space is a continuous function of the real variable t satisfying the condition x(s)x(t) = x(s + t) 74

79 3 Continuos groups and symmetry breaking interactions x(0) = e Then an Hermitic linear operator I exists, defined by Iψ = 1 i lim s 0 x(s) e ψ s such that x(t) = e i I t This procedure can be extended to r-dimensional continuos Lie groups. Again, take, for simplicity, the origin of the parameter space to be the identity element, i.e. e = x(0, 0,...0). The analyticity of the law of composition allows to write, formally, any element in the vicinity of e as power series in the parameter s 1, s 2,..., s r : x(s 1, s 2,...s r ) = e i s 1 I 1 i s 1 I 1... Taking the derivative with respect to s i and putting s i = 0 one obtains a set of r-infinitesimal transformations. 1 x(s 1,..., s r ) I j = s i s j =0 j (the use of the imaginary number i in the various definitions is just a a convention). We can use the operators I j to construct a general element of the group by using the exponential map of the group: x(a 1, a 2,...a r ) = Π r j=1 ex p[ i a j I j ] a 1, a 2,... being a new set of parameters. This expression is called the exponential map of the Lie group and show that one can generate all elements of the Lie group starting form the r-operators I j. The operators I j are therefore called appropriately the generators (German: infinitesimale Erzeugende) of the Lie group. A Lie group with r continuous parameters has r generators. Remark I. The existence of the relation x(s 1, s 2,...s r ) = e i s 1 I 1 i s 1 I

80 3 Continuos groups and symmetry breaking interactions implies that for computing the representation of the elements of the Lie groups one must be able to perform linear combinations of the infinitesimal generators. This means that the infinitesimal generators build a vector space. Remark II. In order to use the exponential expression of the group elements in terms of infinitesimal generators to perform the group operation between any two or more elements of the group, we need to be able to calculate expression such as e A 1 e A 2, A1 and A 2 being, in general, non commuting matrices. This is performed by means of the Baxter-Campbell-Haussdorf (BCH) formula, e A1 e A 2 = ex p(a) A = A 1 + A [A 1, A 2 ] + 1 [A 1, A 2 ], A [A 2, A 1 ], A where [A, B]. = AB BA is the commutator of the two operators A, B. This formula implies that the product of two elements of the group involves commutators between any I l and I k. This establishes a further law of composition within the vector space of the generators, which becomes a Lie algebra. The rule of computations for the commutator are 1. The commutator is closed, i.e. [A, B] L 2. [A, B] = [B, B] 3. (Jacobi identity) [A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0 Remark III. We are able to perform commutator operations within the Lie Algebra if the so called structure relations between the basis elements of the Lie Algebra are specified: [I l, I k ] = r J=1 c j lk I j, 1 l, k r, where c j are certain the so called structure constants of the Lie lk algebra and of the Lie group. Notice that the structure constants are not unique, since the generators of a Lie group are themselves not unique. Theorem: provided the representation of the Lie group is chosen to be unitary, the generators are Hermitic operators and the structure constants are purely imaginary numbers. 76

81 3 Continuos groups and symmetry breaking interactions Proof : We require from any element ex p( ii j a j ) that ex p(ia j I j )ex p( ia ji j ) = ex p(0) which is only possible if I j = I j, i.e. the generators are Hermitic (here we have used the fact that (A n ) t = (A t ) n ). Furthermore, from r j=1 ( c j lk )I j = [I l, I k ] = [I l, I k ] = r j=1 c j lk I j = r j=1 c j lk I j we obtain c j lk = c j, i.e. the structure constants are purely imaginary numbers, lk with c j lk = c j. kl Significance of the Lie Algebra. By virtue of the exponential map, we can formally write, within a small region of the parametric group close to the origin x(a 1, a 2,...a r ) = e ia 1 I 1 ia 2 I ia r I r (a j being small). Accordingly, the problem of finding a matrix representation of G can be reduced to finding a matrix representation of its Lie Algebra: if we are able to find a set of r-square matrices of order p satisfying the structure relations and construct their invariant subspaces we automatically generate a p-dimensional representation of all the elements of the group residing very close to the identity. By means of the exponential map we can export the representation to the entire group. By this exportation the invariant subspaces are maitained. Naturally, those representations of the Lie-Algebra which are irreducible will provide irr. rep. of the Lie-group. In sumary: we can restrict ourselves to studying the representation theory of the Lie- Algebra, which is often less cumbersome to treat than working out the representation of the Lie-group itself. The Peter-Weyl theorem One important tool that we had at disposal when searching for the irreducible representations of finite groups was the fact that the number of irreducible repre- 77

82 3 Continuos groups and symmetry breaking interactions sentations is equal to the number of conjugacy classes. We found that both numbers were finite and countable for finite groups. When dealing with Lie groups, we expect the number of classes to be non-countable infinite, as this is the typical situation for the elements of a continuos group. The question about the number and countability of the irreducible representations for Lie groups was partially answered by the Peter-Weyl theorem: the number of irreducible representations of a continuous Lie groups with average is countably infinite i.e. there is an infinite number of irreducible representations, but they cen be labeled by a set of integer indices Example: the circle group Irreducible representations. In virtue of the Peter-Weyl theorem it might make sense to formally organize the characters of continuous groups with average into a character table as well, taking into account, however, that both the number of columns and the number of rows can be infinite (but the number of columns is certainly countably infinite). As an example we investigate the irreducible representations of the axial rotation group (or circle group) C. The group is Abelian, so each element builds a class on its own. Each class is characterized by a rotation angle ϕ [0, 2π]. The group has an average, so it has infinitely countable inequivalent irreducible representations. First let us show that the dimensionality of finite irreducible representation of any Abelian group is 1. Consider an irreducible matrix representation of the group, and let it be of dimension n > 1. Any of its matrices commute with the matrices representing the group, so, by virtue of Schur Lemma number 1, it is a diagonal matrix. But the diagonal matrix consists of one dimensional blocks into which the original representation can be decomposed, so that the original representation cannot be assumed to be irreducible if n > 1. We now state that. = e imϕ Γ m 78

83 3 Continuos groups and symmetry breaking interactions with m are all irreducible, unitary representations of C. First we prove that the so defined matrix is an homomorphism. This can be proved by considering that e im(ϕ 1+ϕ 2 ) = e im(ϕ1 e imϕ 2 The representation is irreducible, as 1 2π 2π 0 e imϕ 2 dϕ = 1 Γ m represent all irreducible representations. To prove that, assume there exists an irreducible Γ Γ m. In virtue of the orthogonality of the characters we must have 1 2π 2π 0 e imϕ Γ (ϕ) = 0 and this m. But the set {e imϕ is complete in the interval [0, 2π], so that from 1 2π 2π 0 e imϕ Γ (ϕ) = 0 it follows that Γ (ϕ) 0. This contradicts the assumption of Γ being irreducible, because irreducible representation must have average 1, and this average cannot be achieved if Γ (ϕ 0). These results are, of course, in agreement with the Peter-Weyl theorem. On the base of these results we can envisage a character table for C (the column on the right hand side shows that the infinitesimal generators are m ): The regular representation The regular (left) representation of C can be defined as an operator O α acting onto the infinite dimensional Hilbert space consisting of square integrable functions of the variable ϕ subject to periodic boundary conditions f (ϕ + 2π) = f (ϕ): O α f (ϕ) = f (ϕ α) 79

84 3 Continuos groups and symmetry breaking interactions C α I α Γ Γ 1 e iα 1.. Γ m e imα m The operator O α is a representation of the axial rotation group because We find the infinitesimal generator by O α1 O α2 = O α1 +α 2 Ta ylor {}}{ O ε f (ϕ) = f (ϕ ε) = f (ϕ) + f (ϕ) [ ε] = ϕ 1 i ε[ i ϕ ] f (ϕ) From this last equation we can read out the infinitesimal generator as and the operator of the exponential map: I α = i ϕ O α = e iα[ i ϕ ] Writing as i ϕ L z ħh 80

85 3 Continuos groups and symmetry breaking interactions one recognizes that the infinitesimal generator of the left regular representation of the group C is essentially the z-component of the angular momentum operator, z being the rotation axis. Invariant eigenspaces. To find invariant eigenspaces of the left regular representation we solve the differential equation with the boundary conditions i f (ϕ) = λf (ϕ) ϕ f (ϕ + 2π) = f (ϕ) The solution of the differential equation is e iλϕ the eigenvalue λ being determined to be 0, ±1, ±2,... by the boundary condition e iλϕ = e iλ(ϕ+2π) Accordingly, the invariant subspaces are one dimensional functions of the type {1, e ±iϕ, e ±2 iϕ,...} The representation of C constructed over the basis function e i mϕ, m = 0, ±1, ±2,... reads e imϕ, O α e imϕ = e i m α and are the one dimensional irreducible representations. Remark: Notice that C might appear as the subgroup of a larger rotation group, defined onto a larger Hilbert space, with functions involving e.g. the spherical coordinates r, ϑ, alongside ϕ. In this case there are infinitely many basis functions transforming according to Γ m, which can be written as f (r, ϑ) e imϕ. 81

86 3 Continuos groups and symmetry breaking interactions C α Basis I α Γ Γ 1 e iα e iϕ 1... Γ m e imα e imϕ m Remark. We can find invariant eigenspaces to the representation Γ m by the projector method: P Γm f (ϕ) = 1 2π dαe i m α O α f (ϕ) = 1 2π dαe i m α f (ϕ α) 2π 2π 0 0 = e i m ϕ 1 2π dαe i m (ϕ α) f (ϕ α) 2π 0 e i m ϕ f m On the base of these results we can assign to each irreducible representation a symmetry adapted wave function, see the Table. The SO(2) representation of C As SO(2) the group of 2 2 orthogonal matrices with determinant 1 is a oneparameter group, it has only one generator. As the unit matrix is given by ϕ = 0, according to our definition I = 1 i d cos ϕ sin ϕ dϕ sin ϕ cos ϕ ϕ=0 = 0 i +i 0 82

87 3 Continuos groups and symmetry breaking interactions which is just σ y, i.e. one of the Pauli matrices. Accordingly, any 2 2 orthogonal matrix with determinant 1 can be written as follows: cos ϕ sin ϕ = ex p[ i σ y ϕ] sin ϕ cos ϕ The full rotation group Topological properties of SO 3 SO 3 is a continuos group containing all proper rotations about any axis. In E 3 it has a 3 3 matrix representations consisting of all orthogonal matrices with Determinant 1. Every rotation has a rotation axis, characterized by a unit vector n and a rotation angle ϕ. By the cork-screw rule one direction can be defined as positive. The number of parameters required to describe a rotation is 3, so that SO 3 is a three dimensional continuos group. There are three common ways of parameterizing rotation in 3d: Successive rotations about three mutually orthogonal fixed axes Successive rotations about the z-axis, about the new y-axis and finally about the new z-axis. The rotation angles are the Euler angles. The axis-angle representation, defined in terms of a unit vector specifying the rotation axis and an angle ϕ specifying the rotation angle. In this last case the vector ϕ n defines the rotation uniquely: every rotation is determined by the coordinates n 1 ϕ, n 2 ϕ, n 3 ϕ n n2 2 + n2 3 = 1. Accordingly, one can associate every element of SO 3 with a point inside a sphere of radius π. The direction from the origin to the point OA corresponds to the rotation axis, the distance OA from the origin being the rotation angle. A, B, C indicate rotations by the same angle about different axes and belong to the same class: accordingly, the classes of conjugate elements reside onto concentrically spherical sheets. Notice that D and E are diametrically opposite and correspond to the same operation, as rotations by π around ± n are identical elements. This introduces some remarkable features into the topology of SO 3. Connectedness. There are two distinct paths connecting R 1 and R 2 in the param- 83

88 3 Continuos groups and symmetry breaking interactions Figure 3.1 Figure

89 3 Continuos groups and symmetry breaking interactions eter space: a direct path (a) from R 1 to R 2 and a path (b) involving going to x, making a jump to x = x and moving back to R 2. Path (b) cannot be deformed to a path of type (a) by continuos distortions, because as we move the point x on the surface, x remains diametrically opposite to x. We conclude that SO 3 is at least doubly connected. We explore now a further path involving e.h. 2 jumps across the sphere: as we let x approach y x approaches y. Finally, as x and y coincide, so do x and y and the path becomes of type (a). It can be shown that a path which makes an even number of jumps if a type (a) and a path with an uneven number of jumps is of type (b): SO 3 is exactly doubly connected. SO 3 is a compact group and therefore has a group average. The key quantity we would like to determine in order to perform group integration is the density ρ(ϕ, ϑ, φ) of group elements at certain location of the parameter space, i.e. within the infinitesimal volume element ϕ 2 sin ϑdϕdϑdφ ϕ [0, π], ϑ [0, π], φ [0, 2π]. To find ρ(ϕ, ϑ, φ) we adopt a procedure given e.g. by Prof. D.D. Vvedensky, Imperial College. First we notice that ρ(ϕ, ϑ, φ) must be a class function, i.e. depends only on ϕ. In fact, from F we have G F(R)ρ(R) = G F(ARA 1 )ρ(ara 1 )dr ρ(ara 1 ) = ρ(r) Using the procedure indicated by Vvedensky we find the result ρ(ϕ, ϑ, φ) = 4 ϕ 2 sin2 ϕ 2 so that SO3 f (ϕ, ϑ, φ) = 1 2π 2 π 2π 0 0 sin ϑdϑdφ π 0 sin 2 ϕ dϕ f (ϕ, ϑ, φ) 2 85

90 3 Continuos groups and symmetry breaking interactions Irreducible representations To construct the irreducible representations of SO 3 generators. we turn to the infinitesimal We consider infinitesimal rotations e.g. around the z-axis ( n = (0, 0, 1). cos ϕ 3 sin ϕ 3 0 R z (ϕ 3 ) = sin ϕ 3 cos ϕ R z (ϕ 3 ) transforms the coordinates x, y, z. Applying the (left) regular representations onto the function space f (x, y, z) we obtain the infinitesimal generator for rotations around the z-axis. R z (ε 3 )f (x, y, z) = f (x cos ε 3 + y sin ε 3, x sin ε 3 + y cos ε 3, z) = }{{} Ta ylor f (x, y, z) + [i f x y i f y x] [ iε 3] from this relation we can read out the infinitesimal generator of rotations around the z-axis: I z = i[ y x x y ] = L z ħh Similarly one obtains or, summarizing I x = i[z y y z ] = L x ħh I y = i[x z z x ] = L y ħh I = L ħh L being the quantum mechanical operator for the orbital angular momentum. One can refer to standard books on quantum mechanics to compute the structure relations and the structure constants: [I k, I l ] = i ε klm I m k,l,m 86

91 3 Continuos groups and symmetry breaking interactions with ε klm = ±1 for even (uneven) permutation of klm and ε klm = 0 for two or more klm being equal. One can start the algebraic construction of the irreducible representations using these structure relations, but we use a different method, namely the one of obtaining first the invariant subspaces and them constructing the matrix representations over them. As SO 2 is a subgroup of SO 3 a useful starting point to construct invariant subspaces for representations of SO 3 are the basis functions transforming according to the irreducible representations Γ m of SO 2 { f (r, ϑ)e imϕ }. Those element of SO 3 not belonging to SO 2 will mix these basis functions into larger invariant subspaces according to the rule of thumb that expanding the gropup enlarges the invariant subspaces (we will see later that the opposite is also true, i.e. reducing the group restricts the invariant subspaces.) Our task is to find out which basis functions will be mixed. A most significant step toward the construction of invariant subspaces and ultimately, all irr. representations of SO 3 is the Casimir operator, which commutes with all infinitesimal generators of the Lie group. Its invariant subspaces are bound to be useful for finding the irreducible subspaces, in virtue of the Lemma of Schur. According to a theorem by Racah the number of independent Casimir operators of a Lie group is equal to the minimum number of commutating generators (the so called rank of a Lie group). In the case of SO 3 the rank is just one because no two of its generators commute. The one Casimir operator of SO 3 is Within a function space, I 2 x + I 2 y + I 2 z. = I 2 I 2 = Λ =. 1 sin ϑ ϑ (sin ϑ ϑ ) 1 sin 2 ϑ ϕ

92 3 Continuos groups and symmetry breaking interactions and the problem of finding invariant subspaces carrying representations of the group SO 3 reduces to solving the eigenvalue problem Λ f (ϑ, ϕ) = λ f (ϑ, ϕ) winch is known to be solved by f (ϑ + π, ϕ + 2π) = f (ϑ, ϕ) (l = 0, 1, 2, 3,...) E l = {Y l l (ϑ, ϕ), Y l 1 l λ = l(l + 1) (ϑ, ϕ),..., Y l (ϑ, ϕ)} The upper index of the spherical harmonics indicates which Cartan weights m of the infinitesimal generator of SO 2 are mixed by the remaining operations of the group SO 3. Onto these invariant subspaces we can construct the matrix representation of SO 3 : l l I z l = l l i.e. Y m l, I z Y m l = m δ mm 88

93 3 Continuos groups and symmetry breaking interactions Accordingly, the matrix representation of a rotation about the z axis within the subspace E l reads: e ilϕ e i(l 1)ϕ 0... D z l = e ilϕ we summarize the matrix representation of I z, I x, I y according to the convention of Condon and Shortley (only those matrix elements which are different from 0 are given): (Y l,m, I z Y l,m ) = m (Y l,m±1, I x Y l,m ) = 1 (l m)(l ± m + 1) 2 (Y l,m±1, I y Y l,m ) = i (l m)(l ± m + 1) 2 Theorem: The invariant subspace E l is irreducible. This means that the representations D l of SO 3, l = 0, 1, 2,... constructed in the previous paragraph are irreducible representations of SO 3. Proof: Assume there is an invariant subspace of E l. Choose in it the highest value of m, which by assumption is smaller than l. An operator I x + ii y can be defined, which transform Y m<l to Y m+1, so that I l l x + ii y brings us outside the subspace we assumed to be invariant. Accordingly, m < l is not possible. Choose now the lowest m > l. The operator I x ii y brings us to m 1, also m > l is not possible. Accordingly, E l is irreducible. Theorem: the irreducible representations D l represents all finite dimensional irreducible representations of SO 3. Proof: {E l }, l = 0, 1, 2,... build a complete set of basis functions on the sphere. Character Table of SO 3. To construct the character table of SO 3 we notice that the rotations by the same angle ϕ about different axes n belong to the same conjugacy class: in other words, as representative of each class one can choose a rotation 89

94 3 Continuos groups and symmetry breaking interactions SO 3 ϕ Basis functions D 0 1 f (r)y 0 0 D 1 sin(3 ϕ 2 ) sin ϕ 2 g(r) {Y 1 1, Y 0 1, Y 1 1 } D 2 sin(5 ϕ 2 ) sin ϕ h(r) {Y 2 2, Y 1 2, Y 0 2, Y 1 2, Y 2 2 } about the z-axis, which is represented by the matrix e ilϕ e i(l 1)ϕ 0... D z l = The character of this matrix amounts to l m= l e ilϕ e imϕ = sin([(2l + 1)ϕ 2 sin ϕ 2 build irre- and the character table looks like the following scheme: Remark. Using the characters we can immediately prove that the D l ducible representations of SO 3. In fact SO3 χ l (ϕ)χ l(ϕ) = 1 2π 2 sin θ sin 2 ϕ 2 dθ dϕdφ sin2 ([(2l + 1) ϕ 2 sin 2 ϕ 2 We can also prove that D l are all irreducible representations. For let D l be an irreducible representation of SO 3. In virtue of the orthogonality of characters we must have SO3 (ϕ) sin([(2l + 1)ϕ 2 sin ϕ 2 = 0 = 1 90

95 3 Continuos groups and symmetry breaking interactions The integral on the left hand side amounts to 4 π π/2 0 dτ (τ) sin τ sin ((2l + 1)τ) But the set of functions sin τ, sin 3τ, sin 5τ,... builds a complete set in the interval [0, π 2 ] so that the integral can only vanish if (τ) 0, which contradicts the assumption of irreducibility of the hypothetical representation The translation group The 1d translation group is isomorphic to (, +) and is defined by the transformation x = x + t with t. Any representation T(t) obeys the multiplication rule T(t + s) = T(t) T(s) T( t) = T(t) 1 which can be recast in the from of a differential equation d T(t) = i I T(t) d t Theorem: all finite dimensional irreducible representations of are one dimensional. Proof: The group is Abelian. Because of the Lemma of Schur the matrices of any the irreducible representation for any element t must be a diagonal one. Diagonal matrices of any finite dimensional are, by definition, block-diagonalizable into blocks with size 1 1, and these are, by definition, irreducible. Theorem: All unitary, one-dimensional representations of the group are given by t T(t) = e ikt with k being the infinitesimal generator. Proof: The exponential function obeys the differential equation and taking real k 91

96 3 Continuos groups and symmetry breaking interactions ensures unitarity. Within the Hilbert space of square integrable functions we define the left regular representation by T reg (t)f (x). = f (x t) The infinitesimal generator is given by the momentum operator i x as on one side On the other e i ε [ i x ] f (x) = 1 ε x f (x) Ta ylor {}}{ f (x ε) = f (x) ε f (x) In virtue of the identity of the infinitesimal generator with the momentum operator the regular representation is known in Quantum mechanics also as the Schrödinger representation. In the context of group theory, the Schrödinger representation is an infinite dimensional representation of a group without average. In contrast to similar groups with average, Peter-Weyl theorem is not applicable to the translation group and, in addition, the theorems encountered for groups with average must be used with care. Yet the question about the irreducibility of the Schrödinger representation is a central one, e.g. in quantum mechanics, as it is identical with the question of uniqueness or non-uniqueness of the Schrödinger representation of the momentum and position operator. This last question has motivated important works at the very beginning of Quantum mechanics and has culminated in the celebrated Stone-Von-Neumann theorem, proving 1. the irreducibility of the Schrödinger representation and 2. the unitary equivalence of any other representation to the Schrödinger representation. (von Neumann, J. (1931), Die Eindeutigkeit der Schrödingerschen Operatoren, Mathematische Annalen (Springer Berlin / Heidelberg) 104: , doi: /bf , ISSN von Neumann, J. (1932), Ueber Einen Satz Von Herrn M. H. Stone, Annals of Mathematics, Second Series (in German) (Annals of Mathematics) 33 (3): , doi: / , ISSN X, JSTOR

97 3 Continuos groups and symmetry breaking interactions Stone, M. H. (1930), Linear Transformations in Hilbert Space. III. Operational Methods and Group Theory, Proceedings of the National Academy of Sciences of the United States of America (National Academy of Sciences) 16 (2): , doi: /pnas , ISSN , JSTOR Stone, M. H. (1932), On one-parameter unitary groups in Hilbert Space, Annals of Mathematics 33 (3): , doi: / , JSTOR ) Proof of irreducibility of the regular representation The translation group does not have an average but within the Hilbert space of square integrable function, defined on, a scalar product is defined by the usual expression: f, g = f (x) g(x) and can be used to explore the irreducibility of the regular representation. In fact, assume that f belong to a subspace M of L 2 () which is invariant with respect to T reg and consider g M, so that g, T reg f = 0 Accordingly, 0 = 1 2π d te ikt g, T reg f = 1 2π d te ikt d x g (x) f (x t) = g (k) f (k) in virtue of the fact the operation g, T reg f is equivalent to the convolution of g and f and of the fact that the Fourier transform of the convolution of g and f is the product of the Fourier transforms of g and f. The equality g (k) f (k) = 0 has the solution g(k) = g(x) 0 93

98 3 Continuos groups and symmetry breaking interactions so that M coincide with L 2 () and T reg has no invariant space other than the whole L 2 (). But this also implies that the Schrödinger representation is the only possible irreducible representation, as non-equivalent irreducible representation can only appear onto a set of symmetry adapted wave functions which is orthogonal to any function in L 2, but this is impossible as L 2 is complete. 3.3 Cristal field splitting Consider a system with a operator H 0, and suppose that a static perturbation V is introduced giving a new operator H = H 0 + V. Let G 0 be the symmetry group of the operator H 0. If not all transformations of G 0 leave V invariant, then V is a symmetry breaking perturbation and the symmetry group of the total operator H is a subgroup of G 0. Let us call it G V. When limited to the operations of G V the irreducible representations of G 0 might become reducible: new invariant subspaces carrying the irreducible representations of G V appear and the degeneracy of the original eigenvalue E 0 is lifted accordingly. The energy level E 0 splits into states carrying the irreducible representations of G V. In general, we expect D α GV = i n i D i (G V ) i.e. the irreducible representations D α of G 0, when limited to the operations of G V decomposes into irreducible representations of G V. These irreducible representations are said to be compatible with the irreducible rep. of G 0 and the decomposition process can be summarized in the form of compatibility tables. As an example, think of embedding the s, p, d electrons into a crystal potential with O h symmetry. The following compatibility tables illustrate the decomposition process. As a further example, embed the levels with O h symmetry into a crystal potential with C 4v - symmetry. Again, the decomposition process is given in the following compatibility tables. 94

99 3 Continuos groups and symmetry breaking interactions O h E 3C 2 4 6C 4 6C 2 8C 3 I 3IC 2 4 6IC 4 6IC 2 8IC 3 Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ D 0 Oh = Γ 1 D 1 Oh = Γ 15 D 2 Oh = Γ 12 Γ 25 Table 3.1: Character table of the group O h. At the bottom the representations D 0, D 1, D 2 of SO 3 restricted to the operations of O h. C 4v E C 2x C 4x C 1 4x IC 2z IC 2y IC 2yz IC 2y z Γ 1 C4v = 1 Γ 15 C4v = 1 5 Γ 25 C4v = 2 5 Γ 12 C4v = 1 2 Table 3.2: Character table of the group C 4v. At the bottom some representations of O h restricted to the operations of C 4v. 95

100 3 Continuos groups and symmetry breaking interactions SO 3 D 0 D 1 D 2 D 3 O h Γ 1 Γ 15 Γ 12 Γ 25 Γ 2 Γ 25 Γ 15 O h Γ 1 Γ 2 Γ 12 Γ 25 Γ 15 C 4v O h Γ 1 Γ 2 Γ 12 Γ 25 Γ 15 C 4v Table 3.3: Compatibility table between the groups SO 3, O h and C 4v 96

101 3 Continuos groups and symmetry breaking interactions The concept of compatibility establishes a new set of symmetry adapted vectors originating from the invariant subsets belonging to the irreducible representations of G 0. The operations of G V mix only those vectors in the new invariant subspaces. These new invariant subspaces must all belong to different eigenvalues (except in the case of accidental degeneracy) and hence the original energy level E 0 α splits into a number of energy levels due to the lowering of symmetry. The mere knowledge of the irreducible representations of G V and which of these are contained in the decomposition of the representation D α allows an exact labeling of the energy states arising from the splitting of the original level. In addition, supposing we know how to construct symmetry adapted wave functions to G V from the original symmetry adapted wave functions, we are also able to estimate the value of the new eigenvalues of H 0 + V, at least perturbatively. The method of choice to finding the symmetry adapted vectors is the projector technique. The following tables summarize the results for crystal field with selected symmetry. The basis polynomials are constructed on the base of the coordinate system used to describe the symmetry operations of a cube. Example: s and p electrons in C 3v -symmetry We would like to find, within the subspace Y 00, Y 11, Y 10, Y 1 1, the symmetry adapted wave functions transforming according to Λ 1 and Λ 3. We start for instance from Y 00 and we apply the projection operator: P Λ1 Y 00 = Y 00 = Y 00 This means that Y 00 transforms according to Λ 1. The lowest conduction band along the Λ direction of GaAs or Ge is indeed a s-state with Λ 1 symmetry. Applying the projector operator to the same eigenfunctions in oder to project out the wave functions transforming according to Λ 3 will inevitably leads to zero, as Y 00 does not contain parts transforming like Λ 3. Starting from Y 10 we obtain (for convenience, we set the z axis along the Λ direction) P Λ1 Y 10 = Y 10 = Y 10 97

102 3 Continuos groups and symmetry breaking interactions O h E 3C 2 4 6C 4 6C 2 8C 3 I 3IC 2 4 6IC 4 6IC 2 8IC 3 Basis Γ x 4 (y 2 z 2 )+ Γ y 4 (z 2 x 2 )+ z 4 (x 2 y 2 ) Γ (x 2 y 2 ) 2z 2 x 2 y 2 Γ Γ Γ x y z z(x 2 y 2 ) x(y 2 z 2 ) y(z 2 x 2 ) x yz[x 4 (y 2 z 2 )+ y 4 (z 2 x 2 )+ z 4 (x 2 y 2 )] Γ x yz Γ x yz(x 2 y 2 ) x yz(2z 2 x 2 y 2 ) Γ x y(x 2 y 2 ) yz(y 2 z 2 ) zx(z 2 x 2 ) Γ x y yz zx Table 3.4: Character table of O h with symmetry adapted polynomials. C 3v E C 3x yz C 1 3x yz IC 2x ȳ IC 2 xz IC 2y z Basis Λ x + y + z Λ x(y 2. z 2 )+ y(z 2 x 2 )+ z(x 2 y 2 ) Λ x y z y z Table 3.5: Character table of C 3v with symmetry adapted polynomials. 98

103 3 Continuos groups and symmetry breaking interactions C 4v E C 2x C 4x C 1 4x IC 2z IC 2y IC 2yz IC 2 y z Basis , x, 2x 2 y 2 z yz(y 2 z 2 ) y 2 z yz y, z, x y, xz Table 3.6: Character table of C 4v with symmetry adapted polynomials. i.e. Y 10 transforms according to Λ 1 as well. The deep valence Λ 1 band along the Λ- direction in Ga As and Ge has mostly z-character. In order to compute the transformation properties of x, y we use O ϕ f (x, y) = f (x cos ϕ + y sin ϕ, x sin ϕ + y cos ϕ) The projector P Λ3 applied to x and y gives a linear combination of both, so that the invariant subspace belonging to Λ 3 is carried by the two functions x, y. The highest valence band along Λ has this symmetry. Thus, the p-states, which are degenerate at the Γ -point, as they belong to the irreducible representation Γ 15 split into two components, the one containing z (Λ 1 ) and the one containing x, y(λ 3 ). 3.4 Appendix I: Proof of The Peter-Weyl theorem The central theorem for constructing the representations of continuos groups with average is the Peter-Weyl theorem, which essentially establishes that such groups have a countable (albeit infinite) set of finite dimensional irreducible representations. To formulate and prove this theorem we need to introduce some further structures within a group with average. First we define the space of all squareintegrable functions on a group. Definition: Let G be a group with average. Then we define L 2 (G) as the set of complex-valued functions f that satisfy the following conditions: 1. f is measurable under the Haar measure 2. f (g) 2 < 99

104 3 Continuos groups and symmetry breaking interactions 3. The Haar measure induces on L 2 (G) a scalar product (f 1, f 2 ). = g f 1 (g)f 2 (g) With these definitions L 2 (G) becomes a Hilbert space with a complete orthonormal system of basis vectors (COS). Within this Hilbert space we define the regular representations of G as follows: Definition 1: The representation T reg : G L 2 (G); h T(h) with T reg (h)f (g) = f (g h) is called the right regular representation of G. Definition 2 The representation S reg : G L 2 (G); h S(h) with S reg (h)f (g) = f (h 1 g) is called the left regular representation of G. It is important to prove that both representations are 1. unitary: (T(h)f 1, T(h)f 2 ) = g f 1 (gh)f 2(gh) = g f 1 (g)f 2(g) = (f l, f 2 ) and 2. unitary equivalent, i.e. there is a one-to one mapping such that AS(g) = T(g)A for all g G Define Af (g) = f (g 1 ) 100

105 3 Continuos groups and symmetry breaking interactions Then T(h)Af (g) = T(h)f (g 1 ) = f (g 1 h) = f ((h 1 g) l ) == Af (h 1 g) = AS(h)f (g) From now one we will only consider one regular representation, e.g. T reg and transform the vectors of the Hilbert space L 2 (G) and of the conjugation invariant subspace of L 2 (G) containing class functions accordingly. Within this new framework the Peter-Weyl theorem reads: In the decomposition of the regular representation of G each irreducible representation appears at least once and with a multiplicity correspondig to its dimension. Accordingly, L 2 is split into an orthogonal sum of countable invariant subspaces. This theorem generalizes the equation we encountered for finite groups T reg = α l α Γ α to a continuos group with average. Notice that this generalization is not trivial as L 2 (G) (and the corresponding regular representation) are infinite dimensional and the concept of character which we have used to obtain the decomposition of the regular representation in the finite case is no longer necessarily meaningful. The trick aimed at proving the theorem, originally introduced by Peter and Weyl, and since them reused by all authors, is to construct an Hermitic Hilbert- Schmidt operator that commutes with the regular representation. Of such operators it is known that they have an infinite but complete set of countable eigenvectors. As commuting operators have common eigenspaces, we can use the finite invariant eigenspaces of the ad-hoc constructed Hilbert-Schmidt operator to build unitary matrix representations of the regular representation. Once this is done, we might proceed and decompose the unitary finite representations just constructed into irreducible components, if necessary. In order to find the suitable Hilbert-Schmidt operator we define the convolution of two functions f 1, f 2 as f 1 f 2 (h). = g f 1 (g) T reg (h)f 2 (g) = g f 1 (g) f 2 (gh) 101

106 3 Continuos groups and symmetry breaking interactions and associate with it the operator Using the scalar product A f2 f 1 (h). = f 1 f 2 (h) f 1, f 2 = g f 1 (g)f 2(g) we show that A f2 is Hermitic, provided f 2 (ab) = f 2 (ba): f 3, A f2 f 1 = h f 3 (h) g f 1 (g)f 2 (gh) f 2 (gh)=f 2 (hg) {}}{ = h,g (f 3 (h) f 2 (hg)) f 1 (g) = h,g (f 3 (g) f 2 (gh)) f 1 (h) = A f2 f 3, f 1 As the kernel f 2 (gh) is square integrable g,h f 2 (gh) 2 < the operator A f is a Hilbert-Schmidt integral operator and has some welcome properties that are needed to prove the Peter-Weyl theorem, like 1. An integral operator b a G(x, y)f (y)d y. = (Af )(x) with G(x, y) = G (y, x) is of the Hilbert Schmidt type is the kernel is square integrable. 2. If a COS is chosen, it can be represented by an infinite matrix which has the property that a i,j 2 < i,j (the matrix elements are quadratically summable). 3. For Hilbert Schmidt operators there exits a spectral theorem (usable for any compact Hermitic operator on a complex Hilbert space). Accordingly, there exists an at most countable sequence of eigenvalues λ 1, λ 2,... that converge to zero and 102

107 3 Continuos groups and symmetry breaking interactions an orthogonal decomposition L 2 = V 0 V λ1... of L 2 into the eigenspace solving the homogeneous problem Aφ 0 = 0 and the λ i eigenspaces V λi which are all finite dimensional. We now prove the commutativity of A f and T r. [A f T r φ](x) = g φ(gr)f (gr x) On the other side [T r A f φ](x) = T r [ g φ(g)f (g x)] = g φ(gr)f (gr x) showing that the two expressions on the right hand side of these equations are identical. This conclude the proof that the regular representation carries a COS of invariant subspaces. From the Hilbert-Schmidt theory we know that there is a subspace M λ of L 2 (G) that carries a representation T λ of G. As the representation is unitary, we can decompose it into the irreducible components T 1 λ, T 2 λ,..., T p spanning the invariant subspaces M 1 λ, M 2 λ,..., M p. Let us now consider the set of basis functions λ λ e 1 (g), e 2 (g)..., e n in M k carrying the irreducible representation λ T k and denote the λ matrix elements of T k built onto the basis functions as λ t i j(g). As T k is a representation of the right regular representation within the space M k, we have λ λ T k λ (g)e i(h) = e i (hg) = t ji (g)e j (h) Specializing this equation to h = e we obtain e i (g) = t ji (g)e j (e) j As e j (e) are constants, the basis elements e i (g) are therefore linear combinations of the matrix entries t i j (g), j = 1,.., n. This produces exactly n-linear combinations of the n 2 functions t ji (g) that carry the n-dimensional irreducible representation T k λ. j 103

108 3 Continuos groups and symmetry breaking interactions Moreover, there are a total of n linear independent linear combinations of t ji (g) that can be used to build the representation T k λ. Accordingly, the representation T k λ can repeat itself exactly n-times in the decomposition of the regular representation. As the original set of basis function arising from the Hilbert-Schmidt operator build a COS in L 2 (G), so does the the set of all linear combination of the function l λ t λ i j (g), λ running over all irreducible representations of G and l λ being their dimension. The GOT, namely, warrants the ortho normality of the function l λ t λ i j (g). = e λ i j : if if g lλ t,λ i j l λ t λ km = 1 λ = λ, i = k, j = m g lλ t,λ i j l λ t λ km = 0 λ λ, i k, j m The proof of course includes class functions as well, so that a corollary of Peter-Weyl theorem is that The character functions χ λ (g) build a COS within the Hilbert space of class functions A corollary of Peter-Weyl theorem is the following general expansion theorem of functions in L 2 (G) : Every function f (g) in L 2 (G) can be written in the form f (g) = e λ i j (g), f (g) eλ i j (g) λ i,j e λ i j (g), f (g) = ge λ i j (g)f (g) For every function f (g) we have the Plancherel relation f, f = λ i, j e λ i j (g), f (g) 2 104

109 3 Continuos groups and symmetry breaking interactions These equations are a generalization of Fourier theorem to general compact groups. For class functions f (x) we have the corresponding relation: f (x) = χ λ (x), f (x) χ λ (x) 3.5 Appendix II: The Harisch-Chandra character λ When dealing with infinite-dimensional representations of continuos Lie groups, the character of the element x, defined as e n, Γ (x)e n n is typically divergent if considered point wise, even if the group has an average. Accordingly, our theorems involving characters cannot be straightforwardly extended e.g. to the reduction of the regular representation in continuos groups, although, as we have learned in finite groups and within the Peter-Weyl theorem, the regular representations and its decomposition are very useful concepts for applications. Harish-Chandra (see M.F. Atiyah, The harisch-chandra character, in Representation theory of Lie groups, Edited by M.F. Atiya et al., Book DOI: was able to define the notion of characters for infinite dimensional representations of any (semi-simple) Lie group, by showing that e n, Γ (x)e n n converges in the sense of distributions. This distribution is the Harisch-Chandra character of Γ (x). We sketch the ideas of Harisch-Chandra by working out the character theory of the regular representation for compact groups. We first consider T reg (x) and find, within L 2 (G) the eigenvalues of T reg (x), i.e. those matrix elements which fulfills the equation T reg (x)f (g) = f (g x) = λ f (g) 105

110 3 Continuos groups and symmetry breaking interactions (λ being the sought for eigenvalue) and accordingly, in the base provided by the eigenfunctions, appears on the diagonal of the infinite matrix required to represent the regular representation. As T reg is unitary we have λ = 1, i.e. λ lies on the unit circle in the plane. It is plausible that λ can be written as λ(x) = e i Θ n(x) with Θ(x) being a real function of the parameters used to describe the element x (Θ(e) = 0) and the index n describing the eigenfunction used to build the diagonal matrix element T reg (x) nn. In virtue of the corollary to the Peter-Weyl theorem we have the important relation that any function defined on the group G can be expressed as linear combination of the characters e i Θn(x) building the diagonal matrix elements (generalized Fourier-theorem): with f (x) = a n e i Θ n(x) n a n = x e n (x) f (x) We are now ready to compute the character of the regular representation of a group with average taken as a distribution over a suitable test function f (x): From the relation χ Treg (f ) = x e n (x ), T reg (x)e n (x ) f (x) n = x e i Θ n(x) f (x) n = x e i Θn(x) f (x) n = a n = f (0) n χ Treg (f ) = x χ Treg (x) f (x) = f (0) 106

111 3 Continuos groups and symmetry breaking interactions we desume the formal identity χ Treg (x) = δ(x) where δ(x) is the Dirac distribution at the identity e. This result might be compared with the analogous one for finite groups, namely χ reg (e) = G χ reg (x) = 0 if x e. Harisch-Chandra proved rigorously a set of theorems involving character distributions. On the base of these theorems, a generalization of our exact theorems involving characters to include distribution-like character functions might lead to useful results, provided the handling of the distribution is properly performed. One of such tearooms is e.g. the generalization of the decomposition formula n α = g χ α (g)χ(g) to compute the irreducible components of the regular representation. If χ(g) = δ(g) is inserted, the averaging is perfectly meaningful, provided χ α (g) is a sufficiently regular function over G. The result is that n α = l α i.e. the generalization of the formula for finite groups to continuos groups with average, i.e. that every finite irreducible representation of G appears in the decomposition of the regular representation l α -times (see of course Peter-Weyl theorem). 3.6 Appendix III: The Harisch-Chandra character for the non-compact group. Our starting point is the translation group L defined within the interval [ L 2, L 2 ] by imposing periodic boundary conditions (Born-von Karman boundary conditions). 107

112 3 Continuos groups and symmetry breaking interactions This group is isomorphic to the circle group and has an average, 1 L L 2 L 2 f (x)d x so that Peter-Weyl theorem and any other theorem for irreducible representation are applicable. For instance, the orthogonality theorem for characters reads 1 L L 2 L 2 e i 2πκ x e i 2πκ x d x = δ κκ with κ = 1 L n labeling the character of the irreducible, one-dimensional representations. We expect to recover the translation group by letting L, whereby taking care to using only mathematical relations which, in the limit L, remain mathematically meaningful. One important step is to eliminate L from the orthogonality relations, so that it can be sent to infinity. We start from 1 L L 2 L 2 e i 2πκ x e i 2πκ x d x = δ κκ and multiply both sides with a suitable test-function f (κ κ ) and sum over κ : 1 L κ L 2 L 2 e i 2πκ x e i 2πκ x d x f (κ κ ) = δ κκ f (κ κ ) κ The right hand side amounts to f (0), the sum in the left hand side is replaced, by an integral over κ, whereby L is taken to infinity, so that the κ values form a very dense set with density of states L: dκ e i 2πκ x e i 2πκ x d x f (κ κ) = f (0) 108

113 3 Continuos groups and symmetry breaking interactions from which we read out the orthogonality relation for the characters e i 2πκ x of the element x in the irreducible representation of labeled by κ: e i 2πκ x e i 2πκ x d x = δ(κ κ ) δ(...) being the Dirac Delta function. The character is, by construction, no longer a square integrable function but a distribution. Using this relation we can define a generalization of Peter-Weyl theorem: f (x) = dκe i2πκ x f (κ) f (κ) = d xe i2πκx f (x) which are the usual relations defining the mathematical operation of a Fouriertransform. Formally, the scalar product becomes an average over : x f (x). = f (x)d x whereby distributions are allowed to appear as integrand f (x). One immediate application of these relations is the trace of the regular representation T reg (t)f (x). = f (x t) The eigenvalues of the operator T reg (t) are given by e i 2πκ t and the sum over κ, which must be rewritten as an integral, amounts to dκe i 2πκ t = δ(t) 109

114 3 Continuos groups and symmetry breaking interactions For the decomposition of the regular representation into irreducible component this results leads to n κ = d te i2πκt δ(t) = 1 meaning that the irreducible, one-dimensional representations of the group appears just once in the decomposition of the regular representation. We continue along this line and compute the symmetry adapted functions out of any square integrable function f (x) by the projector method: P κ f (x) = d te i2πκt f (x t) = e i2πκx d te i2πκ(x t) f (x t) = e i2πκx f (k) ( f indicating the Fourier transform). The picture emerging from this generalization is one where, if distributions are allowed, the Schrödinger representation is indeed reducible, but the symmetry adapted wave functions are themselves distributions, so that this formal result is in line with the Von Neumann-Stone theorem about the irreducibility of the Schrödinger representation, as one cannot find invariant subspaces built by square integrable functions. 110

115 4 Space groups and their representations 4.1 Elements of crystallography A crystal consists of an infinite number of unit cells that repeat periodically. The unit cell is described by three basic vectors ( a 1, a 2, a 3 ) and the entire lattice can be obtained by linear combinations of the basis vectors: t n = n 1 a 1 + n 2 a 2 + n 3 a 3 (n 1, n 2, n 3 ) being some positive or negative integers or 0 and t n a primitive translation. This means that any unit cell can be brought to coincide with another by a translation t n : the set of translations { t n } builds a symmetry group of the crystal. Theorem: a translationally invariant (3d,2d or 1d) lattice can only have 1, 2, 3, 4or6 fold rotational axes. This restricts possible symmetry operations of a lattice to 32 point groups in 3d, 10 point groups in 2d and 2 point groups in 1d (other than translational symmetries). Proof: for details, see introductory courses on Solid State Physics. Our proof start from the fact that if (R t n ) is a member of the space group and t n is a primitive translation, then R t n must also be a primitive translation. This follows because if (R t n ) is a member of the group, so is (R t n )(E t n )(R t n ) 1 = (E R t n ) which is a pure translation and therefore must be a primitive one. Let us find out which rotational symmetries are compatible with the translational symmetry. Let A (see the Figure) be a point of a crystal lattice, and let us assume that it has a n-fold rotational axis passing though it. In B there is another lattice point, and A and B can be joined by a primitive translation vector. Let, for simplicity, the rotational 111

116 4 Space groups and their representations Figure 4.1: Construction of rotations compatible with translations. axis be perpendicular to the plane formed by A and B. We perform now a rotation by ϕ = 2π/n about A. This bring B into B. Because B is equivalent to A there is also a n fold axis passing though B, and a rotation by ϕ brings A into A. As the rotation is a symmetry of the lattice, A and B are two lattice point, and their distance must be pa, p being some integer and a the lattice constant. From the figure we determine the following trigonometric equation: i.e. a + 2a sin(ϕ π ) = a 2a cos ϕ = pa 2 cos ϕ = 1 p 2 As cos ϕ 1, we can only allow p to be 3, 2, 1, 0. These values lead to the possible n values being 2, 3, 4, 6. A crystal can only have two-three-four-or six fold rotational axes. In particular, five-fold symmetry is strictly speaking prohibited. However, despite this rigorous selection rule, crystal with five-fold symmetry have been found: the so called quasi-crystals. Obviously, they can not have long range translational order, so that one still have problems figuring out what is the actual atomic position occupied in quasi-crystals. We illustrate this theorem in a 2d lattice. The symmetry elements leaving a 2d lattice invariant contain 1, 2, 3, 4or6 fold rotation axes perpendicular to the lattice and vertical reflection planes. This allows to build the following 10 point groups: C 1, C 1v, C 2, C 2v, C 3, C 3v, C 4, C 4v, C 6, C 6v 112

117 4 Space groups and their representations (the C xv -groups are often denoted by D x ). These restrictions to 32 (10,respectively 2) point groups produces following theorem: Theorem: Imposing the 32 (10,2) point groups onto a lattice with low symmetry produces 7 (4,1) crystal systems and 14 (5,1) types of unit cell in E 3 (E 2, E 1 ), the so-called 14 (5,1) Bravais Lattices. They sustain 32 (10,2) crystal classes. Proof: Landau, Lifshitz, Band V Example in 2d: The point group C 1 and C 2 are compatible with the oblique lattice. There is only one type of oblique lattice, defining one crystal system and only one type of oblique Bravais lattice in 2d. The point groups transforming the lattice into itself are C 1 and C 2 : accordingly there are two crystal classes. Notice that the Bravais lattice of a crystal does not include all points of the crystal lattice: atoms or group of atoms might reside in the unit cell, displaced with respect to the Bravais lattice points by some fractional translation. A crystal can therefore be viewed as assembly of identical, inter penetrating Bravais lattices. This means that, in general, there can more space groups associated with one crystal class. In the particular case of the the oblique lattice we have one space group within the C 1 class (called the Wallpaper group p1) and one space group associated with the C 2 crystal class (p2). Starting from an oblique lattice, we impose the point group symmetry C 1v. This Figure 4.2 leads to the equalities a = a b a = 2 sin ϕ b a b {}}{ = n 1 a + n 2 b! 113

118 4 Space groups and their representations (n 1, n 2 ) being. These equations have two solutions: ϕ = 0 n 2 = 1, leading to a rectangular lattice and sin ϕ b = 1 2 a leading to a centered rectangular lattice. These two lattices build one crystal system and provide the two Bravais lattices for the crystal classes C 1v and C 2v. A list of the four 2d crystal systems, their crystal classes and the corresponding space groups is given in the following pages. In summary, for the classification of crystals (3d) therefore one speaks of 7 crystal systems, each of which support one or more of the 14 Bravais lattices and one or more of the 32 point groups, which define 32 crystal classes corresponding to some point group Each crystal class sustains a set of the 230 space groups The following pages summarize the organizational structure of crystallography, including some important examples of Bravais lattices and corresponding space groups An example of space groups Figure

119 4 Space groups and their representations Let us list the symmetry elements of this crystal: Primitive translations: with t n = n 1 a 1 + n 2 a 2. {(E t n ) Two-fold rotation about the origin: (C 2 0) (C 2 t n ) x x1 x = x 2 A reflection at a plane vertical to the crystal followed by a translation along a 1 1 by 1 2 a 2 1 : (σ v : x 1 x = x x (σ v + t n. 0 A reflection to a plane vertical to the crystal followed by a translation along a 1 by 1 2 a 1. (σ v : x x1 x = x 2 1 (σ v 2 + t n. 0 This (2d)-crystal belongs to the crystal class C 2v Some of the operations of the point group C 2v appears in connection with the fractional translation Here are some other elements of crystallography:. 115

120 4 Space groups and their representations Def. The product of two elements (R a) and (S b) reads: (R a)(s b) = (RS R b + a) (R a) 1 = (R 1 R 1 a) Def. Space groups containing only operations of the type (R t n ) ( t n being a primitive translation - are called symmorphic space groups. In 3d there are 70 symmorphic space groups. Def. If some operations of symmetry contain fractional translations (glide planes and/or screw axes), the space group is called non-symmorphic. 4.2 The Translation group and its irreducible representations We consider a crystal terminated at the planes r a 1 = N 1, r a 2 = N 2, r a 3 = N 3 with N 1, N 2, N 3 being large numbers. We then impose Born von Karman boundary conditions (E t (N1,00)) = (E t (0,N2,0)) = (E t (0,0,N3 )) = (E 0) The collection of elements t n with such boundary conditions constitutes a finite, Abelian group T with as many elements as the number of unit cells in the crystal, i.e. g = N 1 N 2 N 3. Def. The reciprocal lattice of a crystal consists of integer linear combinations of the basis vectors b j, defined as those vectors which fulfill the equation a i b j = 2πδ i j i, j = 1, 2, 3. In 3d we have b 1 = 2π a 2 a 3 a 1 a 2 a 3 b 2 = 2π a 3 a 1 a 1 a 2 a 3 116

121 4 Space groups and their representations b 3 = 2π a 1 a 2 a 1 a 2 a 3 Def. Within the reciprocal lattice, one can define the 1st Brillouin zone, containting N 1 N 2 N 3 k-vectors. Example 1: the fcc-cubic lattice. For this lattice we have 113 Figure 4.4 a 1 = a (1, 1, 0) 2 a 2 = a (0, 1, 1) 2 a 3 = a (1, 0, 1) 2 b 1 = 2π a (1, 1, 1) b 2 = 2π a ( 1, 1, 1) 117

122 4 Space groups and their representations b 3 = 2π a (1, 1, 1) The vectors in the first Brillouin zone with some characteristic symmetry (see later) are also indicated: Example 2: the 2d-lattice. Γ k = ( π (0, 0, 0) a L k = ( π (1, 1, 1) a X k = ( π (0, 0, 2) a Γ L Λ k = ( π (δ ]0, 1[, δ ]0, 1[, δ ]0, 1[) a Γ X k = ( π (0, 0, δ ]0, 2[) a Figure 4.5 a 1 = a(1, 0) a 2 = a(0, 1) b 1 = 2π a b 2 = 2π a (1, 0) (0, 1) 118

123 4 Space groups and their representations Example 3: Body centered cubic lattice. For this lattice, b 1 = 2π a b 2 = 2π a b 3 = 2π a (0, 1, 1) (1, 0, 1) (1, 1, 0) The position vectors of the high symmetry points (see later) marked are as follows: k = (0, 0, 0) Γ k = π (2, 0, 0) H a k = π (1, 1, 0) N a k = π (1, 1, 1) P a Theorem: Each element of T is a class on its own. Each vector k in the first Brillouin zone labels an irreducible representation D k of T with matrix element Proof: D k (E t n ) = e i k t n The group is abelian, so we have N 1 N 2 N 3 irr. representations, corresponding to the number of elements. e i k t n is a representation, as e i k t n }{{} D k (E t n ) The representation is irreducible as e i k t m }{{} D k (E t m ) = e i k( t n + t m ) }{{} D k (E t n + t m ) 1 e i k t n 2 = 1 N 1 N 2 N 3 These are all irreducible representations of T. Symmetry adapted were functions 119

124 4 Space groups and their representations T... t n... Basis functions D k... e i k t n... ψ n ( k, r) = e i k r u n ( k, r), n = 0, 1, We use the projector technique: P k ψ( r) = e +i k tn ψ( r t n ) n e i k r e i k( r t n) ψ( r t n ) = n e i k r u( k, r) with u( k, r t n ) = u( k, r) to find that symmetry adapted functions are Bloch-functions (Bloch-theorem) We summarize the results for T within a character table. Figure 4.6 As the Hamiltonian H of a crystal commute with the operator O (E tn ), its eigenfunctions must be Bloch functions. In general, there is an infinite set of energy eigenfunctions and eigenvalues corresponding to each irreducible representation of the group of the Schroedinger equation and hence to each allowed k vector. Eigen- 120

125 4 Space groups and their representations functions and eigenvalues corresponding to the vector k can be indicated by φ n ( k, r) and E n ( k), where the index n labels the members of the set of energies corresponding to the wave vector k. The set of energy eigenvalues E n ( k) corresponding to a particular n are said to form the n th-energy band, and the set of energy bands is said to constitute the band structure. To visualize the band structure, it is convenient to consider one at a time the axes of the Brillouin zone that join the high symmetry points and for every allowed k vector on each axis to plot the energy levels E n ( k). A typical example of such a plot is shown in the figure, which gives the energy bands along and Λ in fcc Cu. Figure 4.7: The occurrence of degenerate values and the apparent opening of gaps within a band of a given index is a consequence of the existence, along some special symmetry directions, of a more complex symmetry group than the translational group and will be explained in the next chapter. 121

126 4 Space groups and their representations 4.3 Irreducible representations of the group of k Consider a symmetry operation (R a) of a space group G. Then 0 R a ϕ n ( k, r) = e i k(r 1 r R 1 a) u n (R 1 r R 1 a) = e ir k r e i kr 1 a u n (R 1 ( r a)) = e ir k r u n ( r) This means that the existence of symmetry elements other than translations transforms a Bloch functions into another, which by the requirements of group theory, must belong to the same energy value. Thus, the existence of symmetry elements other than translations mixes Bloch functions belonging to different k and n into new invariant subspaces that, in general, carry a higher dimensional rep. of the space group. In simple words: Breaking the symmetry lifts degeneracy. Expanding the symmetry introduces degeneracy. The strategy used to find the irr. rep. of a space group involves following steps. a: The small point group of k Def. Given a space group G, the set of elements generating its crystal class is called the point group G 0 of G. Remark: some of the elements of G 0 might appear, considered as symmetry operations of G, with fractional translations. Def. Consider a k-vector in the 1st BZ and those elements of G 0 which transform k into itself or into an equivalent one, i.e. R k k = k + h h being a reciprocal lattice vector. The set { R k }. = G 0 ( k) 122

127 4 Space groups and their representations forms a group G 0 ( k) which is a subgroup of G 0 and is called the small point group of k. Remark: at general k-points G 0 ( k) = {E}, but at so-called high-symmetry locations of the 1 BZ G 0 ( k) is larger than {E}. Examples of high symmetry locations are the Γ, L and X-points of the 1 BZ of the fcc lattice and the, Λ directions. Example: G 0 (Γ ) = G 0 G 0 ( ) = C 4v G 0 (Λ) = C 3v (Lit: L.P. Bouckaert, R. Smoluchowsky, E. Wigner, Phys. Rev. 50, 58 (1936)) b: The group of k Def. The collection of elements of the space group {R k t n + f R k } is called the little group of k or simply the group of k, G k. Remark: At general k, G k T. If k is a high symmetry location, the group G k is given, generally, by (E 0), (R 1 f 1 ),...(R n f n ), (E t 1 ), (R 1 f 1 + t 1 ),...(R n f n + t 1 ),..., (E t i ), (R 1 f 1 + t i ),...(R n f n + t i ),... Theorem: Let k be inside the 1 BZ, i.e. R kk = k. Then from any irreducible representation D p (R k ) of G 0( k) one can obtain an irreducible representation D p, k of the group G k by associating to every element (R k f R + t n ) of G k the matrix D p, k (R k f R + t n ). = e i k( f R + t n ) D p (R k ) This irreducible representations fulfill Bloch theorem, i.e. they are physically 123

128 4 Space groups and their representations G k (E 0), (R 1 f 1 ),...(R n f n )... (E t i ), (R 1 f 1 + t i ),...(R n f n + t i ) G k... (R i f i )... (R i f i + t n ) D p... χ p (R i ) e i k( f i )... χ p (R i ) e i k( f i + t n ) relevant. This theorem means that energy levels in solids at k-points inside the BZ are classified according to the irreducible representations of a point group G 0 ( k). Examples: In the f cc-lattice, along the direction we expect the energy bands to be labeled by the 5 irreducible representations of C 4v, while along the Λ-direction we expect the labeling according to the 3 irreducible representations of C 3v. At the Γ -point the energy levels aquire the quantum numbers corresponding to the irreducible representations of O h. Remark: there are many irr. representations of G k which are not physically relevant as translations are not represented by e i k t n. One of these is certainly the identical respresentation. The structure of the character table of G k is given in the table. Proof: We show first that the matrices are indeed representations: 124

129 4 Space groups and their representations (R i f i + t n ) }{{} a }{{ i } e i k a i D p (R i ) (R j f j + t m ) }{{} a j }{{} e i k a j D p (R j ) The left hand side can be written as The right hand side can be written as = (R i R j R i [ f j + t m ] + f i + t n ) }{{} D p (R i R j )e i k( a i + a j ) D p (R i R j )e i R 1 i k a j + k a i D p (R i R j ) e i k (R i a j + a i ) If k is inside the Brillouin zone, then R 1 i k = k and the two sides become identical, proving the homomorphism. Remark: The phase factor on the right-hand side contains k R i a j, which is necessarily equal to k a j ONLY for k inside the BZ. At the surface of the BZ, however, we might have R 1 i k = k + h and the phase factor becomes k t n + h f j. The presence of the extra term h f j violates, in general, the condition of homomorphism, unless f j 0. Thus, the theorem can be extended at k at the surface of the BZ only for symmorphic crystals. For non-symmorphic crystals and for k points at the surface of the BZ the finding of the irreducible representations becomes a more complex exercise (see later). The representations D p, k are irreducible: in fact, if g 0 ( k) is the number of elements in G 0 ( k), we have 1 χ p (R k g 0 ( k)n 1 N 2 N f + t n ) 2 = 3 R k f + t n 1 g 0 R k χ p (R k ) 2 = 1 Remark: The number of physically relevant irr. representations of G k is given by the number of irr. representations of the small group g 0 ( k). Example: Point Γ in O 5. This is the point at the origin of the first Brillouin zone. The small point group of k is the entire point group O h. Since we h are inside the Brillouin zone, the characters of the irreducible representations at Γ are simply given by the characters of the point group O h, without any modification. 125

130 4 Space groups and their representations G E C 2x C 1 4x IC 2x IC 2x y C 4x IC 2y IC 2y z Line Λ in O 5. k = π h a (λ, λ, λ), λ < 1. This is a point on the line joining the center of the Brillouin zone to the midpoint of an hexagonal face. The small point group of k is made up by the operations whose rotational parts interchange (x, y, z) among themselves, because these are the only operations which leave k invariant. The small point group of k is thus C 3v. Since no fractional translation is associated with the elements of C 3v, the characters of the irreducible representations at Λ are the characters of the group C 3v without any modification. Line in O 5 ( k = 2π h a (δ, 0, 0), with δ < 1). This is a point on the line joining the center of the Brillouin zone to the midpoint of the square face. The small point group of k is given by the operations whose rotational part leave x unchanged. The small point group of k is thus C 4v. Example: the space group 0 7 h (diamond structure) The diamond lattice is invariant with respect to primitive translations (E E n 2 ) with t n = i n i a i and a 1 = a 2 (0, 1, 1); a 2 = a 2 (1, 0, 1); a 3 = a (1, 1, 0) 2 where a is the length of the cube edge. The 1st BZ it the one of the fcc-lattice. There are two equal atoms in the unit cell in the positions d 1 = (0, 0, 0); d 2 = a 4 (1, 1, 1). The diamond lattice can be thought as consisting of two interpenetrating fcc sublattices displaced with respect to each other by the vector d 2 := f. The 126

131 4 Space groups and their representations Figure 4.8 point group G 0 of the diamond lattice is the cubic group 0 h. Notice that some of the symmetry operations of 0 h appear in the space group associated with the fractional translation f = a 4 (1, 1, 1) = 1 4 ( a 1 + a 2 + a 3 ) By choosing a lattice point as the origin of a cubic coordinate system we obtain the symmetry operations given in the following table. The line ( k = 2π a (δ, 0, 0), δ < 1); the small point group of is C 4v. The character table of the group of can be derived from the character table of C 4v by multiplying the character of the point group with the suitable phase factor e ( i k f ) = e i( π 2 δ) For the space group 0 5 (fcc-lattice), there are no fractional translations and the h character table of G, restricted to the operations of G 0 ( ), is identical to the character table of C 4v. 127

132 4 Space groups and their representations Type Operation Coordinate Type Operation Coordinate C 1 (E 0) x yz I (I f ) x + a 4 ȳ + 4 z a + a 4 C 2 4 C 2z x ȳz IC 2 4 (IC 2z f ) x + a 4 y + 4 z a + a 4 C 2x x ȳ z (IC 2x f x + a 4 y + a 4 z + a 4 C 2y x y z (IC 2y f x + a 4 ȳ + a 4 z + a 4 C 4 (C 1 4z f ) ȳ + a 4 x + a 4 z + a 4 IC 4 IC 1 4z y x z (C 4z f ) y + a 4 x + a 4 z + a 4 IC 4z ȳ x z (C 1 4x f ) x + 4 z a + a 4 y + a 4 IC 1 4x xz ȳ (C 4x f ) x + a 4 z + a 4 ȳ + a 4 IC 4x x z y (C 1 4y f ) z + a 4 y + a 4 x + a 4 IC 1 4 y z ȳ x (C 4 y f ) z + a 4 y + a 4 x + a 4 IC 4y z ȳ x C 2 (C 2x y f ) y + a 4 x + 4 z a + a 4 IC 2 IC 2x y ȳ xz (C 2xz f ) z + a 4 ȳ + a 4 x + a 4 IC 2xz z y x (C 2yz f ) x + a 4 z + a 4 y + a 4 IC 2 yz x z ȳ (C 2x ȳ f ) ȳ + a 4 x + 4 z a + a 4 IC 2x ȳ y xz (C 2 xz f ) z + a 4 ȳ + a 4 x + a 4 IC 2 xz z y x (C 2y z f ) x + 4 z a + a 4 ȳ + a 4 IC 2 y z xz y C 3 C 1 3x yz zx y IC 3 (IC 1 3x yz f ) z + a 4 x + a 4 ȳ + a 4 C 3x yz yzx (IC 3x yz f ) ȳ + a 4 z + a 4 x + a 4 C 1 3x ȳz z x ȳ (IC 1 3x ȳz f ) z + a 4 x + a 4 y + a 4 C 3x ȳz ȳ zx (IC 3x ȳz f ) y + a 4 z + a 4 x + a 4 C 1 3x ȳ z z x y (IC 1 3x ȳ z f ) z + a 4 x + a 4 ȳ + a 4 C 3x ȳ z ȳz x (IC 3x ȳ z f ) y + a 4 z + a 4 x + a 4 C 1 3x y z zx ȳ (IC 3x y z f ) z + a 4 x + a 4 y + a 4 C 3x y z y z x (IC 3x y z f ) ȳ + a 4 z + a 4 x + a 4 G (O 7 h ) (E 0) (C 2x 0) (C 1 4x f ) (IC 2x f ) (IC 2x y 0) (C 4x f ) (IC 2y f ) (IC 2y z 0) e i( π 2 δ) e i( π 2 δ) e i( π 2 δ) e i( π 2 δ) e i( π 2 δ) e i( π 2 δ) e i( π 2 δ) e i( π 2 δ)

133 4 Space groups and their representations c: Space groups Def. If (R 0) is a member of G 0 but not of G 0 ( k), then R k is not equivalent to k. the effect of such operations on k is to send k into a number of inequivalent vectors k 1 = k 1, k 2,..., which form the star of k. If g 0 is the number of elements of G 0, g 0 ( k) the number of elements of G 0 ( k), then the number of vectors S k in the star of k is given by g 0 ( k) S k = g 0 Example: For k = 0 (Γ -point), we have only one member of the star of k, and this is k = 0. For k along in a fcc-lattice, 0 h = 48 and C 4v = 8, so that we have 6-vectors in the BZ which are members of the star of k. Def. The reduced volume within the 1st BZ consists of all k-points that do not belong to the same star. The volume of the reduced BZ is the volume of the BZ divided by g 0, the order of G 0. Example: The Figure shows the Brillouin zone of a plane square lattice of lattice constant a. The special points and lines of symmetry and the corresponding small point group are indicated in the table. Figure 4.9 The next figure gives the stars of the various k vectors for a plane square lattice. Every dashed vector is related to one solid vector by a reciprocal lattice vector and is therefore not counted separately. The number of distinct vectors in the star is 129

134 4 Space groups and their representations k Symbol G 0 ( k) (k x, k y ) general C 1 k x, k x ) Σ C 1,h 1 2, k x) Z C 1,h k x, 0 C 1,h 1 2, 0 X C 2,v 1 2, 1 2 M C 4,v (0, 0) Γ C 4,v Table 4.1: The special point and the lines of symmetry in the Brillouin zone of the planar square lattice, in units of 2π a. Figure 4.10 given by s. Notice that the full point group G 0 is C 4v and that the space group is symmorphic. Theorem: Let {ϕ 1,..., ϕlp k k } carry the lp-dimensioned irr. representation Dp k of G k. The S k lp basis functions obtained by transforming the l p-basis functions with the star operations carry a S k l p -dimensional irreducible representation of G. Proof: We limit ourselves to show that k red.vol. (S k l p ) 2 = k red.vol. (S k ) 2 (l p ) 2 p }{{} g 0 k) 130

135 = k red.vol. 4 Space groups and their representations S k (S k g 0 ( k) ) = g }{{} 0 g 0 k 1BZ 1 = g 0 N 1 N 2 N 3 This theorem concludes the discussion of the representation theory for space groups starting from the irreducible representations of the group of k. In summary:the energy spectrum obtained within the reduced BZ is the essential part of the energy spectrum, the remaining part of the BZ increases degeneracy. Fazit: Construct the energy spectrum of the group of k! Example: The one dimensional lattice We consider a chain of N-atoms with lattice constant a. The fundamental translation group has N elements n a, with n = 0,..., N 1. The group has N irreducible representations. In the representation with label p the n-th element has the character e i 2π p n N, with p = 0, 1,..., N 1. This establishes a reciprocal lattice vector of length 2π 2π a hosting the allowed k-vectors k = N a p. The first Brillouin zone of the 1-dimensional lattice is the interval [ π a, π ]. The point group a G 0 of the system is C i, while the small point group of any vector inside the interval is just E. This means that at general k inside the interval k is the only quantum number labeling the energy levels and the symmetry adapted wave functions are of the form e ik x u(x), where u(x) is a periodic function. As possible basis functions we might decide to choose 1 eik x e ig x N a with G = n 2π a, n being a reciprocal lattice vector. For point inside the interval, the inversion operation transforms k into k, which together with k forms the star of k in the one dimensional lattice. As the energy levels belonging to the star of k are degenerate, we obtain that energy levels in the one dimensional lattice are at least double degenerate. The representation theory of the k point at the end of the interval (including k = 0) must be modified because the small group of k contains both operations E and I. For this reason, the energy levels at the edge of the one dimensional Brillouin zone are classified according to the two one-dimensional irreducible representations of 131

136 4 Space groups and their representations C i, namely the one carrying symmetric and the one carrying anti-symmetric wave functions. We are now able to calculate the empty lattice band structure (also free electron band structure), which sets the crystal potential to zero and only consider the kinetic energy of the electrons. Depending of the choice of G we can generate a set of empty lattice energy bands with energy E = ħh2 (k + G)2 2m with G = 0, ± 2π a,... The first band is the one characterized by G = 0, see the figure. Figure

137 4 Space groups and their representations The G = 0 band starts at Γ and ends at the point π a energy level with wave functions 2 N a cos π a x and 2 N a sin π x. Taking a 2π a with a double degenerate as the next G-vector we produce a further band taking off from the Brillouin zone edge, see the figure. This band ends at k = 0 with a double degenerate band with wave functions 2 N a cos 2π a x and 2 N a sin 2π a x. The next band is generated by G = 2π a, and so on. The general expression for the wave functions at k = 0 is 2 N a cos n 2π a x and 2 N a sin n 2π a n = 0, 1, 2,... From this expression it is clear that the deepest lying level (k = G = 0) is not degenerate, while the degeneracy of the higher lying levels it two. The general expression for the wave function at k = π a is x 2 N a cos[π a + n2π a ]x and 2 N a sin[π a + n2π a ]x n = 0, 1,... The band structure calculated for the empty lattice using symmetry adapted wave function is exact. However, the same functions can be used, within in a perturbational approach, to find the first order correction caused by a small crystal potential V (x). The most remarkable feature of switching a small crystal potential is the lifting of the accidental double degeneracy at the center and at the edge of the Brillouin zone. In fact, from general symmetry considerations, we expect that the double degenerate level splits into two single degenerate energy levels, thus opening a gap between the various bands at k = 0 and at k = π a. The energy of the levels at these two singular points of the BZ can be calculated, in first order 133

138 perturbation theory, to be 4 Space groups and their representations E ± (k = 0) = ± ħh2 2m (2π a )2 n N a 1 4N a N a 0 N a 0 d x V (x) d x V (x) cos(2n 2π a )x (the - sign-level is suppressed for n = 0) and E ± (k = π a ) = ± ħh2 2m (2π a )2 ( 2n + 1 ) N a d x V (x) 2 4N a 0 1 4N a N a 0 d x V (x) cos[(2n + 1) 2π a ]x (n = 0, 1, 2,...). Thus, the gap is twice the Fourier-transform of the crystal potential perturbing the empty lattice band structure. Band structure of a 2d square lattice and symmetry analysis of band structures The space group is symmorphic and the essential aspects of symmetry have been discussed previously. The starting point for a quantitative band structure is the empty lattice model, which keeps record of the translational symmetry of the lattice but sets the strength of the crystal potential to zero. A suitable set of basis functions, symmetry adopted to translational symmetry, are plane waves φ k G = 1 N a 2 ei( k+ G) r with G being some reciprocal lattice vectors with coordinates 2π a (m, n), m, n. As an example, we construct the empty lattice band structure along the -direction. The small group of is {E, σ x } = C 1,h and thus has two irr. representations, which we call ( 1, 2 ) The energy bands generated by the numbers (m, n) have energy E(m, n) = ħh2 2m (2π a )2 [(κ + m) 2 + n 2 ] 134

139 4 Space groups and their representations G E m x... (E t n ) (m x t n ) Basis functions e i k t n e i k t n (0, 0), ( 1, 0), (1, 0) e i k t n e i k t n x (0, 1),(0, 1) and the corresponding plane waves are e i 2π a [(κ+m) x+n y] The lowest lying band is the one with m = n = 0 and has 1 symmetry. The next band is characterized by m = 1, n = 0 and has, again 1 symmetry. The degeneracy of these bands is 1. These two bands are followed by a double degenerate band corresponding to the indices (m = 0, n = 1) and (m = 0, n = 1). This band has an accidental degeneracy involving a wave function with 1 symmetry and a wave function with 2 symmetry. The m = 1, n = 0 band has 1 symmetry and has higher energy than the (0, 1)-band, except at the Γ -point where the bands meet. The next reciprocal lattice vectors are the four (1, 1) vectors. The ( 1, 1) and ( 1 1) bands are degenerate along, their energy being equal to the energy of the 0, 1, 0, 1 band at the X -point. The following figure plot E ħh 2 2m ( 2π a )2 versus κ. We now analyze the accidentally degenerate band carrying the pane waves (m = 0, n = 1) and (m = 0, n = 1). The subspace determined by (0, 1) and (0, 1) carries the representation x, which consists of 2 2 matrices: (01, E01) (01, E = (0 1E01) (0 1E0 1) 0 1 (01, mx 01) (01, m x = (0 1m x 01) (0 1m x 0 1) 1 0 where the matrix elements are between the corresponding plane waves e i 2π a (κ x+ y) 135

140 4 Space groups and their representations Figure 4.12 e i 2π a (κ x y) The resulting representation x is twice degenerate and reduces to x = 1 2 The symmetry adapted functions, computed by the projector method, reads P 1 e i 2π a (κ x+ y) e i 2π a (κ x+ y) + e i 2π a (κ x y) e i 2π a (κ x) cos 2π a y P 1 e i 2π a (κ x+ y) e i 2π a (κ x+ y) e i 2π a (κ x y e i 2π a (κ x) sin 2π a y) 136

141 4 Space groups and their representations Notice that the energy band carrying x is doubly degenerate. This is an example of accidental degeneracy, which is bound to be removed when non-symmetry breaking interactions are switched on. To first order perturbation theory, the symmetry adapted wave functions are also eigenfunctions of ħh2 2m + V (x, y) and the energy corrections with respect to the empty lattice band structure will amount to e i 2πκ x a cos 2π a y, V (x, y)ei 2πκ x a cos 2π a y = E 1 e i 2πκ x a sin 2π a y, V (x, y)ei 2πκ x a sin 2π a y = E 2 (see the Figure). Non-diagonal matrix elements between function with different symmetry are vanishing in virtue of the Wigner-Eckart-Koster theorem (see next chapter). Further up in the empty lattice band structure we encounter a further typical situation for band structure computations: the crossing or the meeting of two bands carrying the same irreducible representation, in this case 1. Notice that band crossing is only possible in d 2, i.e. there is no band crossing for 1d-crystals. The question is now: what happens to bands of the same symmetry, crossing at some points of the BZ, when the crystal potential is switched on? At the crossing point, the representation contains more symmetry adapted wave functions carrying the same irreducible rep., i.e. having the same symmetry. As non-diagonal matrix elements of the H-operator between symmetry adapted wave functions belonging to the same irr. repr. are not necessarily vanishing, the eigenvalue problem left to be solved by explicit calculation is a non-trivial one, i.e. it involves computing explicitly the determinant of a non-diagonal matrix (2 nd order perturbation theory). The result will be the formation of an hybridization gap- an anti-crossingan avoided crossing - around which the wave function will be a linear combination of the original symmetry adapted were functions, see the scheme following. Notice the crossing from one symmetry adapted wave function to another type (with the same symmetry) when going through one single band close to the hybridization gap. We now proceed analyzing the Γ and X -points. Both have a larger symmetry group and a larger number of irreducible representations. The plane wave corre- 137

142 4 Space groups and their representations Figure 4.13 Irr.Rep. e C 4, C 3 4 C 2 2m 2σ 1, Γ , Γ , Γ , Γ , Γ sponding to the lowest lying energy level E = 0 is (0, 0) Γ. It is a constant and transforms according to the irreducible representation Γ 1. The next fours waves (1, 0) Γ, ( 1, 0) Γ, (0, 1) Γ and (0, 1) Γ are degenerate with energy 1. They carry some irreducible representations of C 4v, which can be found by decomposing the representation constructed over these four plane waves. We find Γ 1 Γ 3 Γ 5. Using the projector operator technique one can easily find the symmetrized linear combination corresponding to these components. For instance, the function with Γ 1 symmetry is clearly 1 4N a 2 [ei 2π a (x) + e i 2π a (y) + e i 2π a ( x) + e i 2π a ( y) ] The next four plane waves (1, 1) Γ, ( 1, 1) Γ, (1, 1) Γ and ( 1, 1) Γ belong to the four fold degenerate energy level E = 2 and transform according to Γ 1 Γ 4 Γ 5. The symmetry group at X is C 2v = [E, m x, m y, C 2 4 ] and has four irreducible representations which we denote X 1, X 2, X 3, X 4 for the square lattice. We obtain the following set of degenerate plane waves at X : 1. the set (0, 0) X and (0, 1) X generating the representations X 1 X 3 with E = the set (0, 1) X, (0, 1) X, ( 1, 1) X, ( 1, 1) X transforming according to X 1 X 2 X 3 138

143 4 Space groups and their representations Γ 1 Γ 2 Γ 3 Γ 4 Γ X 1 X 2 X 3 X 4 Table 4.2: Compatibility table for the square lattice along the -direction X 4 with E = and so on The connection between the band structure along with the end-points of the BZ can be directly read out in general terms from the compatibility relations given in the Table. 4.4 Non-symmorphic space groups. The strategy for finding the irr. rep. of the groups of k for k at the surface of the BZ in a non-symmorphic crystal is the one described by J.F. Cornwell, "Group theory and electronic energy bands in solids", Selected Topics in Solid State Physics, Vol. 10, edited by G.P. Wohlfarth, North-Holland, Amsterdam, This strategy can be generally applied, although it is really only useful for non-symmorphic space groups. In addition to classes of adjunct elements, there is a further useful way of organizing a group. Def. Let U be a subgroup of the order S of a group G of the order g and suppose that the elements of U are u 1,..., u s. If a is an element of G, then the set {au 1,..., au s } - in short au - is a left coset of U with respect to G (Ua is a right coset). Properties: 1. Right and left cosets do not necessarily contain the same elements (Prove this by e.g. discussing some cosets of C 3v ). 2. Each coset contains s distinct elements 3. If a is a member of U then au = Ua = U 4. If a is not a member of U then au is not a group 139

144 4 Space groups and their representations 5. Two cosets are either identical or have no elements in common 6. If b is a member of au, then bu is identical to au. This shows that any member of the coset may be taken as the coset representative 7. Each group G can be divided into a subgroup U and its cosets, i.e. G = a 1 U(= U) + a 2 U a i U g = s i. Example: Any space group G can be divided into cosets of the type (R 1 f 1 ) T, (R 2 f 2 ) T,..., (R g0 f g0 ) T Def. Consider a subgroup U and a fixed element εg. Then the set of elements U 1 forms a group (prove this). If U 1 = U, the group U is called an invariant (normal) subgroup or normal divisor. Remark: if U is an invariant subgroup, then U x = xu Example: T is an invariant subgroup of a space group. Def. Let U be an invariant subgroup and consider now the i-cosets a 1 U,..., a i U as single entities, the content of the cosets being disregarded for the moment. One can define a product between two such entities as the coset corresponding to the product of the representative elements, i.e. Ua i Ua j = Ua i a j (In order to this definition to be meaningful, one must prove that the result of the multiplication is independent to the choice of cosets representatives.) Def. Once this multiplication is defined, one can define a new, abstract group, the factor group G/U, consisting of elements a 1 U = U, a 2 U,..., a i U. Factor groups are typically smaller in size than the original groups and therefore move manageable. The idea is that one searches for the irr. representations of G k by working with those of the more manageable factor group. Example: We consider a one-dimensional chain with basis 140

145 4 Space groups and their representations Figure 4.14 G (E n) (I m )... (E n ) (E n + n ) (I m + n )... (I m ) (I n + m ) (E m + m )... Table 4.3: Multiplication table for G: G = E, I 1 4 a, E ± a, E ± 2a,..., I 1 a ± a,... 4 G 0 = E, I U = ±a, ±2a,... E ± 0 I 1 4 a ± 0 G/U = {EU, I 1 4 au} = { E ± a I 1 4, a ± a }.... G/U C i = {E, I} Examples: for fcc-lattices we have G /T C 4v, G Γ /T O h. Def. For any k-which might or might not be at the surface of a BZ - we G/U EU I 1 4 U EU EU I 1 4 U I 1 4 U I 1 4 U EU Table 4.4: Multiplication table for G/U 141

146 4 Space groups and their representations define the group T( K) consisting of all primitive translations t n such that e i k t n = 1 T( k) is an invariant subgroup of G k. It is therefore possible to define the factor group G k /T k, which consists of cosets of the type (R f )T( k) Theorem: Let D be an irreducible representation of G k /T ( k). If D has the property Figure 4.15: Example of factor group in a linear chain with basis. (Bloch) D[(E t n )T( k)] = e i k t n D[(E 0)T( k)] for every coset (E t n )T( k) formed from primitive transitions, then the set of matrices defined by D(R t) := D[(R t)t( k)] form an irreducible representation of G k which satisfies the Bloch condition and is thus physically relevant. Furthermore, all physically relevant irr. representations of G k can be constructed in this way starting for those of G k /T( t k ). Remark: Notice that not all irr. representations of G k /T k satisfy the Bloch condition. In particular, for k 0 the identical representation 1 is an important example of physically irrelevant representations. Remark: G k /T( k) is not necessarily isomorphic to a point group, so that the problem of finding the irr.rep. of G k is considerably simplified by this theorem, but 142

147 4 Space groups and their representations cannot always be reduced to a point group problem. Proof. First, we have to prove that the matrix defined above is indeed a representation of G( k). D((R t))d((r t )) = D((R t)t( k))d((r t )T( k)) = D((R t)t( k)(r t )T( k)) = D((R t)(r t )T( k)) = D((R t)(r t )) Next, we have to establish the irreducibility of the representations constructed in this way. We write χ(r t) 2 = t( k) G( k) G( k)/t( k) χ(r t)t( k)) 2 where t( k) is the order of the group T( k). As D((R t)t( k)) is an irr. representation, we have χ(r t)t( k)) 2 = g( k)/t( k) (4.1) G( k)/t( k) which immediately prove the irreducibility of the representation for G( k). The representation constructed in this way obeys the Bloch condition, i.e. D(E t n ) = D((E t n )T( k)) = e i k t n D((E 0)T( k)) = e i k t n D((E 0)) showing that the elements containing primitive translations are represented by e i k t n E as it should be in order to obtain Bloch functions as symmetry adapted wave functions. Example 1: the atomic chain with basis The character table of X = π a example) is given here. in a linear chain of atoms with basis (see previous X 1 and X 2 are not physically relevant, as e.g. (E 2n + 1) is represented by 1 and not by e i π a (2n+1) a = 1, as it should. 143

148 4 Space groups and their representations G π /T π E 0 E +a I 1 a a 4 a I 1 4 a + a X X X X Starting from e ±i π a x we construct the symmetry adapted basis functions to X 3 and X 4 by the projector method: P X3 e i π a x e i π a x e i π a (x+a) + e i π a (x+ a 4 ) e i π a ( x a 4 a) e i π a x + e i π 4 e i π a x and P X4 e i π a x e i π a x e i π 4 e i π a x Figure

149 4 Space groups and their representations Figure 4.17 Example 2: The X -point of O 7 h The small point group of k is D 4h and has 16 elements, it is given by C 4v, IC 4v. In fact, I k = 2π a ( 1, 0, 0) = k ( h 2 + h 3 ) where h 2,3 are reciprocal lattice vectors. As I is associated with fractional translations, the irr.rep of the G X cannot be derived from D 4h. Rather, we have to use the factor group G X /T X which contains 32 elements and is not a point group. The character table of this group is given in the next figure, taken from the book of Bassani and Pastori-Parravicini. The band structure of Si shows explicitely the nonsymmorphic degeneracy at X associated with the peculiarity of the factor group G X /T X. 145

150 5 Kronecker (direct) products Direct products of groups occur in several instances. If the Lagrangian or the Hamiltonian of a system contain different types of coordinates such as those referring to different particles, or spatial and spin coordinates of a particle, then each set of coordinates might have some symmetry group. The symmetry elements acting on different degrees of freedom spatial and spin, for instance commute and the total symmetry group is the direct (Kronecker) product of the groups. By means of the Kronecker product one can add degrees of freedom in a mathematically rigorous way. 5.1 The Kronecker product of matrices Consider two vector spaces L n and L m. The Kronecker or tensor product of the two spaces L p = L n L m consists of a set of all possible ordered pairs ψ n ψ m whose first component is a member of L n and whose second component is a member of L m and for which following rules hold true: 1. λψ n ψ m = ψ n λψ m = λψ n ψ m 2. ψ n (ψ m + ψ m ) = ψ n ψ m + ψ n ψ m 3. (ψ n + ψ n ) ψ m = ψ n ψ m + ψ n ψ m The direct-product space is a space of dimension p = n m and has the p basis vectors e 1 i 1, e 1 i 2,..., e 1 i m, e 2 i 1,...e n i m The scalar product in the spaces L n and L m produces a scalar product in L p according to (u 1 v 1, u 2 v 2 ) = (u 1, u 2 ) (v 1, v 2 ) 146

151 5 Kronecker (direct) products As any vector of L p can be expressed as u = i,j u i je i i j, the scalar product between any vector in L p can be calculated according to (u, v) = ( u i j e i i j, v lm e l i m ) = u i j v i j i,j l,m (because e i i j, e l i k ) = δ il δ jk ). The action of the operator A B on u v is defined as (A B)u v = Au Bv With this definition one can find the action of A B on each vector of L p as (A B)u = i,j u i jae i Bi j. The matrix representation of A B is the direct (or Kronecker) product of the two matrices. i,j Let [A ik ] be a matrix of the order n n and [B pq ] a matrix of order m m. The direct (tensor or Kronecker) product of these two matrices is a matrix C. = A B of the order (nm) 2, which can be written as A 11 B A 12 B... A 1n B A 12 B A 22 B... A 2n B A n1 B A n2 B... A nn B where the element A i j B stands for a matrix of order m m given by A i j B 11 A i j B A i j B 1m A i j B 21 A i j B A i j B 2m A i j B m1 A i j B m2... A i j B mm The Kronecker product of matrices has important properties 1. (A 1 B 1 )(A 2 B 2 ) = (A 1 A 2 ) (B 1 B 2 ) 147

152 5 Kronecker (direct) products 2. Furthermore, if F is the direct product of a number of matrices A, B,C,..., then t r F = (t ra) (t rb) The operation of the direct product of matrices is associative, so that A (B C) = (A B) C = A B C 4. The operation is distributive with respect to the matrix addition, A (B + C) = A B + A C 5. Finally (AB) k = A A A A... = A k B k (A B) 1 = A 1 B 1 (A B) = A B Consider two matrix representations T 1 and T 2 (reducible or irreducible) of a group G. Let us take the direct product of the corresponding matrices for the two representations and denote it by T(a) = T 1 (a) T 2 (a) Theorem: T(a) is also a representation of the group. Proof: T(a)T(b) = (T 1 (a) T 2 (a)(t 1 (b) T 2 (b) = (T 1 (a)t 1 (b)) (T 2 (a)t 2 (b)) = T 1 (ab) T 2 (ab) = T(ab) Thus, we see that the set of matrices constructed by taking the direct product of two representations also forms a representation. An important property of the direct 148

153 product representation is that 5 Kronecker (direct) products χ(a) = χ 1 (a) χ 2 (a) i.e. the characters of the direct product representations are the product of the characters of the individual representations. In general, the direct product representation is reducible, certainly if T 1 or T 2 are reducible. For example, for the reduction of the direct product of irreducible representations T i and T j we expect an expansion of the type T i T j = k n i, j k T k where n i, j are non-negative integers. A sum of this type is also known as the k Clebsch-Gordan series. Example: Product representations of SO(3) Consider the representations D n and D m of SO 3 (take n m for convenience). The task is to find the irreducible components of D n D m, i.e. to establish the Clebsch-Gordan series for D n D m. A practical way of reducing the direct product D n D m is an algorithm established by E. Cartan. First one establishes the matrix representing the infinitesimal generator I 3 of D n D m, according to I 3 (D n D m ) = 1 i d dϕ D z n (ϕ) Dz m (ϕ) ϕ=0 = I n 3 E + E I m 3 149

154 5 Kronecker (direct) products (One can convince oneself that this formula is correct if one explicitly computes the matrix representing rotations about the z axis in the representation D n and D m : e inϕ e i(n 1)ϕ e i( n+1)ϕ e i( n)ϕ e imϕ e i(m 1)ϕ e i( m+1)ϕ e i( m)ϕ and their Kronecker product). We therefore write n + m n + m I 3 (D n D m ) = n m The Cartan weights (multiplicators) appearing along the diagonal of this matrix can be summarized within a Cartan diagram. Among the Cartan weights pick up the largest one, let us call it j k. Then the reduction of D n D m contains D jk at least once. The representation D jk uses the corresponding 2 j k + 1 multiplicators. 150

155 5 Kronecker (direct) products Figure 5.1: Cartan diagram displaying all possible multiplicators appearing along the diagonal of the matrix I 3 (D L 1 D L 2. Each point with coordinate (m1, m 2 ) is associated the weight m 1 + m 2. The remaining multiplicators are sorted out starting from the largest one and repeating the last three steps until all Cartan weights are used. Graphically, one proceeds along the paths given in the following Cartan diagram: Inspection of Figure 5.2: Sampling the weights along the indicated path one recovers all possible values appearing in the reduction of D L 1 D L 2. the Cartan-Diagram gives the Clebsch-Gordan series D n D m = D n+m D n+m 1... D n m 151

156 5 Kronecker (direct) products Notice that the Clebsch-Gordan series provides a rule to sum angular momentum qunatum numbers: n + m n m = n + m 1... n m The Clebsch-Gordan series reducing the direct product can also be found using the theorems on characters. The coefficients c j in D n D m = c j D j are given by π c j = 2 π 0 dϕ sin[(2j + 1) ϕ 2 ]χ n(ϕ)χ m (ϕ) sin ϕ 2 = 8 π = 4 π π 2 0 π 2 0 = 4 π π 0 d x sin[(2j + 1)x] 1 2 dϕ sin[(2j + 1) ϕ m 2 ] sin[(2n + 1)ϕ 2 ] cos kϕ k=0 [sin[(2n k)x] sin[(2n + 1 2k)x]] k d x sin[(2j + 1)x] [sin[(2(n + k) + 1)x] sin[(2(n k) + 1))x]] k The expression on the right is = 1 if i = n + k or n k, with k = 0, 1,..., m and 0 otherwise. Hence we have obtained again the well known Clebsch-Gordan series for the direct product D n D m. 5.2 The Kronecker product of groups Let H = e, h 1, h 2,... and K = e, k 1, k 2,... be two groups such that all the elements h j commute with k j. If we multiply each element of H with each of K we obtain a new set of elements. Theorem: This set build a group: the group is called the direct product of H and K and is denoted by G H K. Proof. With G = e, g 11, g 12,...g 1k, g 21, g 22,..., g i j,..., where g i j = h i k j and h i h m = 152

157 5 Kronecker (direct) products h p and k j k n = k q, then g i j g mn = (h i k j )(h m k n ) = h i h m k j k n = h p k q = g pq Exercise: Prove that H and K are subgroups of G. Example Consider a molecule with atoms arranged at the corners of a square within a plane and with two extra equivalent atoms sticking out of the plane from the center of the square. The symmetry group of the square C 4v must be expanded to consider that the plane where the square resides is a mirror plane σ h. i.e. the original 8 operations do not transform point above and below the plane into each other, but σ h does it. The two extra atoms have symmetry group {E, σ h } and the symmetry group of the entire system consisting of the 6 vertices is C 4v {E, σ h } Representation theory of direct products Let T h be a representation of H and T k a representation of K, i.e. T h (h i )T h (h m ) = T h (h p ) T k (k j )T k (k n ) = T k (k q ) Theorem: the direct product T h T k of the representations of two commuting groups is a representation of the direct product group. Proof: (T h T k )(g i j )(T h T k )(g mn ) = (T h (h i ) T k (k j ))(T h (h m ) T k (k n )) = T h (h p ) T k (k q ) = T h T k (g pq ) 153

158 5 Kronecker (direct) products Theorem: if T h and T k are irreducible representations of H and K, then T h T k T g is an irreducible representation of G. Proof: T h and T k are irreducible representations, i.e. hi Hχ h (h i)χ h (h i ) = 1 ki Kχ k (k i)χ k (k i ) = 1 Taking the product of both sides of these equations leads to 1 = hi,k j χ h (h i)χ h (h i )χ k (k j)χ k (k j ) = gi j χ g (g i j)χ g (g i j ) which proves that T g is indeed an irreducible representation of the product group. Theorem: All irreducible representations of G are the direct product of an irreducible representation of H and one of K. Proof (for finite groups): Let the number of irreducible representations of H be n h, their dimensions be l h i, the number of irreducible representations of K be n k and their dimensions l k j : n h i n g (l h i )2 = h (l k j )2 = k j Taking the product of both sides gives h k = g = (l h i )2 (l k j )2 The irreducible representations of G obtained by direct product will have the dimensions l h i l k j = l g i j. The sum over all such irreducible representations gives i,j (l g i j )2 = (l h i )2 (l k j )2 = g i,j i, j 154

159 5 Kronecker (direct) products O 3 E R(ϕ) I IR(ϕ) Basis functions D f (r)y 0 0 D D cos ϕ cos ϕ D cos ϕ cos ϕ g(r) {Y 1 1, Y 0 1, Y 1 1 } D + sin(5 2 5 ϕ 2 ) sin(5 5 ϕ 2 ) h(r) {Y 2 2, Y 1 2, Y 0 2, Y 1 2, Y 2 2 } sin ϕ 2 sin(5 ϕ 2 ) sin ϕ 2 D sin(5 ϕ sin ϕ 2 ) 2 sin ϕ D + l 2l + 1 sin((2l+1) ϕ 2 ) sin ϕ 2 D l 2l + 1 sin((2l+1) ϕ 2 ) sin ϕ 2 2l + 1 sin((2l+1) ϕ 2 ) sin ϕ 2 2l 1 sin((2l+1) ϕ 2 ) sin ϕ so that the direct product of all irreducible representations exhausts the all irreducible representations of G. The number of such representations is n g = n h n k. This is a very important result as it helps in constructing all irreducible representations of a bigger group from those of smaller groups. Example of group product. The rotation inversion group O 3 is the set containing all proper and improper rotations in 3d. It is isomorphic to the set of 3 3 orthogonal matrices with determinant +1 (proper rotations) and 1 (improper rotations). The elements of O 3 are those of SO 3 and the elements of SO 3 multiplied by the Inversion I. As I commutes with all other elements of the group we have that matrices representing I are of the form λ E. O 3 can be written as O 3 = SO 3 {E, I} The character table of O 3 reads accordingly: Example of group product: C 3v C h. 155

160 5 Kronecker (direct) products C 3v e 2C 3 3σ v Λ Λ Λ C h e σ h g 1 1 u 1 1 C 3v C h e 2C 3 3σ v σ h 2C 3 σ h 3σ v σ h Λ Λ Λ Λ Λ Λ

161 6 Applications of direct products 6.1 An introduction to the universal covering group An important aspect of many-fold connected continuos groups can be illustrated at the example of SO 2. We have shown that Γ m, with m = 0, ±1,... and χ m (ϕ) = e imϕ are irreducible representations of C and that they represent all irreducible representations of C. However, if we allow multivalued representations, it can be easily shown that Γ m k = e m k ϕ with k being any integer obey also the homomorphism condition and are therefore representations of C. Of course, Γ m k is a k-valued representation of C : rotations differing by 0, 2π, 4π,...(k 1) 2π are represented by different matrices, although they correspond to one and the same element of the group if we impose the quite natural 2π-periodicity onto C. In other words: those representations which are constructed from k 1 cannot have 2π-periodical wave functions as symmetry adapted wave functions: their periodicity must be k 2π, i.e. they cannot be made of fuctions of the spherical angle ϕ, they cannot reside within our conventional Euclideal space. A similar problem arises when trying to find symmetry adapted wave functions to e.g. D + of O l 3 for uneven l: they cannot be made of functions of the coordinates x, y, z because for l uneven such functions change sign upon inversion. We conclude that it is the property of k-fold connected continuos groups to have single-value, double valued,...,and k-valued representations. 157

162 6 Applications of direct products C ϕ 2π Γ m e imϕ 1 Γ m 2 e i m 2 ϕ 1 The question that arises naturally is the following: When multivalued representations exist, are they realized in physical systems? What is the meaning of considering them in physical problems? Mathematically speaking, Bethe suggested a way of dealing with such multivalued representations, which we illustrate with the double valued representations of C. In the rep. Γ m 2, E (R(ϕ = 0)) is represented by 1 and R(ϕ = 2π) is represented by 1. Bethe strategy for eliminating the double valuedness of Γ m 2 is to define a new (covering) group SO D 2. = SO 2 R 2π SO 2 with R 2π being a rotation by 2π which is set to be a separate, new element. SO D 2 has double as many elements as SO 2 and is called the double group of SO 2. The identity element of the new group SO D is 2 E R 4π showing that a rotation by 4π is identical to a rotation by 0. The parameter space of SO D 2 is [0, 4π] and the double valued representation of SO 2 becomes a singled-valued representation of SO D. In 2 addition we have the relations (k, k = 1, 2) 1 4π 4π 0 χ m k (ϕ) χ m k (ϕ)dϕ = 1 4π e i( m k m k ) (ϕ)dϕ 4π 0 This last integral is 1 for m = m and k = k or else it is vanishing, showing that the representations Γ m and Γ m k are irreducible representations of the new double group SO D 2. The representations Γ m, m = 0, ±1, ±2,... are called the single group representations of SO D 2. The representations Γ m 2, m = 0, ±1, ±2,... are called the 158

163 6 Applications of direct products SO D 2 E R ϕ R 2π R 2π R ϕ Γ m 1 e imϕ 1 e imϕ Γ m 2 1 e i m 2 ϕ 1 e i m 2 ϕ double group (or extra) representations of SO D 2. (Remark: Of course, SO 2 is an infinitely manyfold connected group and has a universal covering group consisting of an infinite number of summands: SO 2. = SO 2 R 2π SO 2 R 4π SO 2 R 6π SO 2... The universal covering group of SO 2 spans therefore the entire positive real axis obtained by expanding the interval [0, 2π] to n [0, 2π], n = ±1, ±2... ) The question about multivalued representations can now be reformulated as follows: given a k-fold (k can be even infinite) connected group, which group is physically relevant: the group itself or its universal covering? This question is a very crucial one: if one remembers that the index m of an irreducible, singled valued representation of SO 2 is related to the z-component of the qunatum mechanical angular momentum by the relation L z = ħh m, should the covering group be the relevant one, than we expect further values for L z to appears in a two-dimensional system with SO 2 -symmetry, given by L z = ħh m k k being some integer. In particular, we expect fractional quantum numbers for the z-component of the angular momentum. This has implications for the particle statistics: in 2d, one could think of the appearance of particles which are neither Fermions nor Bosons and could be called anyons. The multivalued representations of SO 2 are called accordingly anyons representations (see S. Rao, An Anyon 159

164 Primer, arxiv:hep.th/ ). 6 Applications of direct products 6.2 The universal covering group of SO 3 SO 3 is a doubly connected group, so we expect single and double valued representations to appear. The single valued representations we have already found: D l, with l = 0, 1, 2,... An educated guess for the universal covering group of SO 3 is (Bethe) SO D 3 = SO 3 R(2π) SO 3 where SO D indicates the double group of 3 SO 3. For the double valued representations of SO 3 which will become single valued representations of SO D 3 one can formulate an educated guess, based on the fact that SO D is a subgroup of 2 SOD 3 and that we know that the double valued representations of SO 2 are given by Γ 2m+1 2 : the double valued representations of SO 3 are the so called half-integer representations D l 2 (l = 1, 3,...) that produces the half-integer spin quantum numbers in quantum mechanics. In the single group representations D l (l = 0, 1, 2,...,) R(2π) is represented by the 2l + 1 2l + 1 matrix

165 6 Applications of direct products SO D 3 R 0 R(ϕ) R 2π R 2π R(ϕ) D l (l = 0, 1, 2,..) 2l + 1 sin((2l+1) ϕ 2 ) sin ϕ 2 2l + 1 sin((2l+1) ϕ 2 ) sin ϕ 2 D l 2 (l = 1, 3,...) l + 1 sin((l+1) ϕ 2 ) sin ϕ 2 l 1 sin((l+1) ϕ 2 ) sin ϕ 2 In the double group representations D l 2, (l = 1, 3,...,) R(2π) is represented by the 2l + 1 2l + 1 matrix The characetr table of SO D looks as follows: 3 Dl, l = 0, 1, 2,... are the single group representations of SO D, 3 D l 2, l = 1, 3, 5,... are the double group (or extra) representations of SO D. 3 Remark The half integer representations of SO D 3 introduce half-integer angular momenta in quantum mechanics. In 3d, in contrats to 2d, we have only the angular momentum quantum numbers 0, 1 2, 1,... because SO 3 is only doubly connected. Remark Half-integer spherical harmonics (symmmtery adapted basis functions) cannot be defined on the sphere: they exist only in a space which is different from the conventional Euclidean (or Einstein-like) space. This particular space where periodicity is 4π (for spin 1 2 particles) is the spin space. In the following we take the convention of calling spherical harmonics of the type Y m (ϑ, ϕ) as X m and spin spherical harmonics l l as Y ± It is clear that the spin space only exists at the very small spatial scales 2 corresponding to the world of subatomic physics, where half-integer particles reside. At macroscopic length scales the 2π periodicity is recovered. 161

166 6 Applications of direct products 6.3 SU(2) The D 1 2 representation of SO D is generated by the Pauli Matrices (σ x, σ y, σ z ), where the σ s are i σ z = σ x = σ y = i 0 Using n = (l, m, n) and the rotation angle ϕ [0, 2π] we can write the rotation matrix as n l i m O l,m,n (ϕ) = e i ϕ 2 l + i m n As l 2 + n 2 + m 2 = 1, we have n l + i m l i m n 2 = so that O l,m,n (ϕ) = + = ( iϕ/2)k 1 0 k! 0 1 k even ( iϕ/2)k n l i m k! l + i m n k odd ϕ cos 2 i n sin ϕ 2 ( i l m) sin ϕ 2 ( i l + m) sin ϕ 2 cos ϕ 2 + i n sin ϕ 2 Notice that D 1 2 ( n, ϕ + 2π) = Ē D 1 2 ( n, ϕ) with 1 0 Ē. = R(2π) = 0 1 An alternative way to construct the representation D 1 2 is to parametrize rotations by the Euler angles α, β, γ These angles determine the set of consecutive rotations which are necessary to drive the orthogonal vector system K = (x, y, z) onto the 162

167 6 Applications of direct products Figure 6.1 system K t = (x t, y t, z t ). Three rotations are required: a) a rotation by α(0 α 2π) about the z axis drives K into K 1. We call this rotation R(α, z). b) a rotation by β(0 β π) about the next axis x 1 transforms K 1 into K 2 this operation we call R(β, x 1 ). c) the last rotation is by γ(0 γ 2π) about the z 2 axis operation R(γ, z t ). The composition of the three operations is e i 2 σ zα e i 2 σ xβ e i 2 σ zγ The evaluation of the exponential functions by summing the corresponding series gives e i 2 σ zϕ = = ( i n e i ϕ e i ϕ n! ϕ 2 )n 163

168 6 Applications of direct products and n ( i 2 σ xβ) n n! = ( ( i)n β (σ x n! n 2 )n = σ 0 cos( β 2 ) i σ x sin( β 2 ) β cos 2 i sin β 2 = i sin β 2 cos β 2 Building the product of the three matrices corresponding to the three rotations we obtain β α+γ cos 2 e i 2 i sin β γ α 2 ei 2 O αβγ = i sin β α γ 2 ei 2 cos β α+γ 2 ei 2 Remark: The matrices of D 1 2, independent of their parametrization, all have an important property: they are unitary matrices with determinant 1, and the set of them obtained by running over the parameter space build the group SU(2). This group is known as the special unitary group. It is a 3 parameters continuos Lie group with average. By construction, SU 2 is isomorphic to SO D Symmetry adapted functions in atomic physics: the Clebsch-Gordan coefficients When the spin variable is taken into account, the full symmetry group of an electron with spin embedded in a spherically symmetric potential is SO 3 SU 2 This group corresponds to independent rotations in orbital and spin space. The irreducible representations of SO 3 SU 2 are D l D

169 6 Applications of direct products and the invariant subspaces sustaining D l double their degeneracy acquiring the two spin functions Y ± Y ± : 2 {X l l, X l 1 l,..., X l l } {X l l Y +, X l l Y, X l 1 l Y +, X l 1 l Y,..., X l l Y } This picture is only valid as long as the Hamiltonian does not contain spin-dependent interactions. When e.g. spin-orbit coupling is introduced, i.e. H = H 0 E + α ( L E, E S) then the symmetry group consists of simoultaneous rotations in orbital and spin space. This is a subgroup of SO 3 SU 2 and is isomorphic to SU 2. Accordingly, the direct product representation D l D 1 2 becomes reducible and splits into the components D l D 1 2 = D l+ 1 2 D l 1 2 causing the 2 (2l + 1) degenerate energy level to split into the two sublevels with total angular momentum l+ 1 and 2 l 1 2. The question now arises about the symmetry adapted linear combinations of {X l l Y +, X l l Y, X l 1 l Y +, X l 1 l Y,..., X l l Y } transforming according to D l+ 1 2 and D l 1 2. The coefficients appearing in these linear combinations are known in the literature as the Clebsch-Gordan coefficients (some of them are given in the separate Table made available per pdf file). To compute these symmetry adapted linear combiniations, the use of the projector operator technique is impractical. Rather, one uses an algorithm proposed by Cartan. This algorithm uses the matrix elements of the infinitesimal generators, which are summarized again here (noticing that they can be extended to half-integer 165

170 6 Applications of direct products values!) The Cartan algorithm reads: (Y j,m, I z Y j,m ) = m (Y j,m±1, I x Y j,m ) = 1 (j m)(j ± m + 1) 2 (Y j,m±1, I y Y j,m ) = i (j m)(j ± m + 1) 2 1. Within the D l D m space, find all wave functions carrying the same Cartan weight k. This space V k might have, in general, more than one dimension. For example: in D l D 1 2 the wave functions X m l Y +, X m+1 l Y carry the same Cartan weight k = m Define the auxiliary operators with the property I ± = 1 2 [I x ± i I y ] I ± ψ jm = 1 2 j(j + 1) m(m ± 1)ψ j,m±1 and then solve the equation I + x = 0 within the space V k. The vector x transforms, by construction, according to the top weight of the representation D k, i.e. when I 3 (D k ) is applied to x, x get multiplied by k. 3. The remaining vectors transforming according to D k are found by applying the lowering operator I to the newly found x. Example. The Clebsch-Gordan coefficients. Let us calculate, as an example of this second algorithmus, the invariant subspaces transforming according to the rep- 166

171 6 Applications of direct products resentations D l+1/2 and D l 1/2 appearing in D l D 1/2 = D l+1/2 D l 1/2 The vector space belonging to D l D 1/2 is given by {X l l Y +, X l l Y,..., X l l Y } We construct, using Cartan first algorithm, the matrix l l I 3 (D l D 1/2 ) = l 1 2 from which we can read out the various Cartan weights. Let us pick up the Cartan weight l There is only one function carrying this Cartan weight, X l Y +, which is accordingly the symmetry adapted wave function transforming according to the weight k = l + 1 of the representation 1 2 Dl+ 2. To find all other partner functions in the irr. representation D l+ 1 2 apply the operator I to this function. Let us now pick up the next higher Cartan weight l = l 1 2. The eigenspace carrying the weight is given by {X l Y, X l 1 Y + } There will be one symmetry adapted wave function transforming according to the weight l 1 2 of the irreducible representation Dl+ 1 2 and an orthogonal symmetry adapted wave function transforming according to the SAME weight but belonging 167

172 6 Applications of direct products to the irr. rep. D l 1 2. In order to find this second wave function we solve the equation I + (αx l Y + βx l 1 Y + ) = α[(i + X l ) Y + X l (I + Y )] +β[(i + X l 1 ) Y + + X l 1 (I + Y + )] = [ 1 α + β l]x l Y + = 0 2 which gives and This means that the function α = 2l 2l + 1 X l Y 2l 2l β = 2l l + 1 X l 1 Y + belongs to the invariant subspace of D l 1/2 with k = l 1/2. The application of I to this function produces all partners in the irr. rep. D l 1/2. Remark: In the literature one finds following nomenclature: 1. D l D k = j D j 2. ψ(j, m j ) = m l,m k l k j ψ(l, m l ) ψ(k, m k ) m l m k m j }{{} 3j Wigner Symbol with m l + m k = m j (otherwise the CC-coefficient is vanishing). 6.5 Double point groups When expanding single point groups to include the operation Ē and its product with the other operations of the single point group one must consider that, while the number of elements is exactly doubled, the number of classes does not necessarily 168

173 6 Applications of direct products E Ē Table 6.1: The character Table of C D 1 double. For instance, C 2z in C 4v builds a class on its own. C 2z = ĒC 2z is conjugate to C 2z as and (IC 2x ) 1 = I C 2x (I C 2x )C 2z (IC 2x ) = C 2z so that in C D 4v C 2z and C 2z belong to the same conjugate class. Accordingly, in general the number of irr.rep. of the double group is not necessarily twice that of the single group. Let us now work out some practical examples. The group C D 1 The single group C 1 contains the identity E. The double group contains E and Ē. They form two distinct classes and there are two one-dimensional irreducible representations. One 1 is a single group representation and the next one 1 D is the double group representation. The D 1/2 -representation of E and Ē is ; It is clearly reducible into the sum of two identical one dimensional representations 1 D, with E = 1, Ē = 1. Remark: The extra (or double ) group representations of a double group are the only physically relevant representaions if the spin is considered, as in these representations Ē is properly represented. In order to find what happens to an energy level at a general point of the BZ, carrying the representation 1 when the spin-orbit coupling is introduced, one 169

174 6 Applications of direct products C D i E Ē I Ī g u g D u D must compute the product 1 D 1 2 C D 1 = 1D + 1 D This shows that at a general point of the BZ the introduction of spin-orbit coupling produces generally a split of the originally double degenerate single group energy level. The group C D i The single group C i contains the identity and the inversion I. The double group C D i contains the four elements E, Ē, I, Ī. Because all these elements commute, there are four classes and four one-dimensional irreducible representations given in the table. Notice that, again, the two-dimensional representation D 1/2, when limited to C D i, is clearly reducible: it is the direct sum of two identical one-dimensional representations with 1, 1, 1, 1. To find what happens to a g(u)-band when spin-orbit coupling is switched on we have to consider that g D 1 2 = 2g D u D 1 2 = 2u D The double group C D 3v The single group C 3v contains six elements: E, C 3x yz, C 1 3x yz, IC 2x ȳ, IC 2 xz, IC 2y z, all transforming the k-vector along the Λ- direction into itself ( k Λ = π/a(λ, λ, λ) with 0 < λ < 1). The symbols C 3x yz and 170

175 6 Applications of direct products C 1 3x yz, for instance, means rotations by an angle 2π/3 around an axis with director cosines on the ratio 1 : 1 : 1, see the table and figure for O h. There are three classes: the identity element, the two there-fold rotations and the three reflections. We can easily construt the character table of the single point group C 3v. The double group C D has twice the number of elements and twice the number of classes as 3v C 3v. The matrices representing the 2C 3 group elements in the D 1/2 representation can be constructed by using l = m = n = 1 and a rotation angle 3 ±ϕ = 2π : 3 C 3x yz = 1 i 2 1+i 2 1 i i 2 Similarly, one can obtain the matrices IC 2x ȳ, IC 2 xz, IC 2y z by considering that the inversion is followed by the rotation of 2π/2 around an axis with director cosines in the ration 1 : 1 : 0, i.e. ϕ = π and l = / 2, m = 1/ 2 n = 0. Thus, for instance, IC 2x ȳ = I 0 1 i 2 1+i 2 0 The remaining matrices can be constructed in a similar way. 1 0 E = i 1 i C 3x yz = 1/2 1 i 1 + i 1 + i 1 + i C 1 3x yz = 1/2 1 + i 1 i 0 1 i IC 2x ȳ = 1/2I 1 i 0 i i IC 2 xz = 1/2I i i i 1 IC 2 y z = 1/2I 1 i 1 0 ; Ē = i 1 + i ; C 3x yz = 1/2 1 + i 1 i 1 i 1 i ; C 1 3x yz = 1/2 1 i 1 + i i ; I C 2x ȳ = 1/2I 1 + i 0 i i ; I C 2 xz = 1/2I i i i 1 ; I C 2y z = 1/2I 1 i 171

176 6 Applications of direct products Irr.Rep. E 2C 2 3 3σ v Λ Λ Λ Irr.Rep. E Ē 2C C 2 3 3σ v 3 σ v Λ Λ Λ Λ i -i Λ i i Λ D 1 2 C D 3v Λ 1 D Λ 2 D Λ 3 D The 12 elements are thus divided into 6 classes, and the equation α (l α) 2 = 12 has the solution = 12. Thus, there are a total of four one dimensional representations and two two-dimensional representations. The three irreducible representations of C 3v can be extended as irreducible representations of C D 3v by representing Ē with E and the C elements as the non- C elements. There are three extra additional irreducible representations to be found. We notice that the two dimensional D 1/2 representation is irreducible: in fact with the characters χ(e) = 2, χ(ē) = 2, χ(c 3 ) = 1, χ ĒC 3 = 1, χ(σ v ) = 0 we have A χ(a)2 = 12, which is a necessary and sufficient condition for an irreducible representation. The remaining two-extra one dimensional representations are found by applying the unitarity conditions. Λ 6 is the name of D 1/2 when restricted to this point group. By computing Λ i D 1 2 in terms of the double group representations we are able to find out what happens to single group energy bands when spin-orbit coupling is switched on: Λ 1 D 1 2 = Λ6 172

177 6 Applications of direct products Λ 2 D 1 2 = Λ6 Λ 3 D 1 2 = Λ6 + Λ 4 + Λ 5 Thus, the Λ 3 band split into three sub-bands, each carrying one irreducible representation of the double group. The band Λ 1 remains degenerate and carries the irreducible representation Λ 6 of the double group. Remark: Λ 4 and λ 5 carry complex conjugate wave functions as symmetry adapted wave functions. Because of time reversal symmetry, they are degenerate even if SU 2 symmetry does not impose their degeneracy. Figure The Clebsch-Gordan coefficients for point groups Given an invariant subspace {X 1,...X k } 173

178 6 Applications of direct products transforming according to a single group representation Γ s k the symmetry adapted linear combinations of one would like to compute transforming according to Γ D, where J Γ D J {X 1 Y +, X 1 Y,..., X k Y +, X k Y } is the double group rep. occurring in Γ s k D 1 2 = J n k J Γ D J The symmetry adapted basis functions build up from orbital wave function of the type X 1 and spin functions Y ± can be found from P Λ6 X 1 Y ± X 1 Y ± Thus, the Λ 6 band originating from the coupling of a Λ 1 band with spin is doubly degenerate and its basis functions are simply X 1 Y ±. We construct, as example, now, the symmetry adapted wave functions for Λ 3 D 1 2 = Λ Λ3 4 + Λ3 5 within the space built by the four functions {X 3 1 Y +, X 3 1 Y, X 3 1 Y +, X 3 1 Y } and transforming according to the irreducible representations Λ 3 4, Λ3 5 and Λ3 6. We will use as basis functions for Λ 3 Y ±1 : the matrices representing rotations by an angle ϕ around the z axis in this basis are diagonal, with diagonal elements ex p( iϕ). The matrices representing such rotations in the D 1/2 diagonal, their matrix elements being ex p( i ϕ ). 2 representation are also The symmetry adapted basis functions transforming according to the Λ 3 6 irreducible representation within the subspace containing the four functions X 3 ±1 Y ± are found by the projector method. For instance, we compute P Λ6 X 3 ±1 Y ± to find out the symmetry adapted linear combinations of X 3 1 Y + and X 3 1 Y that transform 174

179 6 Applications of direct products according to Λ 3 6. P Λ6X 3 ±1 Y ± 2 X 3 ±1 Y ± + 2 X 3 ±1 Y ± +1e iϕ X 3 ±1 e i ϕ 2 Y ± + 1e iϕ X 3 ±1 ei ϕ 2 Y ± 1e iϕ X 3 ±1 ( 1)e i ϕ 2 Y ± 1e iϕ X 3 ±1 ( 1)ei ϕ 2 Y ± = 4X 3 ±1 Y ± + 4 cos 3 2 ϕx 3 ±1 Y ± = 0 so that the wave functions X 3 ±1 Y ± do not transform as Λ 3 6. In contrast, P Λ6 X 3 ±1 Y 4 X 3 ±1 Y + 4 cos ϕ 2 X 3 ±1 Y X 3 ±1 Y i.e. the two wave functions X 3 ±1 Y belong to the Λ 3 6 representation. Thus, the Λ 6 band derived from the coupling of Λ 3 orbital wave functions with the spin is doubly degenerate and contains the functions X 3 1 Y and X 3 1 Y +. The remaining two functions must be in the subspace transforming according to Λ 4 and Λ 5. The level scheme along Λ is as follows: X 1 0 Y + X 1 0 Y Λ 1 6 X 3 1 Y + X 3 1 Y Λ X 3 1 Y X 3 1 Y + Λ 3 6 We notice that the time reversal operation is also a symmetry operation of the Hamiltonian, and states which are transformed into each other by the time reversal operator must belong to the same energy eigenvalue. One can show that the two states X 3 ±1 Y ± are exactly such states, i.e. states related by time reversal symmetry, so that the bands belonging to Λ 4 and Λ 5 symmetry are degenerate when it comes to their energy (although they behave differently under the operation of the point group). In the band structure of GaAs and Ge the Λ 3 band is found to generate only two spin-orbit split bands, namely Λ 6 and Λ 4+5. The Γ -point The energy levels at the top of the valence band at the Γ -point of GaAs are analogous to the atomic P3 and P1 levels. We also point out that the full character tables for 2 2 the extra representations of all point groups are given in the review by G.F. Koster, 175

180 6 Applications of direct products Space Groups and Their Representations, in Solid State Physics Vol.5, Academic Press, 1957, Seitz and Turnbull editors. A more complete set of tables on the 32 point groups (including all Clebsch-Gordan coefficient for point groups, is given in Koster, G. F., J. O. Dimmock, R. G. Wheeler and H. Statz,. Properties of the Thirty-Two Point Groups (MIT Press, Cambridge, Mass., 1963). Some of the Table in this book are reproduced and adapted in the book Group Theory, Application to the Physics of Condensed Matter, Dresselhaus, Mildred S., Dresselhaus, Gene, Jorio, Ado, 2008, Springer-Verlag Berlin Heidelberg ebook ISBN , DOI / , Hardcover ISBN , Softcover ISBN The Wigner-Eckart-Koster theorem (G.F. Koster, Phys. Rev. 109, 227 (1958)). The construction of the symmetry adapted wave functions is one of the main applications of of group theory to quantum mechanical problems. The second one relies on the Wigner-Eckart-Koster theorem. The WEK theorem provides a tool for computing matrix elements on the basis of symmetry arguments alone. It is based on following Lemma: Lemma: Let G be a unitary matrix group with measure and ϕ(k, m k, r) be a basis function transforming according to the m k -th column of the irreducible representation D k of the group G. The index r must be introduced because functions having the same symmetry under the operations of G might differ on aspects other then their symmetry. In the case of SO(3), for example, functions which have a certain symmetry under rotations might have a different radial dependence (i.e. they might have a different quantum number n). Let also ϕ(l, m l, s) be a basis function transforming according to the m l -th column of the irreducible representation D l. Then ϕ(k, m k, r), ϕ(l, m l, s) = δ kl δ mk m l C (k) rs i.e. the scalar product of two basis functions is vanishing if the basis functions belong to different irreducible representations or to different columns of the same irreducible representation. Else is the scalar product a constant which does not depend on m k 176

181 6 Applications of direct products but only on k, r, s. Proof: Since the scalar product of two functions is invariant under unitary transformations, for any operation T G ϕ(k, m k, r), ϕ(l, m l, s) = O T ϕ(k, m k, r), O T ϕ(l, m l, s) = l k ϕ(k, m k, r), ϕ(l, m l, s) = = m p m p,m q D k m p m k (T)D l m q m l (T) Taking the average over both sides we obtain ϕ(k, m k, r), ϕ(l, m l, s) = = 1 l k δ kl δ mk m l l l D k m p m (T)ϕ(k, m k p, r), D l m q m (T)ϕ(l, m l q, s) m q ϕ(k, m p, r), ϕ(l, m q, s) (ϕ(k, m p, r), ϕ(l, m q, s)) T G D k m p m (T)D l k m q m (T) l m p,m q = δ kl δ mk m l 1 l k m p m q δ mp m q (ϕ(k, m p, r), ϕ(l, m q, s)) (ϕ(k, m p, r), ϕ(l, m p, s)) m p }{{} Crs k This proves the orthogonality of the basis functions. As the right hand side is independent of m k and m l, provided m k and m l are equal, the scalar product does not depend on the column index but only on k, r and s. The remaining constant 1 i.e. l k m p (ϕ(k, m p, r), ϕ(l, m p, s)) must, in general, be explicitly computed, as symmetry arguments no longer help. This orthogonality Lemma for basis functions can be applied to calculating the matrix elements of operators (this application goes often under the name of Wigner-Eckart theorem). Definition: The set of l l operators Q l 1 ( r),ql 2 ( r),...,ql l l ( r) transform according to the columns of the l l -dimensional irreducible representation D l of the group G if for every T of G O T Q l i ( r)o 1 T = p D l pi Ql p ( r) 177

182 6 Applications of direct products This equation has to be understood as an operator equation. That is, both sides must produce the same result when acting on any scalar function. We also assume that in the left-hand side every operator acts on everything to its right. Examples: The operator r transforms according to the irr.rep. D 1 of SO 3 and D 1 of O 3. p is also a vector operator. In contrast, L transforms according to the rep. D 1+ of O 3, as it is invariant with respect to inversion. In addition, p does not transform according to what we have called physical rep. of the group of k, because it is invariant with respect to translations and therefore does not acquire the multiplicative factor e i k t n upon translations. We consider now matrix elements of one of these operators between scalar functions transforming according to the columns of irreducible representations: O T ϕ(k, m k ) = D k m q m ϕ(k, m k q ) m q O T ϕ(j, m j ) = D j m r m ϕ(j, m j r ) m r and ϕ(j, m j ),Q l i ϕ(k, m k) As both Q l i and ϕ(k, m k ) transform according to representations of the symmetry group, the wave function Q l i ϕ(k, m k) will be in the subspace transforming according to the direct product representation D l D k. The representation constructed over the set of basis function Q l i ϕ(k, m k), i = 1,..., l l, m k = 1,..., l k will be in general reducible, according to D l D k = s n s D s, and the function Q l i ϕ(k, m k) will occur in the subspace of one or more of these irreducible representations. Thus, according to our previous theorem, this function will have non-vanishing matrix element only with wave functions transforming according to the same irreducible representations. This means that A matrix element involving a final state wave function transforming according to a representation D f which is not contained in the direct sum decomposition of the representations D op D i, is certainly vanishing (selection rule theorem). 178

183 6 Applications of direct products Notice that, to apply this selection rule theorem to a concrete example, only the Clebsch-Gordan series is needed, so that the vanishing of a matrix element can be determined with certainty and using symmetry arguments only. Thus, group theory forbid, with absolute certainty, certain matrix elements. Those which are not forbidden by symmetry can still be vanishing by some reason other than symmetry. Symmetry computation of matrix elements This selection rule theorem is, however, the weakest form of a more general matrix element theorem, known as the Wigner-Eckart-Koster theorem (G.F. Koster, Phys. Rev. 109, 227 (1958)). In fact, if we consider more closely the procedure adopted to analyze the matrix element, we will notice that the wave function Q l i ϕ(k, m k) enters the subspace transforming according to D j exactly n j times, n j being specified by the general formulas for the decomposition of product representations. The coefficients of the linear combination determining the various basis functions, for which the representation D l D k is in block form, are the Clebsch-Gordan coefficients and can be determined by group theoretical arguments (for SO(3) see Cartan s algorithm). Clearly, the basis function Q l i ϕ(k, m k) will enter these basis functions with given Clebsch-Gordan coefficients. Thus, the matrix element will be a linear combination of n j Clebsch-Gordan coefficients C(l, k, j, m l, m k, m j, r). Notice that for SO(3) n j = 1 and only one coefficient is needed to describe matrix elements. In summary: if the irreducible representation D j occurs n j times in the direct sum decomposition of D l D k, then ϕ(j, m j ),Q l i ϕ(k, m k) = a 1 C(l, k, j, m l, m k, m j, 1) + a 2 C(l, k, j, m l, m k, m j, 2) a nj C(l, k, j, m l, m k, m j, n j ) The coefficients a i must be calculated explicitly. 179

184 6 Applications of direct products Example 1: Dipole selection rules for spherically symmetric potential. The matrix elements for transitions between the initial state i to the final state f under the action of light writes M i f ψ f, e rψ i where e is the polarization vector of the radiation. As r transforms as D 1, the symmetry of the final state must be contained in the reduction of D 1 D i = D i+1 D i D i 1 This establishes the dipole selection rules: optical transitions are allowed only between states differing by an angular momentum quantum number of 0, ±1. If we include the symmetry with respect to inversion, as the light operator has negative parity, the D i D i transition is forbidden. Consider for instance the optical transition between p and s states. The operator producing a transition in an atom is proportional to ( e r). If we choose linearly polarized light along z and circularly polarized light in the (x y)-plane, we obtain that the light is represented by the operators z( Y 10 ) and x ± i y( Y 1±1 ). These operators transform according to the representation D 1 of SO(3). As a consequence, p s, p p and p d transitions are optically allowed. The p p transition is forbidden if one consider that D 1 -wave functions are uneven with respect to the parity operation, so that an optical transition is only allowed between states of different parity (if the inversion is a symmetry element). Example 2: Spin polarization of p s optical transitions in spherically symmetric potential with spin-orbit coupling. We use e.g. left-hand polarized light propagating along +z, i.e. the operator is Y 1 1. We consider for instance the P3 2 S 1 2 transition. We need to compute the 8 matrix elements Y 0 0 Y ±, Y 1 1 P m j 3 with m j = 3 2, 1 2, 1 2, 3 2. The states on the right-hand side are all possible initial transition. The light states and the state Y 0,0 Y ± is the final state of the P3 2 2 S

185 6 Applications of direct products Figure 6.3: The P3 2 and P1 2 the final state. spin-orbit splitted states build the initials states, the S 1 2 operator does not act in spin space (within the dipole approximation), so that we are left with the problem of finding linear combinations in the space sustained by the rep. D 1 D 1 transforming according to D 0, which is the symmetry of the (orbital part of the) final state. The matrix elements we need to compute are the four matrix elements Y 0, Y 1 1 Y 1,0, 1 1 The basis functions in D 1 D 1 are {Y 1 1 Y 1 1, Y 0 1 Y 1 1, Y 1 1 Y 1 1, Y 0 1 Y 1 1, Y 0 1 Y 0 1, Y 0 1 Y 1 1, Y 1 1 Y 1 1, Y 1 1 Y 0 1, Y 1 1 Y 1 1 } The symmetrized linear combination transforming like D 0 is (see e.g. Condon and Shortley, The theory of Atomic spectra, Cambridge Press, 1935, Table 2, p.76) ψ 0,0 = 1 Y 1 1 Y Y 0 1 Y Y 1 1 Y