Peter Grünberg Institute

Transcription

1 Forschungszentrum Jülich Peter Grünberg Institute strongly correlated systems Julian Mußhoff

2 Infos webpage: tuesday: Lecture 4:5-7:30 thursday: Exercise 6:5-7:45 oral exam (+ Exercises + Lectures), ~ 9 or February e.pavarini@fz-juelich.de language: English exercises are part of the lecture, and help in clarifying some points at the exam, you can also be asked things discussed during the exercises 3 admission to exam Group Theory in Solid State Physics WS 05/6, E. Pavarini Exercise sheet 0. Rules To be admitted to the exam the following conditions should be fulfilled to take part to lectures and exercises most of the times to actively participate in class, both in lectures and exercises to have handed in the solution for most of the exercise sheets. to have gotten at least 40 points in total (0 for mathematicians) to have presented at least 3 times your solution in class 4

3 literature modern approach M. Dresselhaus, G. Dresselhaus, Group Theory: Applications to the Physics of Condensed Matter more rigorous M. Tinkham, Group Theory and Quantum Mechanics M. Hamermesh, Group Theory and its applications to physical problems linear spaces functions 5 more lecture notes Autumn School on Correlated Electrons background, physics back ground physics + basics of group theory advanced topics broken symmetry 6

4 background from previous lectures energy, momentum, angular momentum (concepts) basic quantum-mechanics operators: Hamilton, Laplace, angular momentum,... Schrödinger equation for the hydrogen atom hydrogen-like wavefunctions matrices, matrix algebra linear spaces and their properties basic solid-state physics (unit cells, Bloch theorem, first Brillouin Zone) [second quantization] 7 symmetries in art and nature translation bilateral rotation radial reflection hexagonal 8

5 Kepler and the platonic solids Tetrahedron (four faces) Cube or hexahedron (six faces Octahedron (eight faces) Dodecahedron (twelve faces) Icosahedron (twenty faces) 9 so what? 0

6 symmetries simplify problems H = m r the two-body problem m r + V (r,r ) r r=r-r r H = H cm m r + V (r) however, translational invariance! one-body problem in a central force birefringence symmetries are powerful tool to understand reality

7 symmetries and conservation laws Noether's Theorem: Symmetries = Conservation Law time reversal = energy is conserved translational invariance = momentum is conserved rotational invariance = angular momentum is conserved what are the consequences of these and other symmetries in solid-state physics? symmetries of a free atom point symmetries in molecules and crystals translational symmetries in molecule and crystals symmetries in the many-body problem (broken symmetries) 3 symmetries and particle statistics identical and indistinguishable particles fermions: totally antisimmetric functions fermi-dirac statistic Pauli Principle bosons: totally symmetric functions bose-einstein statistic 4

8 broken symmetry original model have a symmetry but the solution not... ferro and antiferromagnetic order (rotational invariance Heisenberg model) 5 Superconductivity Spontaneous Symmetry Breaking and the Goldstone Theorem Another remarkable development came around 960 when Yôichirô Nambu (Nobel Prize 008) extended ideas from superconductivity to particle physics. He had previously shown that the BCS ground state (named after John Bardeen, Leon Cooper and Robert Schrieffer, Nobel Prize, 97) has spontaneously broken gauge symmetry. This means that, while the underlying Hamiltonian is invariant with respect to the choice of the electromagnetic gauge, the BCS ground state is not. Goldstone Theorem: Goldstone bosons spin waves phonons Higgs mechanism 6

9 broken symmetry An unexplained broken symmetry at the birth of the universe. In the Big Bang, if as much matter as antimatter was created, they should have annihilated each other. But a tiny excess of one particle of matter for every ten billion antimatter particles was enough to make matter win over antimatter. This excess material filled the cosmos with galaxies, stars, planets and eventually life. Spontaneous broken symmetry. The world of this pencil is completely symmetrical. All directions are exactly equal. But this symmetry is lost when the pencil falls over. Now only one direction holds. The symmetry that existed before is hidden behind the fallen pencil. Nobel Prize in Physics 008 Yoichiro Nambu "for the discovery of the mechanism of spontaneous broken symmetry in subatomic physics" Makoto Kobayashi & Toshihide Maskawa "for the discovery of the origin of the broken symmetry which predicts the existence of at least three families of quarks in nature". 7 symmetry and games 8

10 organization of the lecture introduction basics abstract group theory representations and characters orthogonality theorems and consequences symmetries and operators group theory in solid-state physics hydrogen-like atom and group theory from atoms to molecules molecular orbitals, molecular states and symmetry atomic displacements and molecular vibrations equivalence representations from atoms to solids crystal-field double groups Kramers theorem Jahn-Teller theorem crystal translation invariance and the Bloch theorem tight-binding and bands magnetic groups multi-electron states and many-body physics atomic multiplets correlated models in solids broken symmetries topology 9 group A group G is a set of elements {g i } plus an operation,?, which satisfy the following conditions {gi} g operation. G is closed under group multiplication, i.e., g i? g j = g k G 8g i,g j G. the associative law holds, i.e., g i? (g j? g k )=(g i? g j )? g k 8g i,g j,g k G 3. there is an identity element e G, such that g i? e = e? g i = g i 8g i G 4. there is an inverse element g i G to each g i G, such that g i? g i = g i? g i = e. =. ( ) = ( ) 3. = 4. = abelian group: commutative 5. = 0

11 examples real numbers without the zero, operation=multiplication real numbers with operation=addition vectors U(n) unitary n-dimensional matrices SU(n) unitary n-dimensional matrices with det= O(n) orthogonal n-dimensional matrices SO(n) orthogonal n-dimensional matrices with det= S(n) permutation of n object (symmetric group) finite group of matrices the permutation group S(3) E A B C D F

12 NH3, group C3v and S(3) D F A C symmetry group of the equilateral triangle 3 B multiplication table the permutation group S(3) E A B C D F E E A B C D F A A E D F B C B B F E D C A C C D F E A B D D C A B F E F F B C A E D 4

13 E C3 C3 σ σ3 σ E E C3 C3 σ σ3 σ C3 C3 C3 E σ3 σ σ C3 C3 E C3 σ σ σ3 σ σ σ σ3 E C3 C3 σ3 σ3 σ σ C3 E C3 σ σ σ3 σ C3 C3 E σ σ σ3 C3 C3 σ σ σ3 C3 C3 E A B C D F E E A B C D F A A E D F B C B B F E D C A C C D F E A B D D C A B F E F F B C A E D E C3 C3 D F E A B C D F σ A σ3 C σ B some concepts order of a group= number of elements subgroup order of an element: n, such that g n =e period of an element: {,g,...,g n- }, n order of g cyclic groups generators of a finite group 6

14 Lecture 7 maps G' G injective each elements of G' is the image of at most one element of G not all elements of G' are images of elements of G surjective all elements of G are images of elements of G more than one element of G can correspond to the same element of G' bijective=injective+surjective all elements of G' correspond to elements of G and the correspondence is one to one 8

15 homomorphisms structure-preserving mapping between two algebraic structures two groups GA and GB are homomorphic if there is a correspondence f between their elements such that ga ga f f f gb gb ga ga gb gb f (ga)= gb f (ga)= gb f ( ga ga) = f (ga) f(ga) isomorphism: bijective homomorphism epimorphism: surijective homomorphism monomorphism: injective homomorphism 9 homomorphisms H Monomorphism: injective M I E Epimorphism: surjective Isomorphism: bijective 30

16 NH3, group C3v and S(3) σ σ σ3 symmetry group of the equilateral triangle 3 properties of isomorphisms two groups which are isomorphic have the same structure and can be identified if two finite homomorphic groups have the same number of elements they are also isomorphic Cayley theorem: every group of order n is isomorphic to a subgroup of Sn example of isomorphism: S3 and C3v E 3 3 D 3 3 F 3 3 A 3 3 C 3 3 B 3 3 3

17 example of isomorphism C3v S3 E = 0 0 set of 6 matrices E A = 0 0 B = p 3 p 3! C = p 3 p 3! A D = p 3 p 3! F = p 3 p 3! 3 D these matrices describe how a vector (x,y) transforms under C3v operations 33 multiplication table of C3v two isomorphic groups have the same multiplication table E C3 C3 σ σ3 σ E E C3 C3 σ σ3 σ C3 C3 C3 E σ3 σ σ C3 C3 E C3 σ σ σ3 σ σ σ σ3 E C3 C3 σ3 σ3 σ σ C3 E C3 σ σ σ3 σ C3 C3 E E A B C D F E E A B C D F A A E D F B C B B F E D C A C C D F E A B D D C A B F E F F B C A E D 34

18 properties of homomorphisms example of homomorphism: map of G=C3v in G'={e',a'} with {E,C3,C3 } e' and {σ, σ', σ''} a' e' a' the identity of G corresponds to the identity of G' more of one element of G can correspond to the identity of G' if the set {e,a...an} correspond to e' than the set {ge,ga,... gan} correspond to g' the set {e,a...an} of elements which correspond to e' forms a subgroup of G 35 rearrangement theorem: if e, a,a,.., ah are the elements of the group G and ak is an arbitrary element of G than the set ake, aka,..., akah contains each element of the group once and only once E C3 C3 σ σ3 σ E E C3 C3 σ σ3 σ C3 C3 C3 E σ3 σ σ C3 C3 E C3 σ σ σ3 each column/row contains each element once and only once σ σ σ σ3 E C3 C3 σ3 σ3 σ σ C3 E C3 σ σ σ3 σ C3 C3 E 36

19 cosets If H is a subgroup of G and X an element of G, the assembly {ex, hx, hx,.. hnx} where H={e, h, h,..., hn} is the right coset of H in G If H is a subgroup of G and X an element of G, the assembly {Xe, Xh, Xh,.. Xhn} where H={e, h, h,..., hn} is the left coset of H in G A coset is not necessarily a subgroup of G. If however X is an element of H, the cosets XH and HX are subgroups of G. 37 NH3, group C3v and S(3) E D F H A C B G H={E,A} ={E,σ} subgroup of G right cosets of H={E,A} in G {E,A} = {E,σ} {D,B} = {C3,σ} {F,C} = {C3,σ3} 38

20 Theorem: Two right cosets HX and HY either contain no elements in common or have all elements in common. right cosets of H={E,A} in G {E,A} = {E,σ} {B,D} = {C3,σ} {C,F} = {C3,σ3} HE HA HB HD HC HF 39 Lagrange theorem: order of a subgroup is divisor of order of group h elements in the group = n distinct cosets * order of the subgroup if the order of a group is a prime number, only trivial subgroups (identity and G) exist all groups for which the order is a prime number are abelian all cyclic groups are abelian (not viceversa) 40

21 classes an element g Y of G is conjugated to the element gx of G if gy=x gx X - where X is an element of G if g X is conjugated to gy and gy is conjugated to gz than gx is conjugated to gz a class is the totality of elements which can be obtained from a given group element by conjugation an abelian group has as many classes as elements the identity element is always a class by itself all elements in the same class have the same order elements in different classes often are physically distinct symmetries 4 classes g i = g X? g j? g X 4