Lecture 8 Nature of ensemble: Role of symmetry, interactions and other system conditions: Part II

Transcription

1 Lecture 8 Nature of ensemble: Role of symmetry, interactions and other system conditions: Part II We continue our discussion of symmetries and their role in matrix representation in this lecture. An example of continuous symmetry was introduced in previous lecture. Now we consider examples of discrete symmetries: (i) Unitary symmetries In presence of the exact unitary symmetries, it is often convenient to represent the generator in the eigenfunction basis of the symmetries. The structure of the matrix in such a basis turns out to be of the block form[1, 2]. This is due to zero overlap among the basis states corresponding to different eigenvalues of the symmetry operator. The size of each block depends on the other approximate symmetries in the system (lifting the degeneracy). For example, consider the Hamiltonian H of a spinless particle moving in a spherically symmetric potential. This implies [H, J] = 0 with J as the angular momentum operator. Let the eigenvalues of J 2 and J z be h 2 j(j + 1) and hm. The appropriate basis in this case can then be constructed in terms of the angular momentum states jm. Let α be a quantum number (or a set of quantum numbers) corresponding to a partially-violated symmetry (or symmetries), the basis can then be chosen as αjm. The H-matrix in this basis turns out to be of the diagonal block form, with each block matrix H (jm) corresponding to a fixed (j, m): αjm H βj m = δ jj δ mm α H (jm) β (1) Here j, m are referred as good quantum numbers. (ii) Antiunitary symmetries, no geometric symmetries Although an eigenfunction basis of the symmetry is not available in the case of an anti-unitary symmetry, a symmetry-preserving basis is still useful for matrix representation of the generator. The representation in such a basis puts constraints on the allowed class of canonical transformations (also referred as 1

2 invariant classes) for the generator. The three main anti-unitary symmetry classes often found in physical systems, referred as Dyson s threefold way, can briefly be described as follows (see [1, 2] for details): Time-reversal symmetry with integer angular momentum The time-independent Hamiltonian of a system with both time-reversal symetry T and integer angular momentum J (and no geometric symmetry e.g. parity etc present in the system) is subjected to following constraints: [H, T ] = 0 T JT 1 = J. (2) This alongwith T = UK then implies that T 2 = 1. Under these constraints, H can always be given a real-symmetric matrix representation in a generic T - invariant basis. The canonical class of tranformations which preserve the real-symmetric nature of H alongwith its eigenvalues is the orthogonal class, that is, the the transformation OHO 1 = H with O as an orthogonal operator O T.O = 1. (Note, even without T -symmetry, a Hermitian operator is always a real-symmetric matrix in its own eigenfunction basis, however an orthogonal transformation will in general change it to a complex-hermitian one). A well-known example for this cases is the Anderson Hamiltonian H, describing the dynamics of an spinless electron, moving in a lattice with on-site Gaussian disorder and non-random nearest neighbor hopping, in tight-binding approximation: H = n ɛ n a na n + n m b mn a na m. (3) In the site basis, H is a real-symmetric matrix. Exercise 1: Construct a T -invariant basis with the help of constraint T 2 = 1. Exercise 2: Show that under constraints H, T ] = 0, T 2 = 1, H is a realsymmetric matrix in a T -invariant basis. Time-reversal symmetry with half-integer angular momentum The T-symmetry of the Hamiltonian H with half-integer spin corresponds to [ H, T ] = 0, T 2 = 1. This implies that if ψ is an eigenstate of H, then (i) T ψ is also an eigenstate with ψ T ψ = 0, and, (ii) both ψ and T ψ have the same eigenvalues. This pairwise degeneracy of the eigenstates is known as Kramer s degeneracy ; this also requires that the Hilbert-space in which H operates should be even. If T is the only symmetry of the 2N 2N Hamiltonian with half-integer spin, a generic T -preserving basis then consists of the time-reversed pairs of arbitrary states, n and T n with n = 1 N. Defining the complex conjugation operator K relative to this basis, the unitary operator U in T = U.K can be 2

3 written in a block-diagonal form: U (referred as Z from now on) consists of N diagonal block Z mm, each as a 2 2 matrix: ( ) 0 1 Z mm =. (4) 1 0 The off-diagonal blocks are all zero: Z mn = 0 for m n. The matrix elements of H in this basis can be expressed in a quternion form (i.e a 2 2 matrix form): H mn = 4 mn.τ k with τ 1 = 1 and τ s = i σ s 1 for s > 1 with σ s as Pauli matrices: k=1 H(k) τ 1 = ( 0 i i 0 ) ( 0 1, τ 2 = 1 0 ) ( i 0, τ 3 = 0 i ). (5) The time-reversal symmetry along with Hermiticity subjects H mn to be realquaternion (see [1] for details on these matrices): H kl;1 = H lk;1, H kl;s = H lk;s for s > 1 (6) It is easy to show that the real-quaternion nature and eigenvalues of H are invariant under a symplectic transformation S defined by SZ S = Z (7) A common example for this case is an electron moving in a 2-dimensional disordered lattice of linear size L with spin-orbit coupling. Within tight-binding approximation, the Hamiltonian H can be described as follows: H = ɛ n n, σ n, σ + nm n, σ m, σ (8) n,σ n m,σ,σ V σσ with ɛ n as the spin-independent, on site random potential. The nearest neighbor hopping matrix elements V nm are randomly chosen 2 2 matrices describing spin rotation due to the spin-orbit coupling (SOC) on every lattice bond (n, m). In the two component spinor space (associated with each site), they are represented by the quaternions [ ] V V kl = kl;1 + iµv kl;4 µ(v kl;3 + iv kl;2 ) (9) µ(v kl;3 + iv kl;2 ) V kl;1 iµv kl;4 where µ denotes the spin-orbit coupling. Exercise 3: Construct a T -invariant basis with the help of constraint T 2 = 1. Exercise 4: Show that the symplectic transformation SHS 1 = H preserves the real-quaternion nature of H-matrix. (iii) Absence of Time-reversal symmetry The Hamiltonian matrices unrestricted by antiunitary symmetries are in general complex. It is possible to give H a real representation (for example, in 3

4 its own eigenfunction basis) but any such representation will become complex under a general canonical (i.e., unitary) transformation. The canonical class of transformation leaving the transformed H matrix as complex Hermitian is the unitary (U) transformation: consider the transformation UHU 1 = H. For H to be Hermitian, (H ) = H which requires (UHU 1 ) = UHU 1 = [ H, U.U ] = 1. (10) The non-trivial solution of this condition is U.U = 1 [2] which implies unitary nature of U. The tranformation also preserves the eigenvalues of H. Let E and B be the eigenvalue matrix and eigenvector matrix of H, then H = B.E.B. As H is complex-hermitian, this implies B is unitary. Now as H = UHU, this gives H = U.B.E.B.U = C.E.C where C = B.U. (11) Thus both H and H have the same eigenvalues although different eigenvectors. As an example, consider the Hamiltonian in eq.(3) now subjected to an Aharnov Bohm flux φ. The T -symmetry is now broken by φ which also gives rise to a nearest neighbor hopping b mn = exp(iφ) for all nearest-neighbor pairs {m, n}. As a result, H becomes a complex-hermitian matrix. Combination of anti-unitary and geometric symmetries The presence of the geometric symmetries e.g parity R alongwith timereversal changes the symmetry preserving basis and therefore the class of canonical transformations. This can be better explained by an example. Consider a time-reversal system with half-integer angular momentum J which is invariant on rotation by an angle π around an arbitrary direcion, say for example, the x-axis. The system therefore has a parity symmetry R x = exp(iπj x / h) and is subjected to following constraints [R x, T ] = 0, [R x, H] = 0, [T, H], R 2 x = 1, T 2 = 1 (12) The basis preserving all the above constraints can be chosen as follows. As mentioned above, in presence of T only (with T 2 = 1), the appropriate basis is N pairs of states n, T n. In presence of R x only, the symmetry-preserving basis is the one ordered by even (+) and odd ( ) parity. Consequently the basis preserving both the symmeries is n ± which satisfies R x n ± = ±i n ± R x T n ± = T R x n ± = i T n ±. (13) If the basis is organized such that T n ± = ± n, the matrix H then becomes block-structured, with four N N blocks: ( ) H+ 0 H = (14) 0 H 4

5 with block H ± corresponding to matrix elements between states of same parity: H + mn = m + H n+, H mn = m H n. (15) with m, n = 1 N The zero off-diagonal blocks are the result of zero-overlap between states of different parity. It is easy to show that H + mn = (H mn) T and therefore both blocks have same eigenvalues which is again a manifestation of Kramers degeneracy. Further as both H +, H are complex-hermitian, the group of canonical transformations is now the unitary group instead of symplectic group (which was the case in absence of parity). The presence of additional symmetries introduce new constraints on the matrix elements. For example, in presence of another parity R y along with R x and T, a system with half-integer angular momentum is subjected to following conditions besides eq.(12), [R y, T ] = 0, [R y, H] = 0, R x R y + R y R x = 0, R 2 y = 1 (16) Again as in the previous example, the unitary nature of R y and its invariance gives H + = H while antiunitary nature of T gives H + = (H ) T = (H ) ; the simultaneous presence of both therefore reduces both H + and H to real matrices: H ± = (H ± ). The additional parity therefore reduces the admissible class of transformation from unitary to orthogonal. References [1] M.L.Mehta,Random Matrices, (Academic Press, New York, 1991). [2] F.Haake, Quantum Signatures of Chaos (Springer, Berlin, 1991). [3] B.D. Simons, A. Altland, Theories of Mesoscopic Physics, CRM Series in Mathematical Physics (Springer, 2001). 5