Time-Reversal Symmetric Two-Dimensional Topological Insulators: The Bernevig-Hughes-Zhang Model

Transcription

1 Time-Reversal Symmetric Two-Dimensional Topological Insulators: The Bernevig-Hughes-Zhang Model Alexander Pearce Intro to Topological Insulators: Week 5 November 26, / 22

2 This notes are based on J.K. Asbóth, L. Oroszlány, A. Pályi. A Short Course on Topological Insulators (2015). and J.J. Sakurai. Modern Quantum Mechanics, (1994). Alexander Pearce Intro to Topological Insulators: Week 5 November 26, / 22

3 Contents Time Reversal Symmetry Kramers Degeneracy The Bernevig-Hughes-Zhang model Edge States The Z 2 invariant Absence of Back Scattering Alexander Pearce Intro to Topological Insulators: Week 5 November 26, / 22

4 Time Reversal Symmetry We define a time reversal operator which acts as n T n Where T n is the time reversed state. For example: if n = k we expect T k = k ; if n = x we expect T x = x. We also say a Hamiltonian H has time reversal symmetry if T HT 1 = H. How do we define this in a rigorous way? Alexander Pearce Intro to Topological Insulators: Week 5 November 26, / 22

5 Time Reversal Symmetry The infinitesimal time evolution of a state n is generated by the Hamiltonian, and is given by n, t 0 = 0; t = δt = (1 ih δt ) n If we consider applying the several operator at t = 0, then time evolving the state, then at t = δt we have ( 1 ih δt) T n. Intuitively we expect that this should equal T n, t 0 = 0; t = δt. Together this gives (1 ih ) δt T n = T (1 ih ) ( δt) n If this is to be true for any state n we find iht n = T ih n Alexander Pearce Intro to Topological Insulators: Week 5 November 26, / 22

6 Time Reversal Symmetry So we found: iht n = T ih n If T was a unitary operator then we may directly cancel the i s and find HT = T H. This does not work: as it implies HT n = T H n = ɛ n T n Therefore we find that the time reversal operator must be anti unitary (T c n = c T n, where c C), this means T H = HT Alexander Pearce Intro to Topological Insulators: Week 5 November 26, / 22

7 Time Reversal Symmetry What is the effect of time reversal on the wave function? T n = d 3 x T ( x x n ) T n = d 3 k T ( k k n ) = d 3 x T x x n = d 3 k T k k n = d 3 x x x n = d 3 k k k n = d 3 k k k n So this shows that T ψ(x ) ψ(x ) and T ψ(p ) ψ( p ). The representation we choose crucially matters! We also see that two successive applications gives T 2 = 1. We will now show that this story differs for spinfull particles. Alexander Pearce Intro to Topological Insulators: Week 5 November 26, / 22

8 Time Reversal Symmetry The spin eigenstates of a spin half particle can be written as ˆn, + = e iszα/ e isyβ/ + and ˆn, = e iszα/ e isy(π+β)/. Now, noting that T ˆn, + = ˆn, we deduce T = ηe iπjy/ K T ( T j, m j, m n ) = T ( η e iπjy/ j, m j, m n ) = η 2 e 2iπJy/ j, m j, m n Using the properties of the angular momentum eigenstates under rotation we note e 2iπJy/ j, m = ( 1) 2j j, m, therefore we find T 2 j half-integer = j half-integer T 2 j integer = + j integer Alexander Pearce Intro to Topological Insulators: Week 5 November 26, / 22

9 Kramers Degeneracy If we consider a system of charged particles in a static electric field, with V (x) = eφ(x) then [T, H] = 0 will still holds as the electrostatic potential is a real function of the time-reversal operator x. We consider a state n and its time reversed parter T n, these must have the same energy - following HT n = T H n = ɛ n T n. So are these the same state or different states? Alexander Pearce Intro to Topological Insulators: Week 5 November 26, / 22

10 Kramers Degeneracy The states n and T n can only differ by a phase η with η = e iδ. Now applying T once more T n = e iδ n T 2 n = T e iδ n = e iδ T n = e iδ e iδ n = + n Which cannot hold for spin one-half particle. So we must conclude that these are different degenerate states. This holds as long at the system obeys time-reversal symmetry! Alexander Pearce Intro to Topological Insulators: Week 5 November 26, / 22

11 Time Reversal Symmetry of Bulk Hamiltonian Earlier we noted that T ψ(p ) ψ( p ). It is also useful to realise we can split the time reversal operator into two parts T = τk where K is an operator which complex conjugates and τ is a unitary operator which acts on internal degrees of freedom. T H Bulk T 1 = k k k T H(k)T 1 = k k k τh ( k)τ 1 So τh ( k)τ 1 = H(k) in the TRS bulk, and therefore the Hamiltonian must be symmetric to inversion in the Brillouin zone. Alexander Pearce Intro to Topological Insulators: Week 5 November 26, / 22

12 The Bernevig-Hughes-Zhang model How do we construct time reversal invariant Hamiltonians from a lattice model? We take two copies of the system, where ones H = KHK and couple in the same manner as we did with the Chern insulators. H TRI = ( H C C H ) Alexander Pearce Intro to Topological Insulators: Week 5 November 26, / 22

13 The Bernevig-Hughes-Zhang model We look for a coupling with T 2 = 1. We have two options: we choose T = s x K. This gives s x KH TRI (s x K) 1 = H TRI If we chose the coupling such that C = C T. Additionally we may choose an even simpler solution with T = K and not even double the Hilbert space H TRI = H + H 2 = KH TRI K Alexander Pearce Intro to Topological Insulators: Week 5 November 26, / 22

14 The Bernevig-Hughes-Zhang model We look for a coupling with T 2 = 1, therefore we choose T = is y K. This gives is y KH TRI (is y K) 1 = H TRI If we chose the coupling such that C = C T. So we see different types of TRS must be satisfied by different couplings. Alexander Pearce Intro to Topological Insulators: Week 5 November 26, / 22

15 The Bernevig-Hughes-Zhang model We now introduce a toy model for a time-reversal invariant topological insulator. H BHZ (k) = s 0 [ (u+cos k x +cos k y )σ z +sin k y σ y ] +sz sin k x σ x +s x Ĉ For C = 0 this reduces to a 4 band model of HgTe. Alexander Pearce Intro to Topological Insulators: Week 5 November 26, / 22

16 Edge States Alexander Pearce Intro to Topological Insulators: Week 5 November 26, / 22

17 The Z 2 invariant T 2 = 1 demands that we have and equal number of left movers and right moves at any given energy in the bulk gap. N + (E) = N (E) Due to Kramers degeneracy adiabatic deformations of the edge states can only change the number of edge states by 4, a pair of kramers pairs. Therefore we define the Z 2 as the party of the number of Kramers pairs, this is a topological invariant. Z 2 = 0(1) for even(odd) number of Kramers pairs. Alexander Pearce Intro to Topological Insulators: Week 5 November 26, / 22

18 Absence of Back Scattering How does edge state transport behave under the influence of local disorder? Alexander Pearce Intro to Topological Insulators: Week 5 November 26, / 22

19 Absence of Back Scattering Consider a lattice model where the unit cells form a square lattice of size N x N y. If the system is translational invariant in the x-direction we may write the Bloch states as l± = k l,± Φ l,± where the momentum eigenstates are given by k = 1 N x e ikmx m x and Φ l,± incorporates the transverse modes N x m x=1 and the sublattice degree of freedom. We will use these current-normalised states defining 1 l± c = vl,± l± with v l,± being the group velocity. Alexander Pearce Intro to Topological Insulators: Week 5 November 26, / 22

20 Absence of Back Scattering A wave incident on the scattering region is characterized by a vectors of coefficients, in the lead index L, R ( ) a (in) = a (in) L,1, a(in) L,2,..., a(in) L,N, a(in) R,1, a(in) R,2,..., a(in) R,N a (out) = ( a (out) L,1, a(out) L,2,..., a(out) L,N, a(out) The corresponding energy eigenstate reads ψ = N l=1 a (in) R,1, a(out) R,2,..., a(out) R,N L,l l, +, L c + a (out) L,l l,, L c + a (in) R,l l,, R c + a (out) L,l l, +, R c The scattering matrix S relates the two vectors a (out) = Sa (in) S = ( r t t r ) ) Alexander Pearce Intro to Topological Insulators: Week 5 November 26, / 22

21 Absence of Back Scattering Kramers degeneracy gives the connection between states. l,, L c = T l, +, L c l, +, R c = T l,, R c We may write the eigenstates outside the scattering region as ψ = N l=1 a (in) L,l l, +, L c + (Sa (in) ) L,l l,, L c + a (in) R,l l,, R c + (Sa (in) ) L,l l, +, R c Alexander Pearce Intro to Topological Insulators: Week 5 November 26, / 22

22 Absence of Back Scattering We now look for the time reversed state T ψ = N l=1 a (in) L,l l,, L c + (S a (in) ) L,l l, +, L c a (in) R,l l, +, R c + (S a (in) ) L,l l,, R c Comparing ψ and T ψ we must conclude S = We gives r = r therefore r = 0. ( r t ) ( t r = S T r t ) = t r Alexander Pearce Intro to Topological Insulators: Week 5 November 26, / 22