Qi-Wu-Zhang model. 2D Chern insulator. León Martin. 19. November 2015

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1 Qi-Wu-Zhang model 2D Chern insulator León Martin 19. November 2015

2 Motivation Repeat: Rice-Mele-model Bulk behavior Edge states Layering 2D Chern insulators Robustness of edge states

3 Motivation topological invariant number of edge states: bulk-boundary-correspondence 1D topological insulator 2D topological insulator examples Quantum Hall Effect Anomalous Quantum Hall Effect

4 repeat: Rice-Mele-model (1): real space m=n Ĥ(t) =v(t) ( m, B m, A + h.c.) m=1 m=n 1 + w(t) ( m + 1, A m, B + h.c.) m=1 m=n + u(t) ( m, A m, A m, B m, B ) m=1

5 repeat: Rice-Mele-model (2): k-space periodicity: Ĥ(k + 2π, t) = Ĥ(k, t) Ĥ(k, t + T ) = Ĥ(k, t), define: Ω = 2π T Hamiltonian: Ĥ = d σ with ν + cos Ωt + cos k d(k, t) = sin k sin Ωt

6 How to get the Hamiltonian of a 2D TI? 1 Ĥ(k x, k y ) based on RM-model 2 Fouriertransformation

7 Dimensional extension cyclic time t new momentum k y

8 k-space Hamiltonian and bulk energy states Ĥ(k) = sin k x ˆσ x + sin k y ˆσ y + (u + cos k x + cos k y ) ˆσ z sin k x d(k x, k y ) = sin k y u + cos k x + cos k y E ± = ± d Dirac points: u = 2, k x = 0, k y = 0 Γ-point

9 k-space Hamiltonian and bulk energy states Ĥ(k) = sin k x ˆσ x + sin k y ˆσ y + (u + cos k x + cos k y ) ˆσ z sin k x d(k x, k y ) = sin k y u + cos k x + cos k y E ± = ± d Dirac points: u = 0, k x = 0, k y = π u = 0, k x = π, k y = 0 X -points

10 k-space Hamiltonian and bulk energy states Ĥ(k) = sin k x ˆσ x + sin k y ˆσ y + (u + cos k x + cos k y ) ˆσ z sin k x d(k x, k y ) = sin k y u + cos k x + cos k y E ± = ± d Dirac points: u = 2, k x = π, k y = π M-point

11 k-space Hamiltonian and bulk energy states Ĥ(k) = sin k x ˆσ x + sin k y ˆσ y + (u + cos k x + cos k y ) ˆσ z sin k x d(k x, k y ) = sin k y u + cos k x + cos k y E ± = ± d no band gap

12 Chern number (1) surface of d(k) in whole BZ: torus origin contained? u shifts along d z -direction sin k x d(k x, k y ) = sin k y u + cos k x + cos k y

13 Chern number (2) 0, if u < 2 Q =

14 Chern number (2) 0, if u < 2 1, if 2 < u < 0 Q =

15 Chern number (2) 0, if u < 2 1, if 2 < u < 0 Q = 1, if 0 < u < 2

16 Chern number (2) 0, if u < 2 1, if 2 < u < 0 Q = 1, if 0 < u < 2 0, if 2 < u

17 Real space Hamiltonian N x 1 Ĥ = N y m x =1 m y =1 + + u N x N y 1 m x =1 m y =1 N x ( m x + 1, m y m x, m y ˆσ z + i ˆσ x 2 N y m x =1 m y =1 ( m x, m y + 1 m x, m y ˆσ z + i ˆσ y 2 m x, m y m x, m y ˆσ z ) + h.c. ) + h.c.

18 Edge states y: periodic boundary conditions, N y x: open boundary conditions, N x (= 10) FT along y-direction

19 k y -dependent Hamiltonian Ĥ(k y ) = N x 1 m x =1 + N x m x =1 ( m x + 1 m x ˆσ z + i ˆσ x 2 ) + h.c. m x m x (cos k y ˆσ z + cos k y ˆσ y u ˆσ z ) position probability: P N (m x ) = α A,B m y Ψ N m x, α 2 group velocity: de dk y chirality

20 Edge states and edge perturbation (1) Ĥ = N x 1 m x =1 + N x m x =1 + ( m x + 1 m x ˆσ z + i ˆσ x 2 m x {1,N x } µ : onside potential h 2 : second nearest neighbor hopping ) + h.c. m x m x (cos k y ˆσ z + cos k y ˆσ y u ˆσ z ) ( ) m x m x Î µ (mx ) + h (mx ) 2 cos 2k y

21 Edge states and edge perturbation (2) new edge states, but always pairs top. invariant does not change!

22 Higher Chern numbers Hilbert space: H D = H L1 H L2... H LD Hamiltonian: Ĥ D = D d=1 D 1 d d Ĥ Ld + ( d + 1 d + d d + 1 ) CÎ 2Nx N y d=1

24 Robustness of edge states clean bulk part disordered edge region rectangular region: disorder gradually to 0

25 Robustness of edge states edge states with E 0 propagating along the edge leaving the clean part propagating along the edge even disordered sample conducts perfectly

26 Conclusion QWZ-model from RM-model by dimensional extension bulk-boundary-correspondence higher Chern numbers by layering robust edge states

27 Source J.K. Asbóth et al., A Short Course on Topological Insulators: Band-structure topology and edge states in one and two dimensions, arxiv: v1 ( )

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