Qi-Wu-Zhang model 2D Chern insulator León Martin 19. November 2015
Motivation Repeat: Rice-Mele-model Bulk behavior Edge states Layering 2D Chern insulators Robustness of edge states
Motivation topological invariant number of edge states: bulk-boundary-correspondence 1D topological insulator 2D topological insulator examples Quantum Hall Effect Anomalous Quantum Hall Effect
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
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
How to get the Hamiltonian of a 2D TI? 1 Ĥ(k x, k y ) based on RM-model 2 Fouriertransformation
Dimensional extension cyclic time t new momentum k y
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
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
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
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
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
Chern number (2) 0, if u < 2 Q =
Chern number (2) 0, if u < 2 1, if 2 < u < 0 Q =
Chern number (2) 0, if u < 2 1, if 2 < u < 0 Q = 1, if 0 < u < 2
Chern number (2) 0, if u < 2 1, if 2 < u < 0 Q = 1, if 0 < u < 2 0, if 2 < u
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.
Edge states y: periodic boundary conditions, N y x: open boundary conditions, N x (= 10) FT along y-direction
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
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
Edge states and edge perturbation (2) new edge states, but always pairs top. invariant does not change!
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
Robustness of edge states clean bulk part disordered edge region rectangular region: disorder gradually to 0
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
Conclusion QWZ-model from RM-model by dimensional extension bulk-boundary-correspondence higher Chern numbers by layering robust edge states
Source J.K. Asbóth et al., A Short Course on Topological Insulators: Band-structure topology and edge states in one and two dimensions, arxiv:1509.02295v1 (9.9. 2015)