qi-wu-zhang model - 2d chern...
TRANSCRIPT
Qi-Wu-Zhang model2D Chern insulator
Leon Martin
19. November 2015
Motivation
Repeat: Rice-Mele-model
Bulk behavior
Edge states
Layering 2D Chern insulators
Robustness of edge states
Motivation
I topological invariant ⇔ number of edge states:
bulk-boundary-correspondence
I 1D topological insulator → 2D topological insulator
examples
I Quantum Hall Effect
I Anomalous Quantum Hall Effect
repeat: Rice-Mele-model (1): real space
H(t) =v(t)m=N∑m=1
(|m,B〉 〈m,A|+ h.c .)
+ w(t)m=N−1∑m=1
(|m + 1,A〉 〈m,B|+ h.c .)
+ u(t)m=N∑m=1
(|m,A〉 〈m,A| − |m,B〉 〈m,B|)
repeat: Rice-Mele-model (2): k-space
periodicity:
I H(k + 2π, t) = H(k , t)
I H(k , t + T ) = H(k, t), define: Ω = 2πT
Hamiltonian:
H = d · σ
with
d(k , t) =
ν + cos Ωt + cos ksin k
sin Ωt
How to get the Hamiltonian of a 2D TI?
1 H(kx , ky ) based on RM-model
2 Fouriertransformation
Dimensional extension
cyclic time t −→ new momentum ky
k-space Hamiltonian and bulk energy states
H(k) = sin kx σx + sin ky σy + (u + cos kx + cos ky ) σz
d(kx , ky ) =
sin kxsin ky
u + cos kx + cos ky
E± = ± |d|
Dirac points:u = −2, kx = 0, ky = 0Γ-point
k-space Hamiltonian and bulk energy states
H(k) = sin kx σx + sin ky σy + (u + cos kx + cos ky ) σz
d(kx , ky ) =
sin kxsin ky
u + cos kx + cos ky
E± = ± |d|
Dirac points:u = 0, kx = 0, ky = πu = 0, kx = π, ky = 0X -points
k-space Hamiltonian and bulk energy states
H(k) = sin kx σx + sin ky σy + (u + cos kx + cos ky ) σz
d(kx , ky ) =
sin kxsin ky
u + cos kx + cos ky
E± = ± |d|
Dirac points:u = 2, kx = π, ky = πM-point
k-space Hamiltonian and bulk energy states
H(k) = sin kx σx + sin ky σy + (u + cos kx + cos ky ) σz
d(kx , ky ) =
sin kxsin ky
u + cos kx + cos ky
E± = ± |d|
no band gap
Chern number (1)
I surface of d(k) in whole BZ:torus
I origin contained?
I u shifts along dz -direction
d(kx , ky ) =
sin kxsin ky
u + cos kx + cos ky
Chern number (2)
Q =
0, if u < −2
Chern number (2)
Q =
0, if u < −2
−1, if − 2 < u < 0
Chern number (2)
Q =
0, if u < −2
−1, if − 2 < u < 0
1, if 0 < u < 2
Chern number (2)
Q =
0, if u < −2
−1, if − 2 < u < 0
1, if 0 < u < 2
0, if 2 < u
Real space Hamiltonian
H =Nx−1∑mx=1
Ny∑my=1
(|mx + 1,my 〉 〈mx ,my | ⊗
σz + i σx2
+ h.c .
)
+Nx∑
mx=1
Ny−1∑my=1
(|mx ,my + 1〉 〈mx ,my | ⊗
σz + i σy2
+ h.c.
)
+ uNx∑
mx=1
Ny∑my=1
|mx ,my 〉 〈mx ,my | ⊗ σz
Edge states
I y: periodic boundaryconditions, Ny →∞
I x: open boundaryconditions, Nx(= 10)
⇒ FT along y-direction
ky -dependent Hamiltonian
H(ky ) =Nx−1∑mx=1
(|mx + 1〉 〈mx | ⊗
σz + i σx2
+ h.c .
)
+Nx∑
mx=1
|mx〉 〈mx | ⊗ (cos ky σz + cos ky σyu ⊗ σz)
position probability: PN(mx) =∑
α∈A,B∑
my|〈ΨN |mx , α〉|2
group velocity: dEdky⇒ chirality
Edge states and edge perturbation (1)
H =Nx−1∑mx=1
(|mx + 1〉 〈mx | ⊗
σz + i σx2
+ h.c.
)
+Nx∑
mx=1
|mx〉 〈mx | ⊗ (cos ky σz + cos ky σyu ⊗ σz)
+∑
mx∈1,Nx
|mx〉 〈mx | ⊗ I(µ(mx ) + h
(mx )2 cos 2ky
)µ : onside potentialh2 : second nearest neighbor hopping
Edge states and edge perturbation (2)
I new edge states, but always pairs
I top. invariant does not change!
Higher Chern numbers
Hilbert space:
HD = HL1 ⊕HL2 ⊕ . . .⊕HLD
Hamiltonian:
HD =D∑
d=1
|d〉 〈d | ⊗ HLd +D−1∑d=1
(|d + 1〉 〈d |+ |d〉 〈d + 1|)⊗ C I2NxNy
Robustness of edge states
I clean bulk part
I disordered edge region
I rectangular region:disorder gradually to 0
Robustness of edge states
I edge states with E ≈ 0
I propagating along the edge
I leaving the clean part
I propagating along the edge
⇒ even disordered sampleconducts perfectly
Conclusion
I QWZ-model from RM-model by dimensional extension
I bulk-boundary-correspondence
I higher Chern numbers by layering
I robust edge states
Source
J.K. Asboth et al., A Short Course on Topological Insulators:Band-structure topology and edge states in one and twodimensions, arXiv:1509.02295v1 (9.9. 2015)