Topological Insulators - GDR Meso

Anderson Univ. classes and Topological Order
Non-Linear Sigma Model
S = 1 Z
d d x Tr [@ µ Q(x)@ µ Q(x)]
g
Not Topologically Protected Topologically Protected
Dirac Fermions
↵ = ~ + scalar
2⇡V Re Tr ⇥ potential
j ↵ G R j G A⇤
:
H = ~v F ( y .k x x .k y )+V (x)
Electrons + random spin orbit :
1/k F l e
H = (~k)2
‣ Orthogonal / AI Class (T 2 =+1,C=0)
- spin and T symmetries
- 4 Diffuson + 4 Cooperon
‣ Symplectic / AII Class (T 2 =-1,C=0)
- no spin symmetry / T symmetry
- 1 Diffuson + 1 Cooperon
‣ Unitary / A Class (T=0,C=0)
- no spin and no T symmetries
- 1 Diffusion / 0 Cooperon
SO(2n)/SO(n)xSO(n)
Sp(4n)/Sp(2n)xSp(2n)
U(2n)/U(n)xU(n)
2m + iV SO ~ .ˆk ⇥ ˆk 0 ‣ AII Class in d=2 : Topological Term allowed
But : Topological Term does not contribute to the weak
localization of surface states
➡ both AII / Symplectic classes equivalent
➡ known results for electrons + random SO sufficient
➡ Prevents Anderson localization
➡ 2 subclasses
- top. term (cf Berry Phase) : Top. Protection
from d+1 Bulk
- no top. term : standard AII class
‣ Classification of all (d+1) topological phases from their d-
surface states properties
S. Ryu et al., NJP 12 (2010)
mardi 3 janvier 12