Topological Insulators - GDR Meso

Weak localization : Universality classes
Coherent Diffusive Regime
Non-Linear Sigma Model
S = 1 Z
d d x Tr [@ µ Q(x)@ µ Q(x)]
g
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
Classification : what is the target manifold
➡ i.e. : how many Cooperon / Diffuson
(encoded in field Q(x) )
Based on symmetry argument (T-reversal / C conjugation)
2m + iV SO ~ .ˆk ⇥ ˆk 0 ‣ 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 Diffuson / 0 Cooperon
SO(2n)/SO(n)xSO(n)
Sp(4n)/Sp(2n)xSp(2n)
U(2n)/U(n)xU(n)
Based on symmetry arguments
mardi 3 janvier 12
Hikami, PRB (1981)
Altshuler, Kravtsov and Lerner, (1991)
Altland, Zirnbauer, PRB (1997)