Topological Insulators - GDR Meso

Weak localization : Universality classes
Diagrammatic techniques
Kubo formula
↵ = ~
2⇡V Re Tr ⇥ j ↵ G R j G A⇤
Coherent Diffusive Regime
Non-Linear Sigma Model
S = 1 g
Z
d d x Tr [@ µ Q(x)@ µ Q(x)]
Perturbative expension in
1/k F l e
Correlate sequences of scatterers in same / opposite order
➡Diffusons / Cooperon propagators
Average conductance
Fluctuations of conductance
⃗k ′
⃗ k
⃗ Q − ⃗ k, EF − Ω
Γ C (Ω, ⃗ Q)
H C ⃗Q − K, ⃗
Q ⃗ − k ⃗′ ,EF − Ω
EF − Ω
Γ C (−Ω, Q) ⃗
Based on microscopic Hamiltonian
⃗Q − ⃗K ′ ,
EF − Ω
⃗k ′ k ⃗′
⃗ k ⃗ k k ⃗′ k ⃗′
⃗ k
k ⃗′
⃗
⃗ k ⃗ k ⃗ k ⃗ k k ⃗ k
H C
⃗K
⃗K ′
⃗k ′
⃗ k
H C
H D
⃗ k − ⃗q, EF − Ω
⃗ k ′ − ⃗q, EF − Ω
Γ D (Ω, ⃗q)
Γ D (−Ω, −⃗q)
Corrections of Hikami boxes (anisotropic scattering !)
mardi 3 janvier 12
Γ C ( ⃗ Q)
H D = + +
⃗k ′
⃗ k
H0 D H1 D H2
D
End Formula depends only on number of
Cooperons / Diffusons : How many Singlet / Triplets ?
See e.g. Book by Akkermans and Montambaux
⃗K ′ − ⃗q,
EF − Ω
⃗K − ⃗q,
EF − Ω
⃗k ′
H D
⃗k ′
⃗K
⃗K ′
⃗ k
⃗ k
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)
‣ 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
Based on symmetry arguments
SO(2n)/SO(n)xSO(n)
Sp(4n)/Sp(2n)xSp(2n)
U(2n)/U(n)xU(n)
Hikami, PRB (1981)
Altshuler, Kravtsov and Lerner, (1991)
Altland, Zirnbauer, PRB (1997)