Topological Insulators - GDR Meso
Topological Insulators - GDR Meso 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)
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 with magnetic disorder : +J~. V ~ m ‣ 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)
- Page 1 and 2: Some Transport properties of Topolo
- Page 3 and 4: Surface States of 3D Topological In
- Page 5 and 6: Transport measurement on TI Bi2Se3
- Page 7 and 8: Transport measurement on TI Bi2Se3
- Page 9 and 10: Transport measurement on TI ‣ Bi2
- Page 11 and 12: 2D Dirac Matter A B H = ~v F ( y .k
- Page 13 and 14: Diffusion of 2D Dirac states • Tr
- Page 15 and 16: Study of diffusion of 2D Dirac stat
- Page 17 and 18: (Weak) Localization • Probability
- Page 19 and 20: (Weak) Localization / Quantum trans
- Page 21: Weak localization : Universality cl
- Page 25 and 26: Anderson Univ. classes and Topologi
- Page 27 and 28: Anderson Univ. classes and Topologi
- Page 29 and 30: Anisotropic Scattering of Dirac sta
- Page 31 and 32: Anisotropic Scattering of Dirac sta
- Page 33: Conclusion ‣ New playground for u
Weak localization : Universality classes<br />
Coherent Diffusive Regime<br />
Non-Linear Sigma Model<br />
S = 1 Z<br />
d d x Tr [@ µ Q(x)@ µ Q(x)]<br />
g<br />
Dirac Fermions<br />
↵ = ~ + scalar<br />
2⇡V Re Tr ⇥ potential<br />
j ↵ G R j G A⇤<br />
:<br />
H = ~v F ( y .k x x .k y )+V (x)<br />
Electrons + random spin orbit :<br />
1/k F l e<br />
H = (~k)2<br />
Classification : what is the target manifold<br />
➡ i.e. : how many Cooperon / Diffuson<br />
(encoded in field Q(x) )<br />
Based on symmetry argument (T-reversal / C conjugation)<br />
2m + iV SO ~ .ˆk ⇥ ˆk 0<br />
with magnetic disorder : +J~. V ~ m<br />
‣ Orthogonal / AI Class (T 2 =+1,C=0)<br />
- spin and T symmetries<br />
- 4 Diffuson + 4 Cooperon<br />
‣ Symplectic / AII Class (T 2 =-1,C=0)<br />
- no spin symmetry / T symmetry<br />
- 1 Diffuson + 1 Cooperon<br />
‣ Unitary / A Class (T=0,C=0)<br />
- no spin and no T symmetries<br />
- 1 Diffuson / 0 Cooperon<br />
SO(2n)/SO(n)xSO(n)<br />
Sp(4n)/Sp(2n)xSp(2n)<br />
U(2n)/U(n)xU(n)<br />
Based on symmetry arguments<br />
mardi 3 janvier 12<br />
Hikami, PRB (1981)<br />
Altshuler, Kravtsov and Lerner, (1991)<br />
Altland, Zirnbauer, PRB (1997)
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