C dk k ln t1 t2 eik 2 i t 1 t2 e ik i2 0 t1 t2 su

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Unformatted text preview: ik I2 = 0 t1 > t2 Su, Schrieffer, Heeger (1978) Thursday, January 6, 2011 Simplest d=1 chiral topological insulator t1 ˆ H= ￿￿ t∗ 1 † t1 ax+1 ax ˆ ˆ + † t2 ax+2 ax+1 ˆ ˆ x even I2 ∼ ￿ π −π t1 t2 t∗ 2 ￿ t∗ 1 + h.c. ￿ ￿ dk ∂k ln t1 + t2 eik 2π i Zero mode (edge state) satisfy t1 ψx + t2 ψx+2 = 0 ￿ ￿x t1 2 If x>0, exist only if t1<t2. ψx = − t2 Thursday, January 6, 2011 21 t 1 + t2 e ik I2 = 0 t1 > t2 Su, Schrieffer, Heeger (1978) 22 Topological invariants and the edge states Thursday, January 6, 2011 insulators (superconductors) are characterized by an integer invariant (Z) or a G. Volovik, 1980s Z2 quantity, respectively. The symbol ‘0’ denotes the case when there exists no topological insulator (superconductor), i.e. when all quantum ground states are topologically equivalent to the trivial state. d=2; Classes A, D, C Cartan 0 1 2 3 4 5 d 6 Complex case: A IQHE AIII Z 0 0 Z Z 0 0 Z Z 0 0 Z Z 0 0 Z Z 0 0 Z Z 0 0 Z ... ... Real case: AI BDI D p-wave s.c. DIII 3He B AII CII C CI singlet s.c. Z Z2 Z2 0 2Z 0 0 0 0 Z Z2 Z2 0 2Z 0 0 0 0 Z Z2 Z2 0 2Z 0 0 0 0 Z Z2 Z2 0 2Z 2Z 0 0 0 Z Z2 Z2 0 0 2Z 0 0 0 Z Z2 Z2 Z2 0 2Z 0 0 0 Z Z2 Z2 Z2 0 2Z 0 0 0 Z Z Z2 Z2 0 2Z 0 0 0 0 Z Z2 Z2 0 2Z 0 0 0 0 Z Z2 Z2 0 2Z 0 0 0 0 Z Z2 Z2 0 2Z ... ... ... ... ... ... ... ... 7 8 9 10 11 ... ¯ ons if and only if the target space of the NLσ M on the d -dimensional boundary allows r (i) aChiral Z2 topological term, which is the case when πd¯ (G / H ) = πd −1 (G / H ) = Z2 , or Nonchiral ZW term, which is the case when πd (G / H ) = πd¯+1 (G / H ) = Z. By using this rule in tion with 6, 2011 2 of homotopy groups, we arrive with the help of table 1 at table 3 of table Thursday, January 23 d=2; Classes A, D, C NL Domain wall Thursday, January 6, 2011 NR G. Volovik, 1980s 23 d=2; Classes A, D, C NL Domain wall Thursday, January 6, 2011 NR G. Volovik, 1980s Green’s function 23 d=2; Classes A, D, C NL NR G. Volovik, 1980s Green’s function Inverse green’s function Domain wall Thursday, January 6, 2011 23 d=2; Classes A, D, C NL NR G. Volovik, 1980s Green’s function Inverse green’s function Domain wall Construct the simplest topological invariant Thursday, January 6, 2011 23 d=2; Classes A, D, C NL NR G. Volovik, 1980s 23 Green’s function Inverse green’s function Domain wall Construct the simplest topological invariant zero mode Thursday, January 6, 2011 d=2; Classes A, D, C G. Volovik, 1980s NL Domain wall Thursday, January 6, 2011 24 NR d=2; Classes A, D, C G. Volovik, 1980s Wigner transformed Green’s function NL Domain wall Thursday, January 6, 2011 24 NR d=2; Classes A, D, C G. Volovik, 1980s Wigner transformed Green’s function Gradient (Moyal product) expansion Thursday, January 6, 2011 NL Domain wall 24 NR d=2; Classes A, D, C G. Volovik, 1980s Wigner transformed Green’s function Gradient (Moyal product) expansion Thursday, January 6, 2011 NL Domain wall 24 NR d=2; Classes A, D, C G. Volovik, 1980s Wigner transformed Green’s func...
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