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Lecture13

# Volovik 1980s wigner transformed greens function

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Unformatted text preview: tion Gradient (Moyal product) expansion NR NL Domain wall 4-dim vector, space ω, px, ps, s Thursday, January 6, 2011 24 d=2; Classes A, D, C G. Volovik, 1980s Wigner transformed Green’s function Gradient (Moyal product) expansion 24 NR NL Domain wall 4-dim vector, space ω, px, ps, s #(zero modes) = NR − NL Thursday, January 6, 2011 d=2; Classes A, D, C NL G. Volovik, 1980s NR Domain wall #(zero modes) = NR − NL Thursday, January 6, 2011 24 d=1, classes AIII, BDI, CII IL 25 IR s Q(ω , ps , s) = G (ω , ps , s) Σ G(ω , ps , s) ￿∞ ￿∞ 1 I (s) = tr dω dps Q (∂ω Q∂ps Q − ∂ps Q∂ω Q) 16π i 0 −∞ ￿ 1 I (L) − I (−L) = lim tr dxdk Q (∂x Q∂ps Q − ∂ps Q∂x Q) 16π i ω→0 −1 Thursday, January 6, 2011 d=1, classes AIII, BDI, CII IL 25 IR s Q(ω , ps , s) = G (ω , ps , s) Σ G(ω , ps , s) ￿∞ ￿∞ 1 I (s) = tr dω dps Q (∂ω Q∂ps Q − ∂ps Q∂ω Q) 16π i 0 −∞ ￿ 1 I (L) − I (−L) = lim tr dxdk Q (∂x Q∂ps Q − ∂ps Q∂x Q) 16π i ω→0 −1 #(zero modes) = lim ω tr Σ K ∂ω G ω →0 Thursday, January 6, 2011 d=1, classes AIII, BDI, CII IL 25 IR s Q(ω , ps , s) = G (ω , ps , s) Σ G(ω , ps , s) ￿∞ ￿∞ 1 I (s) = tr dω dps Q (∂ω Q∂ps Q − ∂ps Q∂ω Q) 16π i 0 −∞ ￿ 1 I (L) − I (−L) = lim tr dxdk Q (∂x Q∂ps Q − ∂ps Q∂x Q) 16π i ω→0 −1 #(zero modes) = lim ω tr Σ K ∂ω G ω →0 gradient expansion # (zero modes) = I (L) − I (−L) Thursday, January 6, 2011 Other classes of topological insulators Relationship between the edge states and the Green’s function topological invariant 1. All nonchiral classes in even d higher than 2: A.W.W. Ludwig, A. Essin, VG, 2010 (in preparation) 2. Chiral classes in odd d higher than 1. A. Essin, VG, 2010 (in preparation) 3. Z2 topological invariants, A. Essin, VG, 2010 (in preparation) Thursday, January 6, 2011 26 27 Topological invariants in the presence of interactions Thursday, January 6, 2011 The invariant at d=0, D=1 with interactions No interactions Thursday, January 6, 2011 28 VG, 2010 The invariant at d=0, D=1 with interactions No interactions In the presence of interactions Thursday, January 6, 2011 28 VG, 2010 Matrix elements Dh 1 No interactions iω − ￿n VG, 2010 Matrix elements G= Df The invariant at d=0, D=1 with interactions 28 In the presence of interactions det G = Thursday, January 6, 2011 ￿Dh −Df n=1 ￿ Dh n=1 (iω − rn ) (iω − ￿n ) zeroes of the Green’s function poles of the Green’s function The invariant at d=0, D=1 with interactions 28 VG, 2010 No interactions In the presence of interactions det G = Thursday, January 6, 2011 ￿Dh −Df n=1 ￿ Dh n=1 (iω − rn ) (iω − ￿n ) zeroes of the Green’s function poles of the Green’s function The invariant at d=0, D=1 with interactions 29 VG, 2010 Switching on interactions det G = Thursday, January 6, 2011 ￿Dh −Df n=1 ￿ Dh n=1 (iω − rn ) (iω − ￿n ) The invariant at d=0, D=1 with interactions G. Volovik, 2006 VG, 2010 Switching on interactions det G = Thursday, January 6, 2011 29 ￿Dh −Df n=1 ￿ Dh n=1 (iω − rn ) (iω − ￿n ) The invariant at...
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