Single particle Green s functions and interacting topological insulators
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1 1 Single particle Green s functions and interacting topological insulators Victor Gurarie Nordita, Jan 2011
2 Topological insulators are free fermion systems characterized by topological invariants. 2 In this talk 1. All the invariants can be constructed out of single particle Green s functions of these insulators 2. It is generally believed that at the boundaries of topological insulators there must be zero energy edge states. The Green s functions provide a very simple proof of this statement. 3. In the presence of interactions, edge states can disappear and get replaced by the zeroes of the Green s functions. VG, arxiv: A. Essin, VG, work in progress Discussions with A.W.W. Ludwig
3 3 Noninteracting topological insulators
4 Topological insulators 4 Topological insulators are free fermion systems fermionic creation and annihilation operators which happen to be band insulators of a special type
5 Band insulators 5 Spectrum is essentially the same regardless of whether the boundary conditions in the y-direction periodic or hard wall.
6 Integer quantum Hall effect as a topogical insulator 6 Periodic boundary conditions in the y-direction Same tight binding model but with 2π/3 magnetic flux through each plaquette
7 Integer quantum Hall effect as a topogical insulator 6 Hard wall boundary conditions in the y-direction Same tight binding model but with 2π/3 magnetic flux through each plaquette
8 Integer quantum Hall effect as a topogical insulator 6 Hard wall boundary conditions in the y-direction Same tight binding model but with 2π/3 magnetic flux through each plaquette Edge states
9 Laughlin s argument 7 increasing flux though a point outward current Outward current deposits charge somewhere. There must be zero energy edge states to absorb the charge
10 TKNN invariant 8 Thouless, Kohmoto, Nightingale, Den Nijs, 1982 { This is a topological invariant (always integer times 2πi) Bloch waves
11 TKNN invariant 9 Band structure topological invariant quantized Hall conductance Laughlin argument edge states
12 Other topological insulators? 10
13 Other topological insulators? D px+ipy superconductor ( insulator since its Bogoliubov quasiparticles have a gap in the spectrum). Kopnin, Salomaa, Read and Green, 2000.
14 Other topological insulators? D px+ipy superconductor ( insulator since its Bogoliubov quasiparticles have a gap in the spectrum). Kopnin, Salomaa, Read and Green, D quantum Hall effect. Zhang, Hu, 2001
15 Other topological insulators? D px+ipy superconductor ( insulator since its Bogoliubov quasiparticles have a gap in the spectrum). Kopnin, Salomaa, Read and Green, D quantum Hall effect. Zhang, Hu, Solitons in 1D chains. Su, Schrieffer, Heeger (1978)
16 Other topological insulators? D px+ipy superconductor ( insulator since its Bogoliubov quasiparticles have a gap in the spectrum). Kopnin, Salomaa, Read and Green, D quantum Hall effect. Zhang, Hu, Solitons in 1D chains. Su, Schrieffer, Heeger (1978) 4. Modern 2D and 3D topological insulators. Kane and Mele (2005); Zhang, Hughes, Bernevig, (2006); Moore, Balents, (2007); Fu, Kane, Mele (2007)
17 Altland-Zirnbauer s tenfold way 11
18 Altland-Zirnbauer s tenfold way 11 Symmetries: 1. Time-reversal
19 Altland-Zirnbauer s tenfold way 11 Symmetries: 1. Time-reversal 2. Particle-hole
20 Altland-Zirnbauer s tenfold way 11 Symmetries: 1. Time-reversal 2. Particle-hole 3. Chiral (sublattice)
21 Altland-Zirnbauer s tenfold way 11 Symmetries: 1. Time-reversal 2. Particle-hole 3. Chiral (sublattice) localization physics at long w Cartan label T C S A (unitary) AI (orthogonal) AII (symplectic) AIII (ch. unit.) BDI (ch. orth.) CII (ch. sympl.) D (BdG) C (BdG) DIII (BdG) CI (BdG) From: Ryu, Schnyder, Furusaki, Ludwig, 2010
22 Altland-Zirnbauer s tenfold way 11 Symmetries: 1. Time-reversal 2. Particle-hole 3. Chiral (sublattice) IQHE spin-orbit coupling coupling: modern top. ins. Su-Schrieffer-Heeger solitons p-wave spin-polarized superconductors 3 He, phase B s-wave superconductors localization physics at long w Cartan label T C S A (unitary) AI (orthogonal) AII (symplectic) AIII (ch. unit.) BDI (ch. orth.) CII (ch. sympl.) D (BdG) C (BdG) DIII (BdG) CI (BdG) From: Ryu, Schnyder, Furusaki, Ludwig, 2010
23 Classification table of topological 12 insulators and superconductors topologically equivalent to the trivial state. Table from Ryu, Schnyder, Furusaki, Ludwig, 2010 d space dimensionality Cartan Complex case: A Z 0 Z 0 Z 0 Z 0 Z 0 Z 0... AIII 0 Z 0 Z 0 Z 0 Z 0 Z 0 Z... Real case: AI Z Z 0 Z 2 Z 2 Z BDI Z 2 Z Z 0 Z 2 Z 2 Z D Z 2 Z 2 Z Z 0 Z 2 Z 2 Z 0... DIII 0 Z 2 Z 2 Z Z 0 Z 2 Z 2 Z... AII 2Z 0 Z 2 Z 2 Z Z 0 Z 2 Z 2... CII 0 2Z 0 Z 2 Z 2 Z Z 0 Z 2... C 0 0 2Z 0 Z 2 Z 2 Z Z 0... CI Z 0 Z 2 Z 2 Z Z... symmetry classes Kitaev, 2009; Ludwig, Ryu, Schnyder, Furusaki, 2009.
24 Classification table of topological 12 insulators and superconductors topologically equivalent to the trivial state. Table from Ryu, Schnyder, Furusaki, Ludwig, 2010 d space dimensionality Cartan IQHE Complex case: A Z 0 Z 0 Z 0 Z 0 Z 0 Z 0... AIII 0 Z 0 Z 0 Z 0 Z 0 Z 0 Z... Real case: AI Z Z 0 Z 2 Z 2 Z BDI Z 2 Z Z 0 Z 2 Z 2 Z D Z 2 Z 2 Z Z 0 Z 2 Z 2 Z 0... DIII 0 Z 2 Z 2 Z Z 0 Z 2 Z 2 Z... AII 2Z 0 Z 2 Z 2 Z Z 0 Z 2 Z 2... CII 0 2Z 0 Z 2 Z 2 Z Z 0 Z 2... C 0 0 2Z 0 Z 2 Z 2 Z Z 0... CI Z 0 Z 2 Z 2 Z Z... symmetry classes Kitaev, 2009; Ludwig, Ryu, Schnyder, Furusaki, 2009.
25 Classification table of topological 12 insulators and superconductors topologically equivalent to the trivial state. Table from Ryu, Schnyder, Furusaki, Ludwig, 2010 d space dimensionality Cartan IQHE Su, Schrieffer, Heeger Complex case: A Z 0 Z 0 Z 0 Z 0 Z 0 Z 0... AIII 0 Z 0 Z 0 Z 0 Z 0 Z 0 Z... Real case: AI Z Z 0 Z 2 Z 2 Z BDI Z 2 Z Z 0 Z 2 Z 2 Z D Z 2 Z 2 Z Z 0 Z 2 Z 2 Z 0... DIII 0 Z 2 Z 2 Z Z 0 Z 2 Z 2 Z... AII 2Z 0 Z 2 Z 2 Z Z 0 Z 2 Z 2... CII 0 2Z 0 Z 2 Z 2 Z Z 0 Z 2... C 0 0 2Z 0 Z 2 Z 2 Z Z 0... CI Z 0 Z 2 Z 2 Z Z... symmetry classes Kitaev, 2009; Ludwig, Ryu, Schnyder, Furusaki, 2009.
26 Classification table of topological 12 insulators and superconductors topologically equivalent to the trivial state. Table from Ryu, Schnyder, Furusaki, Ludwig, 2010 d space dimensionality Cartan IQHE Su, Schrieffer, Heeger D p-wave supercond uctor Complex case: A Z 0 Z 0 Z 0 Z 0 Z 0 Z 0... AIII 0 Z 0 Z 0 Z 0 Z 0 Z 0 Z... Real case: AI Z Z 0 Z 2 Z 2 Z BDI Z 2 Z Z 0 Z 2 Z 2 Z D Z 2 Z 2 Z Z 0 Z 2 Z 2 Z 0... DIII 0 Z 2 Z 2 Z Z 0 Z 2 Z 2 Z... AII 2Z 0 Z 2 Z 2 Z Z 0 Z 2 Z 2... CII 0 2Z 0 Z 2 Z 2 Z Z 0 Z 2... C 0 0 2Z 0 Z 2 Z 2 Z Z 0... CI Z 0 Z 2 Z 2 Z Z... symmetry classes Kitaev, 2009; Ludwig, Ryu, Schnyder, Furusaki, 2009.
27 Classification table of topological 12 insulators and superconductors topologically equivalent to the trivial state. Table from Ryu, Schnyder, Furusaki, Ludwig, 2010 d space dimensionality Cartan IQHE Su, Schrieffer, Heeger D p-wave supercond uctor Complex case: A Z 0 Z 0 Z 0 Z 0 Z 0 Z 0... AIII 0 Z 0 Z 0 Z 0 Z 0 Z 0 Z... Real case: AI Z Z 0 Z 2 Z 2 Z BDI Z 2 Z Z 0 Z 2 Z 2 Z D Z 2 Z 2 Z Z 0 Z 2 Z 2 Z 0... DIII 0 Z 2 Z 2 Z Z 0 Z 2 Z 2 Z... AII 2Z 0 Z 2 Z 2 Z Z 0 Z 2 Z 2... CII 0 2Z 0 Z 2 Z 2 Z Z 0 Z 2... C 0 0 2Z 0 Z 2 Z 2 Z Z 0... CI Z 0 Z 2 Z 2 Z Z... symmetry classes 3 He, phase B Kitaev, 2009; Ludwig, Ryu, Schnyder, Furusaki, 2009.
28 Classification table of topological 12 insulators and superconductors topologically equivalent to the trivial state. Table from Ryu, Schnyder, Furusaki, Ludwig, 2010 d space dimensionality Cartan IQHE Su, Schrieffer, Heeger D p-wave supercond uctor Complex case: A Z 0 Z 0 Z 0 Z 0 Z 0 Z 0... AIII 0 Z 0 Z 0 Z 0 Z 0 Z 0 Z... Real case: AI Z Z 0 Z 2 Z 2 Z BDI Z 2 Z Z 0 Z 2 Z 2 Z D Z 2 Z 2 Z Z 0 Z 2 Z 2 Z 0... DIII 0 Z 2 Z 2 Z Z 0 Z 2 Z 2 Z... AII 2Z 0 Z 2 Z 2 Z Z 0 Z 2 Z 2... CII 0 2Z 0 Z 2 Z 2 Z Z 0 Z 2... C 0 0 2Z 0 Z 2 Z 2 Z Z 0... CI Z 0 Z 2 Z 2 Z Z... symmetry classes New Kane-Mele topological insulators 3 He, phase B Kitaev, 2009; Ludwig, Ryu, Schnyder, Furusaki, 2009.
29 Chiral symmetry 13
30 Chiral symmetry 13 Often realized as hopping on a bipartite lattice
31 Chiral symmetry 13 Often realized as hopping on a bipartite lattice Properties of chiral systems All levels come in pairs
32 Chiral symmetry 13 Often realized as hopping on a bipartite lattice Properties of chiral systems All levels come in pairs { right zero modes left zero modes
33 Chiral symmetry 13 Often realized as hopping on a bipartite lattice Properties of chiral systems All levels come in pairs { right zero modes left zero modes #R-#L is a topological invariant (index theorem)
34 Chiral vs nonchiral systems 14 Non-chiral systems Chiral systems can be characterized by an integer topological invariant in even spacial dimensions only can be characterized by an integer topological invariant in odd spacial dimensions only
35 Chiral vs nonchiral systems 15 topologically equivalent to the trivial state. d Cartan Complex case: A Z 0 Z 0 Z 0 Z 0 Z 0 Z 0... AIII 0 Z 0 Z 0 Z 0 Z 0 Z 0 Z... Real case: AI Z Z 0 Z 2 Z 2 Z BDI Z 2 Z Z 0 Z 2 Z 2 Z D Z 2 Z 2 Z Z 0 Z 2 Z 2 Z 0... DIII 0 Z 2 Z 2 Z Z 0 Z 2 Z 2 Z... AII 2Z 0 Z 2 Z 2 Z Z 0 Z 2 Z 2... CII 0 2Z 0 Z 2 Z 2 Z Z 0 Z 2... C 0 0 2Z 0 Z 2 Z 2 Z Z 0... CI Z 0 Z 2 Z 2 Z Z... Chiral Nonchiral
36 16 Topological invariants via single particle Green s functions
37 Topological invariants for nonchiral insulators 17
38 Topological invariants for nonchiral insulators 17 space-time space
39 Topological invariants for nonchiral insulators 17 space-time space
40 Topological invariants for nonchiral insulators 17 space-time space Translational invariance
41 Topological invariants for nonchiral insulators 17 space-time space Translational invariance map { D-dim space-time
42 Topological invariants for nonchiral insulators 17 space-time space Translational invariance map { D-dim space-time Notes: 1. d must be even. If d=2 this coincides with the TKNN invariant Niu, Thouless, Wu (1985) 2. Subsequently used by Volovik in a variety of contexts (80 s and 90 s)
43 The meaning of the invariant at d=0, D=1 18 Some parameter As long as the system remains gapful, N1 is an invariant
44 Topological invariants for chiral insulators 19 space-time space VG, 2010
45 Topological invariants for chiral insulators 19 space-time space Translational invariance VG, 2010
46 Topological invariants for chiral insulators 19 space-time space Translational invariance VG, 2010 Skyrmion number pic. by N. Cooper
47 Topological invariants for chiral insulators 19 space-time space Translational invariance VG, 2010 Skyrmion number pic. by N. Cooper
48 The meaning of the invariant at D=0 20 I 0 =trq =trσ =# R # L Properties of chiral systems All levels come in pairs { right zero modes left zero modes #R-#L is a topological invariant (index theorem)
49 Simplest d=1 chiral topological insulator 21 t 1 t 2 t 1 Ĥ = x even t 1 t 2 t 1 â x+1âx + t 2 â x+2âx+1 t 1 + h.c. Su, Schrieffer, Heeger (1978)
50 Simplest d=1 chiral topological insulator 21 t 1 t 2 t 1 Ĥ = x even t 1 t 2 t 1 â x+1âx + t 2 â x+2âx+1 t 1 + h.c. t 1 + t 2 e ik I 2 π π dk 2πi k ln t 1 + t 2 e ik I 2 =1 t 1 <t 2 Su, Schrieffer, Heeger (1978)
51 Simplest d=1 chiral topological insulator 21 t 1 t 2 t 1 Ĥ = x even t 1 t 2 t 1 â x+1âx + t 2 â x+2âx+1 t 1 + h.c. t 1 + t 2 e ik I 2 π π dk 2πi k ln t 1 + t 2 e ik I 2 =0 t 1 >t 2 Su, Schrieffer, Heeger (1978)
52 Simplest d=1 chiral topological insulator 21 t 1 t 2 t 1 Ĥ = x even t 1 t 2 t 1 â x+1âx + t 2 â x+2âx+1 t 1 + h.c. t 1 + t 2 e ik I 2 π π dk 2πi k ln t 1 + t 2 e ik Zero mode (edge state) satisfy t 1 ψ x + t 2 ψ x+2 =0 ψ x = t x 2 1 t 2 If x>0, exist only if t1<t2. I 2 =0 t 1 >t 2 Su, Schrieffer, Heeger (1978)
53 22 Topological invariants and the edge states
54 d=2; Classes A, D, C G. Volovik, 1980s 23 topologically equivalent to the trivial state. d Cartan Complex case: A IQHE Z 0 Z 0 Z 0 Z 0 Z 0 Z 0... AIII 0 Z 0 Z 0 Z 0 Z 0 Z 0 Z... Real case: AI Z Z 0 Z 2 Z 2 Z BDI Z 2 Z Z 0 Z 2 Z 2 Z D p-wave s.c. Z 2 Z 2 Z Z 0 Z 2 Z 2 Z 0... DIII 3 He B 0 Z 2 Z 2 Z Z 0 Z 2 Z 2 Z... AII 2Z 0 Z 2 Z 2 Z Z 0 Z 2 Z 2... CII 0 2Z 0 Z 2 Z 2 Z Z 0 Z 2... C 0 0 2Z 0 Z 2 Z 2 Z Z 0... CI singlet s.c Z 0 Z 2 Z 2 Z Z... Chiral Nonchiral
55 d=2; Classes A, D, C G. Volovik, 1980s 23 N L N R Domain wall
56 d=2; Classes A, D, C G. Volovik, 1980s 23 N L N R Green s function Domain wall
57 d=2; Classes A, D, C G. Volovik, 1980s 23 N L N R Green s function Inverse green s function Domain wall
58 d=2; Classes A, D, C G. Volovik, 1980s 23 N L N R Green s function Inverse green s function Domain wall Construct the simplest topological invariant
59 d=2; Classes A, D, C G. Volovik, 1980s 23 N L N R Green s function Inverse green s function Domain wall Construct the simplest topological invariant zero mode
60 d=2; Classes A, D, C G. Volovik, 1980s 24 N L N R Domain wall
61 d=2; Classes A, D, C G. Volovik, 1980s 24 Wigner transformed Green s function N L N R Domain wall
62 d=2; Classes A, D, C G. Volovik, 1980s 24 Wigner transformed Green s function N L N R Gradient (Moyal product) expansion Domain wall
63 d=2; Classes A, D, C G. Volovik, 1980s 24 Wigner transformed Green s function N L N R Gradient (Moyal product) expansion Domain wall
64 d=2; Classes A, D, C G. Volovik, 1980s 24 Wigner transformed Green s function N L N R Gradient (Moyal product) expansion Domain wall 4-dim vector, space ω, px, ps, s
65 d=2; Classes A, D, C G. Volovik, 1980s 24 Wigner transformed Green s function N L N R Gradient (Moyal product) expansion Domain wall 4-dim vector, space ω, px, ps, s #(zero modes) = N R N L
66 d=2; Classes A, D, C G. Volovik, 1980s 24 N L N R Domain wall #(zero modes) = N R N L
67 d=1, classes AIII, BDI, CII 25 I L I R s Q(ω, p s,s)=g 1 (ω, p s,s) Σ G(ω, p s,s) I(s) = 1 16πi tr 0 dω I(L) I( L) = 1 16πi lim tr ω 0 dp s Q ( ω Q ps Q ps Q ω Q) dxdk Q ( x Q ps Q ps Q x Q)
68 d=1, classes AIII, BDI, CII 25 I L I R s Q(ω, p s,s)=g 1 (ω, p s,s) Σ G(ω, p s,s) I(s) = 1 16πi tr 0 dω I(L) I( L) = 1 16πi lim tr ω 0 #(zero modes) = lim ω 0 ω tr Σ K ω G dp s Q ( ω Q ps Q ps Q ω Q) dxdk Q ( x Q ps Q ps Q x Q)
69 d=1, classes AIII, BDI, CII 25 I L I R s Q(ω, p s,s)=g 1 (ω, p s,s) Σ G(ω, p s,s) I(s) = 1 16πi tr 0 dω I(L) I( L) = 1 16πi lim tr ω 0 #(zero modes) = lim ω 0 ω tr Σ K ω G gradient expansion #(zero modes)=i(l) I( L) dp s Q ( ω Q ps Q ps Q ω Q) dxdk Q ( x Q ps Q ps Q x Q)
70 Other classes of topological insulators 26 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)
71 27 Topological invariants in the presence of interactions
72 The invariant at d=0, D=1 with interactions 28 VG, 2010 No interactions
73 The invariant at d=0, D=1 with interactions 28 VG, 2010 No interactions In the presence of interactions
74 The invariant at d=0, D=1 with interactions 28 VG, 2010 G = Df Matrix elements D h 1 No interactions iω n Matrix elements In the presence of interactions det G = Dh D f n=1 (iω r n ) Dh n=1 (iω n) zeroes of the Green s function poles of the Green s function
75 The invariant at d=0, D=1 with interactions 28 VG, 2010 No interactions In the presence of interactions det G = Dh D f n=1 (iω r n ) Dh n=1 (iω n) zeroes of the Green s function poles of the Green s function
76 The invariant at d=0, D=1 with interactions 29 VG, 2010 det G = Switching on interactions Dh D f n=1 (iω r n ) Dh n=1 (iω n)
77 The invariant at d=0, D=1 with interactions 29 VG, 2010 G. Volovik, 2006 det G = Switching on interactions Dh D f n=1 (iω r n ) Dh n=1 (iω n)
78 The invariant at d=0, D=1 with interactions 29 VG, 2010 G. Volovik, 2006 Poles and zeroes can emerge and disappear in pairs det G = Switching on interactions Dh D f n=1 (iω r n ) Dh n=1 (iω n)
79 The invariant at d=0, D=1 with interactions 29 VG, 2010 G. Volovik, 2006 Poles and zeroes can emerge and disappear in pairs energy det G = Switching on interactions Dh D f n=1 (iω r n ) Dh n=1 (iω n) pole parameter
80 The invariant at d=0, D=1 with interactions 29 VG, 2010 G. Volovik, 2006 Poles and zeroes can emerge and disappear in pairs energy det G = Switching on interactions Dh D f n=1 (iω r n ) Dh n=1 (iω n) pole zero parameter parameter
81 The invariant at d=0, D=1 with interactions 29 VG, 2010 G. Volovik, 2006 Poles and zeroes can emerge and disappear in pairs energy det G = Switching on interactions Dh D f n=1 (iω r n ) Dh n=1 (iω n) pole zero parameter parameter parameter
82 The invariant at d=0, D=1 with interactions 29 VG, 2010 G. Volovik, 2006 Poles and zeroes can emerge and disappear in pairs energy det G = Switching on interactions Dh D f n=1 (iω r n ) Dh n=1 (iω n) pole zero parameter parameter parameter parameter
83 Fidkowski-Kitaev model (2010) 30 t 1 t 2 t 1 t 1 t 2 t 1 quartic interactions g t 1 t 2 t 1 = t 2
84 Fidkowski-Kitaev model (2010) 30 t 1 t 2 t 1 t 1 t 2 t 1 quartic interactions g t 1 t 2 t 1 = t 2 Phase transition
85 Fidkowski-Kitaev model (2010) 30 t 1 t 2 t 1 t 1 t 2 t 1 quartic interactions g No phase transition t 1 t 2 t 1 = t 2 Phase transition
86 Fidkowski-Kitaev model (2010) 30 t 1 t 2 t 1 t 1 t 2 t 1 quartic interactions g A. Essin, VG: Green s function has No phase transition a zero at zero energy t 1 t 2 t 1 = t 2 Phase transition
87 Conclusions and open questions Single particle Green s functions - a powerful tool to understand topological insulators without or even with interactions. 2. Zeroes of the Green s functions. What are they, when do they appear, how can they be detected, why are they important for interacting topological insulators?
88 32 The end
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