C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 1 / fermio 93. quartic interactions

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1 The breakdown of the topological classification Z for gapped phases of noninteracting fermions by quartic interactions Christopher Mudry 1 Takahiro Morimoto 2,3 Akira Furusaki 2 1 Paul Scherrer Institut, Switzerland 2 RIKEN, Japan 3 University of California at Berkeley UIUC, October C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 1 / fermio 93

2 1 Main results 2 Introduction 3 Strategy 4 Examples 5 Application to the surfaces of SnTe 6 Summary 7 Appendices C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 2 / fermio 93

3 Main results Breakdown of the tenfold way (noninteracting topological classification of fermions) by quartic contact interactions: Class T C Γ 5 V d d=1 d=2 d=3 d=4 d=5 d=6 d=7 d=8 A C 0+d 0 Z 0 Z 0 Z 0 Z AIII C 1+d Z 4 0 Z 8 0 Z 16 0 Z 32 0 AI R 0 d Z 0 Z 2 Z 2 Z BDI R 1 d Z 8, Z Z 16, Z 8 0 Z 2 Z 2 D R 2 d Z 2 Z Z 0 Z 2 DIII R 3 d Z 2 Z 2 Z Z 32 0 AII R 4 d 0 Z 2 Z 2 Z Z CII R 5 d Z 2, Z 2 0 Z 2 Z 2 Z 16, Z C R 6 d 0 Z 0 Z 2 Z 2 Z 0 0 CI R 7 d 0 0 Z 4 0 Z 2 Z 2 Z 32 0 Breakdown of the three-dimensional AII+R topological crystalline classification Z to Z 8 by quartic contact interactions. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 2 / fermio 93

4 1 Main results 2 Introduction 3 Strategy 4 Examples 5 Application to the surfaces of SnTe 6 Summary 7 Appendices C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 3 / fermio 93

5 Introduction: What is the tenfold way? The tenfold way applies to noninteracting fermions that realize a noninteracting manybody ground state, a Slater determinant, such that, (i) it is unique and it is separated from all excitations by a gap, when space is boundaryless, (ii) while it supports gapless excitations that are extended along but bounded away from any boundary of space otherwise. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 4 / fermio 93

6 Comment 1: This definition does not require translation invariance in the direction parallel to a boundary. In particular, it applies when static impurities are present. Comment 2: This definition implies a notion of topology since two spectral properties (the existence of gapless and extended modes) depend on the topological distinction between manifolds without (e.g., sphere, torus) and with (e.g., cube, cylinder) boundaries. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 5 / fermio 93

7 Introduction: Chronological examples 1 Zero modes at domain walls in polyacetylene (Jackiw and Rebbi 1976; Su, Schrieffer, and Heeger 1979). 2 Chiral edge states along the boundary of an heterojunction displaying the IQHE (Laughlin 1981; Halperin 1982). 3 Chiral Majorana zero modes along the boundary of a two-dimensional (p + ip) superconductor (Read and Green 2000). 4 A pair of helical zero modes along the boundary of a two-dimensional Z 2 topological insulator (Kane and Mele 2005). 5 The tenfold way (Schnyder, Ryu, Furusaki, and Ludwig; Kitaev 2008): Class T C Γ 5 V d d=1 d=2 d=3 d=4 d=5 d=6 d=7 d=8 A C 0+d 0 Z 0 Z 0 Z 0 Z AIII C 1+d Z 0 Z 0 Z 0 Z 0 AI R 0 d Z 0 Z 2 Z 2 Z BDI R 1 d Z Z 0 Z 2 Z 2 D R 2 d Z 2 Z Z 0 Z 2 DIII R 3 d Z 2 Z 2 Z Z 0 AII R 4 d 0 Z 2 Z 2 Z Z CII R 5 d Z 0 Z 2 Z 2 Z C R 6 d 0 Z 0 Z 2 Z 2 Z 0 0 CI R 7 d 0 0 Z 0 Z 2 Z 2 Z 0 C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 6 / fermio 93

8 Robustness of the gapless extended boundary states arises iff certain symmetries hold. The distinct combinations of symmetries produce 10 symmetry classes. These symmetries are local in the table Class T C Γ 5 V d d=1 d=2 d=3 d=4 d=5 d=6 d=7 d=8 A C 0+d 0 Z 0 Z 0 Z 0 Z AIII C 1+d Z 0 Z 0 Z 0 Z 0 AI R 0 d Z 0 Z 2 Z 2 Z BDI R 1 d Z Z 0 Z 2 Z 2 D R 2 d Z 2 Z Z 0 Z 2 DIII R 3 d Z 2 Z 2 Z Z 0 AII R 4 d 0 Z 2 Z 2 Z Z CII R 5 d Z 0 Z 2 Z 2 Z C R 6 d 0 Z 0 Z 2 Z 2 Z 0 0 CI R 7 d 0 0 Z 0 Z 2 Z 2 Z 0 but this need not be so. Crystalline topological insulators are noninteracting fermions that support gapless extended boundary states iff crystalline symmetries hold. An all inclusive terminology is Short-ranged entangled (SRE) topological phases of fermionic matter Symmetry-protected topological (SPT) phases of fermionic matter C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 7 / fermio 93

9 Examples of diagnostic for SRE topological phases For band insulators on a torus H = k BZ H(k). Single-particle observables for topological SRE phases can then be chosen to be Polyacetylene: H(k) := ( ) 0 q(k) q, ν = i (k) 0 2π +π π ( dk tr q ) k q (k). IQHE for a Chern band insulator: H(k) := a u a (k) u a (k), A ab µ (k) := ua (k) k µ ub (k), ν = i +π d 2 k tr [ F 12 (k) ] 2π occupied bds. π Z 2 topological insulator in two-dimensional space: H(k) := a u a (k) u a (k), w ab µ (k) := ua ( k) Θ u b (+k), ν = K = K BZ Pf [w(k )] occupied bds. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 8 / fermio 93

10 The problem is that such observables are not useful in the presence of static disorder or in the presence of interactions. Alternative observables can be constructed in the presence of static disorder. The only observable that I understand in the presence of static disorder and interactions is the Hall conductivity averaged over twisted boundary conditions, Niu and Thouless 1984; Niu, Thouless, and Wu 1985 C = i 2π 2π 0 dφ 2π 0 [ Ψ dϕ φ Ψ ϕ Ψ ϕ ] Ψ φ with Ψ the manybody ground state obeying twisted boundary conditions. If C is an integer in the noninteracting limit, and if the interactions are not too strong (compared to the band gap), then C is unchanged by interactions. This is so because there is no backscattering allowed among the chiral edge states C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 9 / fermio 93

11 Laughlin 1981: The Hall conductivity must be rational and if it is not an integer, the ground state manifold must be degenerate and support fractionally charged excitations x z B y Halperin 1982: Chiral edges are immune to backscattering within each traffic lane (a) (b) Integer Quantum Hall Effect Fractional Quantum Hall Effect C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 10 / fermio 93

12 This argument (no backscattering) is tied to the boundary being a one-dimensional manifold. Question: Are all the topological entries of the tenfold way robust to manybody interactions? C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 11 / fermio 93

13 Partial answer for strong topological insulators Class T C Γ 5 V d d=1 d=2 d=3 d=4 d=5 d=6 d=7 d=8 A C 0+d 0 Z 0 Z 0 Z 0 Z AIII C 1+d Z 4 0 Z 8 0 Z 16 0 Z 32 0 AI R 0 d Z 0 Z 2 Z 2 Z BDI R 1 d Z 8, Z Z 16, Z 8 0 Z 2 Z 2 D R 2 d Z 2 Z Z 0 Z 2 DIII R 3 d Z 2 Z 2 Z Z 32 0 AII R 4 d 0 Z 2 Z 2 Z Z CII R 5 d Z 2, Z 2 0 Z 2 Z 2 Z 16, Z C R 6 d 0 Z 0 Z 2 Z 2 Z 0 0 CI R 7 d 0 0 Z 4 0 Z 2 Z 2 Z 32 0 Fidkowsky and Kitaev 2010;... Kitaev 2011 (unpublished); Fidkowski, Chen, and Vishwanath 2014;... Tang and Wen 2012 Wang and Senthil 2014 C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 12 / fermio 93

14 Partial answer for crystalline topological insulators There also are examples of crystalline topological insulators that are unstable to quartic interactions: Z Z 8 for DIII+R in 2d (Yao and Ryu 2013; Qi 2013) topological insulators with inversion symmetry (You and Xu 2014). Z Z 8 for AII+R in 3d (Isobe and Fu 2015; Yoshida and Furusaki; this work 2015) C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 13 / fermio 93

15 Many different approaches were used to reach these conclusions. Is there an approach, perhaps a simpler one, that reproduces all known results and extends readily to all entries of the tenfold way? C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 14 / fermio 93

16 1 Main results 2 Introduction 3 Strategy 4 Examples 5 Application to the surfaces of SnTe 6 Summary 7 Appendices C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 15 / fermio 93

17 Strategy The strategy that we use to study the robustness of ν boundary modes to quartic contact fermion-fermion interactions is related to the approached advocated by You and Xu 2014 and (unpublished) Kitaev It consists of three steps. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 16 / fermio 93

18 Step 1 A noninteracting topological phase is represented by the manybody ground state of a massive Dirac Hamiltonian with a matrix dimension that depends on ν. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 17 / fermio 93

19 Step 2 A Hubbard-Stratonovich transformation is used to trade a generic quartic contact interaction in favor of dynamical Dirac mass-like bilinears coupled to their conjugate fields (that will be called dynamical Dirac masses). These dynamical Dirac masses may violate any symmetry constraint other than the particle-hole symmetry (PHS). C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 18 / fermio 93

20 Step 3 The ν boundary modes that are coupled with a suitably chosen subset of dynamical Dirac masses are integrated over. The resulting dynamical theory on the (d 1)-dimensional boundary is a bosonic one, a quantum nonlinear sigma model (QNLSM) in [(d 1) + 1] space and time, with a target space that depends on ν. These QNLSM can have the following phase diagrams: Without topological term With topological term d 2 g=0 g s g=0 gs d > 2 g=0 g c g s g=0 g c g s C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 19 / fermio 93

21 Step 4 The reduction pattern is then obtained by identifying the smallest value of ν for which: This QNLSM cannot be augmented by a topological term that is compatible with local equations of motion. The quantum fluctuations driven by the interactions have restored all the symmetries broken by the saddle point. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 20 / fermio 93

22 Comment 1: The question that we address in this paper is whether the topological classification of noninteracting fermions is reduced by interactions. A complete classification of fermionic SRE phases (combined with that for the bosonic SRE phases) is beyond the scope of our approach. Comment 2: The presence or absence of topological terms in the relevant QNLSM is determined by the topology of the spaces of boundary dynamical Dirac masses, i.e., the topology of classifying spaces. Now, K-theory provides a systematic way to study the topology of the classifying space. Comment 3: This is why the same approach that was used to obtain the tenfold way of noninteracting fermions can be relied on to deduce a classification of topological short-range entangled (SRE) phases for interacting fermions. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 21 / fermio 93

23 1 Main results 2 Introduction 3 Strategy 4 Examples 5 Application to the surfaces of SnTe 6 Summary 7 Appendices C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 22 / fermio 93

24 Example: Symmetry class AII in d = 3 Consider the single-particle static massive Dirac Hamiltonian [this is THE Dirac Hamiltonian (Dirac 1928)] H (0) (x) := i x X 21 i y X 11 i z X 02 + m(x) X 03, where X µµ := σ µ τ µ for µ, µ = 0, 1, 2, 3. Because (1a) (1b) H (0) (x) = +T H (0) (x) T 1, T := ix 20 K, (1c) we interpret this Hamiltonian as realizing a noninteracting topological insulator in the three-dimensional symmetry class AII. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 23 / fermio 93

25 The static domain wall in the mass (Jackiw and Rebbi 1976) m(x, y, z) = m sgn(z) (2a) binds a zero mode to the boundary z = 0 that is annihilated by the boundary single-particle Hamiltonian H (0) bd (x, y) = i x σ 2 i y σ 1 = T bd H (0) 1 bd (x, y) Tbd, (2b) where T bd := iσ 2 K. (2c) C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 24 / fermio 93

26 The boundary dynamical Dirac Hamiltonian H (dyn) bd (τ, x, y) = i x σ 2 i y σ 1 + M(τ, x, y) σ 3 (3) belongs to the symmetry class A, as the Dirac mass Mσ 3 breaks TRS unless M( τ, x, y) = M(τ, x, y). The space of normalized boundary dynamical Dirac mass matrices {±σ 3 } is homeomorphic to the space of normalized Dirac mass matrices in the symmetry class A V ν=1 = 1 U(ν)/ [U(k) U(ν k)] (4) k=0 when space is two dimensional. The domain wall in imaginary time (the instanton) M(τ, x, y) = M sgn(τ) (5) prevents the gapping of the boundary zero mode. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 25 / fermio 93

27 We conclude that the noninteracting topological classification Z 2 of three-dimensional insulators in the symmetry class AII is robust to the effects of interactions under the assumption that only fermion-number-conserving interacting channels are included in the stability analysis. The logic used to reach this conclusion is summarized by Table D π D (C 0 ) ν Topological obstruction 0 Z 1 Domain wall 1 0 once the line corresponding to ν = 1 has been identified. This line is fixed by the first integer D that accommodates a non-trivial entry for the corresponding homotopy group. Moreover, one verifies by introducing a BdG (Nambu) grading that this robustness extends to interaction-driven dynamical superconducting fluctuations. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 26 / fermio 93

28 Example: Symmetry class AII in d = 3 with reflection symmetry R We consider again the bulk, boundary, and dynamical boundary Hamiltonian defined in Eqs. (1) (4). We observe that the single-particle Hamiltonian (1a) has the symmetry R x H (0) ( x, y, z) ( R x ) 1 = +H (0) (x, y, z), (6a) where in addition to the TRS (1c). R x := ix 10, R 2 x = 1, [T, R x ] = 0, (6b) C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 27 / fermio 93

29 The presence of the additional reflection symmetry allows one to define a mirror Chern number (n + Z) for the sector with the eigenvalue R x = +i on the two-dimensional mirror plane (k x = 0) in the three-dimensional Brillouin zone (Hsieh, Lin, Liu, Duan, Bansil, and Fu 2012). Thus, the ν linearly independent zero modes that follow from tensoring the single-particle Hamiltonian (1a) with the ν ν unit matrix along the domain wall (2a) are stable to strong one-body perturbations on the boundary that preserve the reflection symmetry. (If we forget the reflection symmetry and keep only the TRS, it is only the parity of ν that is stable to strong one-body perturbations on the boundary.) C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 28 / fermio 93

30 If we only consider dynamical masses that preserve the fermion-number U(1) symmetry, the space of normalized boundary dynamical Dirac mass matrices after tensoring the boundary dynamical Dirac Hamiltonian (4) with the unit ν ν matrix is homeomorphic to the space of normalized Dirac masses in the symmetry class A V ν = ν U(ν)/[U(k) U(ν k)] (7) k=0 when space is two dimensional. The limit ν of these spaces is the classifying space C 0. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 29 / fermio 93

31 Integrating the boundary Dirac fermions delivers a QNLSM in (2+1)-dimensional space and time. In order to gap out dynamically the boundary zero modes without breaking the symmetries, this QNLSM must be free of topological obstructions. We construct explicitly the spaces for the relevant normalized boundary dynamical Dirac mass matrices [M(τ, x, y) in Eq. (3)] of dimension ν = 2 n with n = 0, 1, 2, 3 in the following. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 30 / fermio 93

32 Case ν = 1: There is a topological obstruction of the domain-wall type as the target space is S 0 = {±1} (8) and π 0 (S 0 ) 0. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 31 / fermio 93

33 Case ν = 2: There is a topological obstruction of the monopole type as the target space is S 2 = {c 1 X 1 + c 2 X 2 + c 3 X 3 c c2 2 + c2 3 = 1} (9) and π 2 (S 2 ) = Z. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 32 / fermio 93

34 Case ν = 4: There is a topological obstruction of the WZ type as the target space is { S 4 = c 1 X 13 + c 2 X 23 + c 3 X 33 + c 4 X 01 + c 5 X 02 and π 4 (S 4 ) = Z. 5 i=1 c 2 i } = 1, c i R (10) C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 33 / fermio 93

35 Case ν = 8: There is no topological obstruction as one can find more than five pairwise anticommuting matrices such as the set {X 133, X 233, X 333, X 013, X 023, X 001, X 002 }. (11) C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 34 / fermio 93

36 Reduction from Z to Z 8 due to interactions for the topologically equivalent classes of the three-dimensional topological insulators with time-reversal and reflection symmetries (AII+R). We denote by V ν the space of ν ν normalized Dirac mass matrices in boundary (d = 2) Dirac Hamiltonians belonging to the symmetry class A. The limit ν of these spaces is the classifying space C 0. The second column shows the stable D-th homotopy groups of the classifying space C 0. The third column gives the number ν of copies of boundary (Dirac) fermions for which a topological obstruction is permissible. The fourth column gives the type of topological obstruction that prevents the gapping of the boundary (Dirac) fermions. D π D (C 0 ) ν Topological obstruction 0 Z 1 Domain wall Z 2 Monopole Z 4 WZ term Z 8 None 7 0 C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 35 / fermio 93

37 This Z 8 classification is unchanged if all boundary dynamical masses that break the fermion-number U(1) symmetry are accounted for. The corresponding target spaces for the boundary dynamical masses and their topological obstructions are derived by extending the single-particle Hamiltonian [Eq. (3)] to a BdG Hamiltonian H (dyn) bd = ( i x σ 2 ρ 3 i y σ 1 ρ 0 ) Iν ν + γ (τ, x, y), (12a) where ρ 0 and ρ µ are unit 2 2 and Pauli matrices, respectively, acting on the particle-hole (Nambu) space, I ν ν is the unit ν ν matrix, and the particle-hole symmetry is given by C = ρ 1 K. In this case, the target spaces of the QNLSM made of normalized boundary dynamical Dirac mass matrices γ of dimension ν = 2 n with n = 0, 1, 2, 3 are modified as listed in the following with the notation X µµ µ µ... = σ µ ρ µ τ µ τ µ.... (12b) C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 36 / fermio 93

38 The relevant homotopy groups are given in D π D (R 0 ) ν Topological obstruction 0 Z 1 Z 2 1 Vortex 2 Z 2 2 Monopole Z 4 WZ term Z 8 None We note that these target spaces are closed under the global U(1) transformation generated by ρ 3. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 37 / fermio 93

39 1 Main results 2 Introduction 3 Strategy 4 Examples 5 Application to the surfaces of SnTe 6 Summary 7 Appendices C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 38 / fermio 93

40 Application to the surfaces of SnTe The crystal SnTe is a three-dimensional topological crystalline insulator protected by time-reversal and reflection symmetries (AII+R). SnTe supports four Dirac cones on the [001] surface and six Dirac cones on the [111] surface. If strong interaction effects are present, we expect the following. On the [001] surface, the ν = 4 phase described by a QNLSM with a WZ term should be realized. On the [111] surface, the ν = 6 = phase should be realized, whereby the effective field theory is that of a QNLSM with a WZ term for 4 out of the six surface Dirac cones and that of a QNLSM with a topological term arising from a gas of monopoles for the remaining two surface Dirac cones. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 39 / fermio 93

41 1 Main results 2 Introduction 3 Strategy 4 Examples 5 Application to the surfaces of SnTe 6 Summary 7 Appendices C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 40 / fermio 93

42 Summary Breakdown of the tenfold way (noninteracting topological classification of fermions) by quartic contact interactions: Class T C Γ 5 V d d=1 d=2 d=3 d=4 d=5 d=6 d=7 d=8 A C 0+d 0 Z 0 Z 0 Z 0 Z AIII C 1+d Z 4 0 Z 8 0 Z 16 0 Z 32 0 AI R 0 d Z 0 Z 2 Z 2 Z BDI R 1 d Z 8, Z Z 16, Z 8 0 Z 2 Z 2 D R 2 d Z 2 Z Z 0 Z 2 DIII R 3 d Z 2 Z 2 Z Z 32 0 AII R 4 d 0 Z 2 Z 2 Z Z CII R 5 d Z 2, Z 2 0 Z 2 Z 2 Z 16, Z C R 6 d 0 Z 0 Z 2 Z 2 Z 0 0 CI R 7 d 0 0 Z 4 0 Z 2 Z 2 Z 32 0 Breakdown of the three-dimensional AII+R topological crystalline classification Z to Z 8 by quartic contact interactions. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 41 / fermio 93

43 1 Main results 2 Introduction 3 Strategy 4 Examples 5 Application to the surfaces of SnTe 6 Summary 7 Appendices C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 42 / fermio 93

44 The classifying spaces and their homoytopy groups are Class T C Γ Extension V d A Cl d Cl d+1 C 0+d AIII Cl d+1 Cl d+2 C 1+d AI Cl 0,d+2 Cl 1,d+2 R 0 d BDI Cl d+1,2 Cl d+1,3 R 1 d D Cl d,2 Cl d,3 R 2 d DIII Cl d,3 Cl d,4 R 3 d AII Cl 2,d Cl 3,d R 4 d CII Cl d+3,0 Cl d+3,1 R 5 d C Cl d+2,0 Cl d+2,1 R 6 d CI Cl d+2,1 Cl d+2,2 R 7 d Class V π 0 (V ) π 0 (V ) π 0 (V ) π 0 (V ) π 0 (V ) π 0 (V ) π 0 (V ) π 0 (V ) A C 0+d Z 0 Z 0 Z 0 Z 0 AIII C 1+d 0 Z 0 Z 0 Z 0 Z AI R 0 d Z Z 0 Z 2 Z 2 BDI R 1 d Z 2 Z Z 0 Z 2 D R 2 d Z 2 Z 2 Z Z 0 DIII R 3 d 0 Z 2 Z 2 Z Z AII R 4 d Z 0 Z 2 Z 2 Z CII R 5 d 0 Z 0 Z 2 Z 2 Z 0 0 C R 6 d 0 0 Z 0 Z 2 Z 2 Z 0 CI R 7 d Z 0 Z 2 Z 2 Z C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 43 / fermio 93

45 C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 44 / fermio 93

46 Main results (AZ stands for Altland and Zirnbauer) Case d = 1 Five AZ symmetry classes Three AZ symmetry classes (A) (B) N=1 (B) N=2 and so on (C) g Insulator Two AZ symmetry classes g g Insulator g ν=0 Insulator Insulator Insulator Insulator Insulator ν= 1/2 ν=+1/2 ν= 1 ν=+1 ν=0 ν=+1 Insulator 0 m 0 m 0 m 0 m Case d = 3 Five AZ symmetry classes Three AZ symmetry classes Two AZ symmetry classes (A) (B) N=1 (B) N=2 and so on (C) g Metal Insulator g g Metal Metal ν=0 ν= 1/2 ν=+1/2 ν= 1 ν=+1 ν=0 ν=+1 0 m 0 m 0 m 0 m g Metal C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 45 / fermio 93

47 Recent related works Case d = 1 chiral classes: A. Altland, D. Bagrets, L. Fritz, A. Kamenev, and H. Schmiedt, Quantum criticality of quasi one-dimensional topological Anderson insulators, Phys. Rev. Lett. 112, (2014), Case d = 1 class D: M.-T. Rieder and P. W. Brouwer, Density of states at disorder-induced phase transitions in a multichannel Majorana wire, arxiv: ; M.-T. Rieder, P. W. Brouwer, and I. Adagideli, Reentrant topological phase transitions in a disordered spinless superconducting wire, Phys. Rev. B 88, (R) (2013); and taken from A. Altland, D. Bagrets, and A. Kamenev, Topology vs. Anderson localization: non-perturbative solutions in one dimension, arxiv: C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 46 / fermio 93

48 1 Main results 2 Introduction 3 Strategy 4 Examples 5 Application to the surfaces of SnTe 6 Summary 7 Appendices C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 47 / fermio 93

49 The Dirac equation: an application of Clifford algebra Dirac (1928) [taken from P. A. M. Dirac, Principles of Quantum Mechanics (4th ed.), Clarendon, page 255 (1958)]: where {α i, α j } = 2 δ ij, {α i, β} = 0, i, j = 1, 2, 3, β 2 = 1. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 47 / fermio 93

50 1931: Dirac introduces topology in physics C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 48 / fermio 93

51 : Tamm and Schockley surface states I. Tamm, On the possible bound states of electrons on a crystal surface, Phys. Z. Soviet Union 1, 733 (1932). W. Shockley, On the Surface States Associated with a Periodic Potential, Phys. Rev. 56, 317 (1939). P-Y. Chang, C. Mudry, and S. Ryu 2014 Direct sum of a p x + ip y and of a p x ip y BdG superconductor in a cylindrical geometry. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 49 / fermio 93

52 Anderson localization before 1981 P. W. Anderson, Absence of Diffusion in Certain Random Lattices, Phys. Rev. 109, 1492 (1957): What is the fate of a single-particle Bloch state when perturbed by a static potential that breaks all space-group symmetries? In a noninteracting many-body setting, there are three possible phases: The metallic phase: σ = at T = 0. The insulating phase: σ = 0 at T = 0. A quantum critical phase: σ = σ c at T = 0. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 50 / fermio 93

53 1953: Dyson s exception F. J. Dyson, The Dynamics of a Disordered Linear Chain, Phys. Rev. 92, 1331 (1953). C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 51 / fermio 93

54 1963: The threefold way for random matrices F. J. Dyson, The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics, J. Math. Phys. 3, 1199 (1962): P(θ 1,, θ N ) 1 j<k N e iθ j e iθ k β, β = 1, 2, 4. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 52 / fermio 93

55 1976: Jackiw and Rebbi introduce Fermion number fractionalization R. Jackiw and C. Rebbi, Solitons with fermion number 1/2, Phys. Rev. D 13, 3398 (1976): Dirac equation with a single point defect in a background field supports a single zero mode that carries the fermion number 1/2. W. P. Su, J. R. Schrieffer, and H. J. Heeger, Soliton excitations in polyacetylene, Phys. Rev. B 22, 2099 (1980): Propose polaycetylene as a realization of fermion fractionalization ε(k) k π/2 +π/2 C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 53 / fermio 93

56 1981: Nielsen-Ninomiya theorem H. B. Nielsen and M. Ninomiya, A no-go theorem for regularizing chiral fermions, Phys. Lett. B105, 219 (1981): The Nielsen-Ninomiya theorem is a no-go theorem that prohibits defining a theory of chiral fermions on a lattice in odd-dimensional space. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 54 / fermio 93

57 1981: Jackiw and Rossi introduce a localized Majorana zero mode in a two-dimensional relativistic superconductor R. Jackiw and P. Rossi, Zero modes of the vortex-fermion system, Nucl. Phys. B190, 681 (1981). L. Fu and C. L. Kane, Superconducting Proximity Effect and Majorana Fermions at the Surface of a Topological Insulator, Phys. Rev. Lett. 100, (2008) have proposed to realize this localized Majorana zero mode on the core of a superconducting vortex induced by proximity effect on the surface of a strong Z 2 topological insulator. TI s wavesc C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 55 / fermio 93

58 Anderson localization after 1981 The discovery of the Integer Quantum Hall effect, Klitzing, Dorda, and Pepper (1980), lead to the understanding that there are topologically dictinct insulating phases separated by critical point at which a plateau transition takes place. Example: Graphene deposited on SiO 2 /Si, T =1.6 K and B=9 T (inset T =30 mk): ν = ±2, ±6, ±10, = ±2(2n + 1), n N after Zhang et al., C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 56 / fermio 93

59 The Integer Quantum Hall Effect ε m= 1 m h ω c Without disorder ε F ε m= 1 m h ω c With disorder but no interactions εf 2 6 m= 0 m= 1 DOS σ xy m= 0 m= 1 DOS 6 2 σ xy [e^2/h] At integer fillings of the Landau levels, the noninteracting ground state is unique and the screened Coulomb interaction V int can be treated perturbatively, as long as transitions between Landau levels or outside the confining potential V conf along the magnetic field are suppressed by the single-particle gaps: V int ω c V conf, ω c = e B/(m c). C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 57 / fermio 93

60 : Laughlin and Halperin introduce the bulk-edge correspondence in the Quantum Hall Effect Laughlin 1981: The Hall conductivity must be rational and if it is not an integer, the ground state manifold must be degenerate and support fractionally charged excitations Halperin 1982: Chiral edges are immune to backscattering within each traffic lane x z (a) B y (b) Integer Quantum Hall Effect Fractional Quantum Hall Effect C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 58 / fermio 93

61 1982: TKNN relate linear response to topology in the Quantum Hall Effect Thouless, Kohmoto, Nightingale, and den Nijs, (1982), Avron, Seiler, and Simon (1983); Simon (1983), Niu and Thouless (1984); Niu, Thouless, and Wu (1985) The Hall conductance is proportional to the first Chern number C = i 2π 2π 0 dφ 2π 0 [ Ψ dϕ φ Ψ ϕ Ψ ϕ ] Ψ φ with Ψ the many-body ground state obeying twisted boundary conditions. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 59 / fermio 93

62 : Khmelnitskii and Pruisken introduce the scaling theory of the Integer Quantum Hall Effect A topological term modifies the scaling analysis of the gang of four: σ Khmelnitskii 1983 xx Pruisken 1985 d ln g d ln L φ=π σ xy ln g d=2 φ=0 C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 60 / fermio 93

63 : Haldane introduces the θ term for spin chains and Witten achieves non-abelian bosonization F. D. M. Haldane, Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets: Semiclassically Quantized Solitons of the One-Dimensional Easy-Axis Néel State, Phys. Rev. Lett. 50, 1153 (1983), Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model, Phys. Lett. A 93, 464 (1983) : Topological θ term modifies the RG flow in the two-dimensional O(3) non-linear-sigma model. O(3) NLSM O(3) NLSM+ θ=π g SU(2) 1 E. Witten, Nonabelian bosonization in two dimensions, Commun. Math. Phys. 92, 455 (1984): Non-Abelian bosonization. g C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 61 / fermio 93

64 1988: Haldane model for a Chern band insulator F. D. M. Haldane, Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the Parity Anomaly, Phys. Rev. Lett. 61, 2015 (1988). C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 62 / fermio 93

65 1994: Random Dirac fermions in two-dimensional space A. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G. Grinstein, Integer quantum Hall transition: An alternative approach and exact results, Phys. Rev. B 50, 7526 (1994): Effects of static disorder on a single Dirac fermion in two-dimensional space. AIII IQHE D AII C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 63 / fermio 93

66 1997: The tenfold way for random matrices A. Altland and M. R. Zirnbauer, Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B 55, 1142 (1997). Cartan label T C S Hamiltonian G/H (ferm. NLSM) A (unitary) U(N) U(2n)/U(n) U(n) AI (orthogonal) U(N)/O(N) Sp(2n)/Sp(n) Sp(n) AII (symplectic) U(2N)/Sp(2N) O(2n)/O(n) O(n) AIII (ch. unit.) U(N + M)/U(N) U(M) U(n) BDI (ch. orth.) O(N + M)/O(N) O(M) U(2n)/Sp(2n) CII (ch. sympl.) Sp(N + M)/Sp(N) Sp(M) U(2n)/O(2n) D (BdG) SO(2N) O(2n)/U(n) C (BdG) Sp(2N) Sp(2n)/U(n) DIII (BdG) SO(2N)/U(N) O(2n) CI (BdG) Sp(2N)/U(N) Sp(2n) The column entitled Hamiltonian lists, for each of the ten symmetry classes, the symmetric space of which the quantum mechanical time-evolution operator exp(it H) is an element. The last column entitled G/H (ferm. NLσM) lists the (compact sectors of the) target space of the NLσM describing Anderson localization physics at long wavelength in this given symmetry class. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 64 / fermio 93

67 : Brouwer et al. establish that there are five symmetry class that display quantum criticality in disordered quasi-one-dimensional wires The radial coordinate of the transfer matrix M from the Table below makes a Brownian motion on an associated symmetric space. Class TRS SRS m o m l D M H δg ln g ρ(ε) for 0 < ετ c 1 O Yes Y CI AI 2/3 2L/(γl) ρ 0 U No Y(N) 2 1 2(1) AIII A 0 2L/(γl) ρ 0 S Y N DIII AII +1/3 2L/(γl) ρ 0 cho Y Y AI BDI 0 2m o L/(γl) ρ 0 ln ετ c chu N Y(N) 2 0 2(1) A AIII 0 2m o L/(γl) πρ 0 ετ c ln ετ c chs Y N AII CII 0 2m o L/(γl) (πρ 0 /3) (ετ c ) 3 ln ετ c CI Y Y C CI 4/3 2m l L/(γl) (πρ 0 /2) ετ c C N Y CII C 2/3 2m l L/(γl) ρ 0 ετ c 2 DIII Y N D DIII +2/3 4 L/(2πγl) πρ 0 / ετ c ln 3 ετ c D N N BDI D +1/3 4 L/(2πγl) πρ 0 / ετ c ln 3 ετ c C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 65 / fermio 93

68 2000: Read and Green introduce the chiral p-wave topological superconductor N. Read and D. Green, Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries and the fractional quantum Hall effect, Phys. Rev. B 61, (2000). C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 66 / fermio 93

69 2005: Kane and Mele introduce the strong Z 2 topological insulator The spin-orbit coupling is ignored in the QHE as the breaking of time-reversal symmetry provides the dominant energy scale. C. L. Kane and E. J. Mele, Z 2 Topological Order and the Quantum Spin Hall Effect, Phys. Rev. Lett. 95, (2005): Combine a pair of time-reversed Haldane models with a small Rashba coupling and find protected helical edge states. E (for a k 0 E F E 0 k y Γ S k x Bi x Pb 1 x /Ag(111) surface alloy, say) C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 67 / fermio 93

70 2008: The tenfold way for topological insulators and superconductors Schnyder, Ryu, Furusaki, and Ludwig 2008 and 2010; Kitaev 2008 complex case: Cartan\d 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: Cartan\d AI Z Z 0 Z 2 Z 2 Z BDI Z 2 Z Z 0 Z 2 Z 2 Z 0 0 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 C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 68 / fermio 93

71 1 Main results 2 Introduction 3 Strategy 4 Examples 5 Application to the surfaces of SnTe 6 Summary 7 Appendices C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 69 / fermio 93

72 Definition and question to be addressed Consider the random Dirac Hamiltonian in d-dimensional space of rank r given by H d,r = d i=1 α i i x i +M(x)+, {α i, α j } = 2 δ ij, i, j = 1,, d, where 1 r tr[m(x) M] [M(y) M] =: g2 e x y /ξ dis, with all higher cumulants vanishing. Is the single-particle energy eigenstate at ε = 0 delocalized or localized? A necessary but not sufficient condition for localization is that a Dirac mass matrix is premitted i.e., M(x) 0. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 69 / fermio 93

73 Why necessary but not sufficient? Choose d = 1 and r = 2: H d=1,r=2 = τ 1 [ i + A 1 (x)] + τ 0 A 0 (x) + τ 2 m 2 (x) + τ 3 m 3 (x). There is a single localized phase. Choose d = 2 and r = 2: H d=2,r=2 = τ 1 [ i 1 + A 1 (x)] + τ 2 [ i 2 + A 2 (x)] + τ 0 A 0 (x) + τ 3 m(x). There are two topological distinct phases and a critical line separating them if m(x) = 0. Choose d = 3 and r = 2: H d=3,r=1 = τ 1 [ i 1 + A 1 (x)] + τ 2 [ i 2 + A 2 (x)] + τ 3 [ i 3 + A 3 (x)] + τ 0 A 0 (x). Localization is prohibited. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 70 / fermio 93

74 A normalized Dirac mass matrix β is defined by the condition {α, β} = 0, β 2 = 1. When d = 1 and r = 2, the normalized Dirac mass matrix is β(x) = τ 2 cos θ(x) + τ 3 sin θ(x), tan θ(x) := m 3 (x) m 2 (x). As a space, it is the unit circle. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 71 / fermio 93

75 A normalized Dirac mass matrix β is defined by the condition {α, β} = 0, β 2 = 1. When d = 2 and r = 2, the normalized Dirac mass matrix is β(x) = m(x) m(x) τ 3, As a space, it is two points. If m(x) = m, then τ xy = 1 2sgn m. The domain wall m(x, y) = m(x) with lim x ± m(x) = m supports the boundary states Ψ(x, y; k) e ik y iτ y x dx m(x ) ( ) 1 i m(x) y with the dispersion x x ε(k) = k. This is the minimal model for the IQHE. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 72 / fermio 93

76 A normalized Dirac mass matrix β is defined by the condition {α, β} = 0, β 2 = 1 r. When d = 3 and r = 2, the normalized Dirac mass matrix is not allowed. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 73 / fermio 93

77 1 Main results 2 Introduction 3 Strategy 4 Examples 5 Application to the surfaces of SnTe 6 Summary 7 Appendices C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 74 / fermio 93

78 The tenfold way for the classifying spaces when d = 1 Symmetry class A: H(k) := τ 3 k + τ 3 A 1 + τ 2 M 2 + τ 1 M 1 + τ 0 A 0, V A d=1;r=2 = S1. Symmetry class AII: H(k) = +τ 2 H ( k) τ 2, Symmetry class AI: H(k) = +τ 1 H ( k) τ 1, Symmetry class AIII: V AII d=1;r=2 =. V AI d=1;r=2 = S1. H(k) = τ 1 H(k) τ 1, V AIII d=1;r=2 = {±τ 2 }. Symmetry class CII: No Dirac Hamiltonian when r = 2. Symmetry class BDI: H(k) = τ 1 H(k) τ 1 = +τ 1 H ( k) τ 1, V BDI d=1;r=2 = {±τ 2 }. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 74 / fermio 93

79 Symmetry class D: Symmetry class DIII: H(k) = H ( k), V D d=1;r=2 = {±τ 1 }. H(k) = H ( k) = +τ 2 H ( k) τ 2, V DIII d=1;r=2 =. Symmetry class C: H(k) = τ 2 H ( k) τ 2 = τ 3 A 1 + τ 2 M 2 + τ 1 M 1. PHS squaring to minus unity prohibits a kinetic energy in the symmetry class C. Symmetry class CI: H(k) = τ 2 H ( k) τ 2 = +τ 1 H ( k) τ 1 = τ 2 M 2 + τ 1 M 1. PHS squaring to minus unity prohibits a kinetic energy in the symmetry class CI. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 75 / fermio 93

80 1 Main results 2 Introduction 3 Strategy 4 Examples 5 Application to the surfaces of SnTe 6 Summary 7 Appendices C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 76 / fermio 93

81 Existence and uniqueness of normalized Dirac masses For each symmetry class, there exists a minimum rank N r min (d) > 2, for which the d-dimensional Dirac Hamiltonian supports a mass matrix and below which either no mass matrix or no Dirac kinetic contribution are allowed by symmetry. Suppose that the rank of the d-dimensional massive Dirac Hamiltonian is r = r min (d) N. When there exists a mass matrix squaring to unity (up to a sign) that commutes with all other symmetry-allowed mass matrices, we call it the unique mass matrix. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 76 / fermio 93

82 Existence and uniqueness of normalized Dirac masses There are the following three cases: (a) π 0 (V d ) = {0}: For any given mass matrix, there exists another mass matrix that anticommutes with it for N 1. (b) π 0 (V d ) = Z: There always is a unique mass matrix for N 1. (c) π 0 (V d ) = Z 2 : There is a unique mass matrix when N is an odd integer. However, when N is an even integer, there is no such unique mass matrix. When N = 1 and when the Dirac Hamiltonian has a unique mass matrix, then topologically distinct ground states are realized for different signs of its mass. When the mass is varied smoothly in space, domain boundaries where the sign-of-mass changes are accompanied by massless Dirac fermions, whose low-energy Hamiltonian is of rank r min (d)/2. It follows that r min (d 1) = r min (d). C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 77 / fermio 93

83 Path connectedness of the normalized Dirac masses Case (a): π 0 (V ) = {0} Trivial phase Odd N ν= 1/2 ν=+1/ Case (b): π 0 (V ) = Z Even N ν= 1 ν=0 ν= Case (c): π 0 (V ) = Z 2 ν=0 ν=1 C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 78 / fermio 93

84 Network model and criticality For each realization of a random potential perturbing the massless Dirac Hamiltonian defined in d-dimensional Euclidean space, it may be decomposed into open sets (domains) of linear size ξ dis. In each of these domains, the normalized Dirac masses correspond to a unique value of their zeroth homotopy group. At the boundary between domains differing by the values taken by their zeroth homotopy group, the Dirac masses must vanish. Such boundaries support zero-energy boundary states. B A A B B B A A B B B A When it is possible to classify the elements of the zeroth homotopy group by the pair of indices A and B, we may assign the letters A or B to any one of these domains as is illustrated. When the typical volume of a domain of type A equals that of type B, quasi-zero modes undergo quantum percolation through the sample and thus establish either a critical or a metallic phase of quantum matter in d-dimensional space. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 79 / fermio 93

85 The solution for the classifying spaces is Class T C Γ Extension V d A Cl d Cl d+1 C 0+d AIII Cl d+1 Cl d+2 C 1+d AI Cl 0,d+2 Cl 1,d+2 R 0 d BDI Cl d+1,2 Cl d+1,3 R 1 d D Cl d,2 Cl d,3 R 2 d DIII Cl d,3 Cl d,4 R 3 d AII Cl 2,d Cl 3,d R 4 d CII Cl d+3,0 Cl d+3,1 R 5 d C Cl d+2,0 Cl d+2,1 R 6 d CI Cl d+2,1 Cl d+2,2 R 7 d Class V π 0 (V ) π 0 (V ) π 0 (V ) π 0 (V ) π 0 (V ) π 0 (V ) π 0 (V ) π 0 (V ) A C 0+d Z 0 Z 0 Z 0 Z 0 AIII C 1+d 0 Z 0 Z 0 Z 0 Z AI R 0 d Z Z 0 Z 2 Z 2 BDI R 1 d Z 2 Z Z 0 Z 2 D R 2 d Z 2 Z 2 Z Z 0 DIII R 3 d 0 Z 2 Z 2 Z Z AII R 4 d Z 0 Z 2 Z 2 Z CII R 5 d 0 Z 0 Z 2 Z 2 Z 0 0 C R 6 d 0 0 Z 0 Z 2 Z 2 Z 0 CI R 7 d Z 0 Z 2 Z 2 Z C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 80 / fermio 93

86 where Label Classifying space V C 0 N { [ ]} n=0 U(N)/ U(n) U(N n) C 1 U(N) R 0 N { [ ]} n=0 O(N)/ O(n) O(N n) R 1 O(N) R 2 O(2N)/U(N) R 3 U(2N)/Sp(N) R 4 N { [ ]} n=0 Sp(N)/ Sp(n) Sp(N n) R 5 Sp(N) R 6 Sp(N)/U(N) R 7 U(N)/O(N) Label π 0 (V ) π 1 (V ) π 2 (V ) π 3 (V ) π 4 (V ) π 5 (V ) π 6 (V ) π 7 (V ) C 0 Z 0 Z 0 Z 0 Z 0 C 1 0 Z 0 Z 0 Z 0 Z R 0 Z Z 2 Z 2 0 Z R 1 Z 2 Z 2 0 Z Z R 2 Z 2 0 Z Z Z 2 R 3 0 Z Z Z 2 Z 2 R 4 Z Z Z 2 Z 2 0 R Z Z 2 Z 2 0 Z R Z Z 2 Z 2 0 Z 0 R 7 0 Z Z 2 Z 2 0 Z 0 0 C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 81 / fermio 93

87 The parameters for the phase diagram: π 0 (V ) = {0} For any of the five AZ symmetry classes with V = C 1, R 3, R 5, R 6, R 7, we can always choose the parameter space (m, g) R [0, [ by selecting α = α min 1 N, β 0 := β min 1 N, and M(x) =: m β 0, 1 r tr { [M(x) m β 0 ][M(y) m β 0 ] } =: g 2 e x y /ξ dis. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 82 / fermio 93

88 The parameters for the phase diagram: π 0 (V ) = Z For any of the three AZ symmetry classes with V = C 0, R 0, R 4, we can always choose the parameter space (m, g) R [0, [ by selecting and M(x) =: m β 0, α = α min 1 N, β 0 := β min 1 N, 1 r tr { [M(x) m β 0 ][M(y) m β 0 ] } =: g 2 e x y /ξ dis. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 83 / fermio 93

89 Moreover, we can also always write M(x) = β min M(x), where the condition B B det[m(x)] = 0 A A defines the boundaries of the domains from B the quantum network model, each of which can be assigned the index ν := 1 2 tr{ sgn[m(x)] } { 1/2, +1/2}, if N = 1, { 1, 0, +1}, if N = 2, and so on, if N > 2, A B A B B B A with [U(x) is unitary] M =: U diag (λ 1,..., λ N ) U, ( sgn[m(x)] := U λ1 diag λ 1,..., λ ) N λ N U. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 84 / fermio 93

90 The parameters for the phase diagram: π 0 (V ) = Z 2 For any of the two AZ symmetry classes with V = R 1, R 2, we can always choose the parameter space (m, g) R [0, [ by selecting α = α min 1 N, M(x) = ρ min M(x), with ρ min σ 2 delivering a representation of the Clifford algebra of rank r min, M(x) = M (x), M(x) = M T (x), m := Pf [im(x)], and M(x) = 0, 1 r tr [M(x) M(y)] =: g2 e x y /ξ dis. C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 85 / fermio 93

91 B A A B B B A A B B B A Moreover, the condition det[m(x)] = 0 defines the boundaries of the domains for the quantum network model, each of which can be assigned the index ( 1) ν := Pf [im(x)]. det [im(x)] C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 86 / fermio 93

92 1 Main results 2 Introduction 3 Strategy 4 Examples 5 Application to the surfaces of SnTe 6 Summary 7 Appendices C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 87 / fermio 93

93 Application to 1D space AZ symmetry class r min V d=1,r π 0 (V d=1,r ) Phase diagram Cut at m = 0 A 2 C 1 0 (a) insulating AIII 2 C 0 Z (b) even-odd AI 2 R 7 0 (a) insulating BDI 2 R 0 Z (b) even-odd D 2 R 1 Z 2 (c) critical DIII 4 R 2 Z, (c) critical 2 AII 4 R 3 0 (a) insulating CII 4 R 4 Z (b) even-odd C 4 R 5 0 (a) insulating CI 4 R 6 0 (a) insulating (a) A, AI, AII, C, CI (b) AIII, BDI, CII N odd N even (c) D, DIII Insulator... ν= 1/2 ν=1/ ν= 1 ν=0 ν=1... Insulator Insulator ν=0 ν=1 ν= N/2 ν=n/2 ν= N/2 ν=n/ C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 87 / fermio 93

94 Application to 2D space AZ symmetry class r min V d=2,r π 0 (V d=2,r ) π 1 (V d=2,r ) Phase diagram Cut at m = 0 A 2 C 0 Z 0 (a) even-odd AIII 4 C 1 0 Z (e) Gade singularity AI 4 R (b) insulating BDI 4 R 7 0 Z (e) Gade singularity D 2 R 0 Z Z 2 (c) even-odd DIII 4 R 1 Z 2 Z 2 (d) metallic AII 4 R 2 Z 2 0 (d) metallic CII 8 R 3 0 Z (f) Gade singularity C 4 R 4 Z 0 (a) even-odd CI 8 R (b) insulating (a) A, C (N=1) (b) A, C (N=2) (g) AII Metal Insulator ν= 1/2 Insulator ν=1/2 Insulator ν= 1 Insulator ν=0 Insulator ν=1 Insulator Insulator ν=0 ν= (c) D (N=1) Metal (d) D (N=2) Metal (h) AI, CI Insulator ν= 1/2 Insulator ν=1/2 Insulator ν= 1 Z2 singularity Insulator ν=0 Insulator ν=1 Insulator C. Mudry (PSI) (e) DIII (N=1) The breakdown of(f) the DIII topological (N=2) classification Z for gapped (i) AIII, BDI, phases CII of noninteracting 88 / fermio 93

95 Application to 2D space (a) A, C (N=1) (b) A, C (N=2) (g) AII Metal Insulator ν= 1/2 Insulator ν=1/2 Insulator ν= 1 Insulator ν=0 Insulator ν=1 Insulator Insulator ν=0 ν= (c) D (N=1) Metal (d) D (N=2) Metal (h) AI, CI Insulator ν= 1/2 Insulator ν=1/2 Insulator ν= 1 Z2 singularity Insulator ν=0 Insulator ν=1 Insulator (e) DIII (N=1) Metal (f) DIII (N=2) Metal (i) AIII, BDI, CII Gade singularity Insulator Insulator ν=0 ν=1 Insulator Insulator ν=0 ν=1 Z2 singularity Z2 singularity Insulator C. Mudry (PSI) The breakdown of the topological classification Z for gapped phases of noninteracting 89 / fermio 93