Interactions in Topological Matter

Transcription

1 Interactions in Topological Matter Christopher Mudry 1 1 Paul Scherrer Institute, Switzerland Harish-Chandra Research Institute, Allahabad, February C. Mudry (PSI) Interactions in Topological Matter 1 / 108

2 Outline 1 Introduction Tutorials 2 The tenfold way in quasi-one-dimensional space 3 Fractionalization from Abelian bosonization Tutorials 4 Stability analysis for the edge theory in the symmetry class AII 5 Construction of two-dimensional topological phases from coupled wires C. Mudry (PSI) Interactions in Topological Matter 2 / 108

3 1931: Dirac introduces topology in physics Paul Dirac, Quantised Singularities in the Electromagnetic Field, Proc. Roy. Soc. (London) A 133, 60 (1931). C. Mudry (PSI) Interactions in Topological Matter 3 / 108

4 : Tamm and Schockley surface states I. Tamm (1932), 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) Interactions in Topological Matter 4 / 108

5 1950 s: Anderson localization, Dyson s exception P. W. Anderson, Absence of Diffusion in Certain Random Lattices, Phys. Rev. 109, 1492 (1957). F. J. Dyson, The Dynamics of a Disordered Linear Chain, Phys. Rev. 92, 1331 (1953). C. Mudry (PSI) Interactions in Topological Matter 5 / 108

6 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) Interactions in Topological Matter 6 / 108

7 : Berezinski-Kosterlitz-Thouless transition Topology acquires a mainstream status in physics as of 1973 with the disovery of Berezinskii and of Kosterlitz and Thousless that topological defects in magnetic classical textures can drive a phase transition. V. L. Berezinskiǐ, Destruction of Long-range Order in One-dimensional and Two-dimensional Systems having a Continuous Symmetry Group I. Classical Systems, Soviet Journal of Experimental and Theoretical Physics, , (1971). J. M. Kosterlitz and D. J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems,, J. Phys. C, 6, 1181 (1973). J. M. Kosterlitz, The critical properties of the two-dimensional xy model, J. Phys. C, 7, 1046 (1974). C. Mudry (PSI) Interactions in Topological Matter 7 / 108

8 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) Interactions in Topological Matter 8 / 108

9 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) Interactions in Topological Matter 9 / 108

10 1980 s: The Quantum Hall Effect K. von Klitzing, G. Dorda, and M. Pepper, New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall, Phys. Rev. Lett. 45, 494 (1980). 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., Nature 438, 201 (2005). C. Mudry (PSI) Interactions in Topological Matter 10 / 108

11 The Integer Quantum Hall Effect ε m= 1 m h ω c Without disorder n εm h ωc m= 1 or ε F With disorder but no interactions n 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) Interactions in Topological Matter 11 / 108

12 1982: The Fractional Quantum Hall Effect D. C. Tsui, H. L. Stormer, and A. C. Gossard, Two-Dimensional Magnetotransport in the Extreme Quantum Limit, Phys. Rev. Lett. 48, 1559 (1982). At fractional fillings of a Landau level, r s is effectively : A landau level is a massively degenerate flat band of single-particle states. Naively, one would expect a Wigner crystal (or more exotic ground states with broken symmetry) to be selected by the interaction out of all possible degenerate Slater determinants. Instead, for magic filling fractions, featureless (i.e., liquid like) ground states are selected by the screened Coulomb interaction. For example, whenever 1/ν is an odd integer, the featureless ground state is an incompressible ground state called a Laughlin state. C. Mudry (PSI) Interactions in Topological Matter 12 / 108

13 Distinctive signature The conductivity tensor is given by the classical Drude formula ( lim j = 0 + ( ) ) 1 B R H τ ( ) 1 E, R 1 H := n e c. B R H 0 in the ballistic regime when Galilean invariance is not broken. In the presence of moderate static disorder, all but one single-particles are localized in a Landau level whereas many-body groundstates such as the Wigner crystal are pinned. In the presence of moderate static disorder the magic filling fractions turn into plateaus at which σ xx = 0, as a function of B for fixed n. σ xy = ν e2 h C. Mudry (PSI) Interactions in Topological Matter 13 / 108

14 : 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) Interactions in Topological Matter 14 / 108

15 1982: TKNN relate linear response to topology in the Quantum Hall Effect 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. D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized Hall Conductance in a Two-Dimensional Periodic Potential, Phys. Rev. Lett. 49, 405 (1982). J. E. Avron, R. Seiler and B. Simon, Homotopy and Quantization in Condensed Matter Physics, Phys. Rev. Lett. 51, 51 (1983); Holonomy, the Quantum Adiabatic Theorem, and Berry s Phase, B. Simon, ibid. 51, 2167 (1983). Q. Niu and D. J. Thouless, Quantised adiabatic charge transport in the presence of substrate disorder and many-body interaction, J. Phys. A 17, 2453 (1984). Q. Niu, D. J. Thouless, and Y. S. Wu, Quantized Hall conductance as a topological invariant, Phys. Rev. B 31, 3372 (1985). C. Mudry (PSI) Interactions in Topological Matter 15 / 108

16 : 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) Interactions in Topological Matter 16 / 108

17 : 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) Interactions in Topological Matter 17 / 108

18 1988: Haldane model 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) Interactions in Topological Matter 18 / 108

19 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) Interactions in Topological Matter 19 / 108

20 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) Interactions in Topological Matter 20 / 108

21 : Brouwer et al. establish that there are five symmetry class that are quantum critical 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) Interactions in Topological Matter 21 / 108

22 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) Interactions in Topological Matter 22 / 108

23 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 k 0 E F E 0 k y Γ S k x (for a Bi x Pb 1 x /Ag(111) surface alloy, say) C. Mudry (PSI) Interactions in Topological Matter 23 / 108

24 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) Interactions in Topological Matter 24 / 108

25 2011: Fractional Chern and fractional Z 2 topological insulators a) 1 T. Neupert, L. Santos, C. Chamon, and C. Mudry, Fractional Quantum Hall States at Zero Magnetic Field, Phys. Rev. Lett. 106, (2011) and T. Neupert, L. Santos, S. Ryu, C. Chamon, and C. Mudry, Fractional topological liquids with time-reversal symmetry and their lattice realization, Phys. Rev. B 84, (2011). λ fold 9-fold U/V b) c) d) U/V = 0, λ = 0 U/V = 0, λ = 1 U/V = 3, λ = 1 E/V γ x γ 2 x γ x C. Mudry (PSI) Interactions in Topological Matter 25 / 108

26 The goals of these lectures are the following. First, we would like to rederive the tenfold way for non-interacting fermions in the presence of local interactions and static local disorder. Second, we would like to decide if interactions between fermions can produce topological phases of matter with protected boundary states that are not captured by the tenfold way. We will apply this program in two-dimensional space. C. Mudry (PSI) Interactions in Topological Matter 26 / 108

27 Main result TABLE I. (Color online) Realization of a two-dimensional array of quantum wires in each symmetry class of the tenfold way. 2 Θ 2 Π 2 C 2 Short-range entangled (SRE) topological phase Long-range entangled (LRE) topological phase A Z AIII NONE AII 0 0 Z 2 DIII + + Z 2 D Z BDI NONE AI NONE CI + + NONE C 0 0 Z CII + NONE C. Mudry (PSI) Interactions in Topological Matter 27 / 108

28 Organization of the lectures Introduction The tenfold way in quasi-one-dimensional space Fractionalization from Abelian bosonization Stability analysis for the edge theory in the symmetry class AII Construction of two-dimensional topological phases from coupled wires C. Mudry (PSI) Interactions in Topological Matter 28 / 108

29 Tutorial on Galilean transformation A Galilean transformation is a transformation to a new frame of reference that leaves the time difference t 1 t 2 and space separation x 1 x 2 unchanged as well as the equation of motion m ẍ = 0 of a free particle form invariant. It is given by t = t + a, x = Ox + vt + w, a R, v, w R 3, O O(3). The transformation law of momentum p mẋ and kinetic energy E kin (1/2)mẋ 2 under a Galilean transformation are p = Op + mv, E kin = E kin + m (Oẋ) v + (1/2)m v 2. C. Mudry (PSI) Interactions in Topological Matter 29 / 108

30 Tutorial on reversal of time: Reversal of time A Galilean transformation can be composed with reversal of time t = t, x = x. Under reversal of time, the momentum p := m ẋ = p is odd and so is the angular momentum L := x p = L. C. Mudry (PSI) Interactions in Topological Matter 30 / 108

31 Tutorial on reversal of time: The case of classical electromagnetism Maxwell equations are separated into the pair of homogeneous equations B = 0, E + 1 c t B = 0, and into the pair of inhomogeneous equations E = 4π ρ, B 1 c They are invariant under the transformation laws t E = 4π c j. t = t, E(x, t ) = +E(x, t ), B(x, t ) = B(x, t ), ρ(x, t ) = +ρ(x, t ), j(x, t ) = j(x, t ). C. Mudry (PSI) Interactions in Topological Matter 31 / 108

32 Tutorial on reversal of time: The case of quantum mechanics for a spinless particle We start with the non-relativistic Schrödinger equation for a spinless particle i t ψ(x, t) = { 1 2m [ ˆp e ] 2 } c A(x, t) + e A0 (x, t) + V (x, t) ψ(x, t). How to implement reversal of time? We demand that is odd under reversal of time This is achieved by defining ˆp := i x t = t, x = +x. ˆp = K ˆp K = ˆp where K represents the antilinear operation of complex conjugation. C. Mudry (PSI) Interactions in Topological Matter 32 / 108

33 We conclude that i { 1 [ t ψ(x, t) = ˆp e ] 2 } 2m c A(x, t) + e A0 (x, t) + V (x, t) ψ(x, t) is invariant under the transformations t = t, x = +x, ψ (+x, t ) K ψ(+x, t ) = ψ(x, t ), ˆp K ˆp K = ˆp, A(+x, t ) = A(x, t ), A 0 (+x, t ) = +A 0 (x, t ), V (+x, t ) = +V (x, t ). C. Mudry (PSI) Interactions in Topological Matter 33 / 108

34 Tutorial on reversal of time: The case of quantum mechanics for a spin-1/2 particle We start with the Pauli equation for a spin-1/2 particle i { 1 [ ( t Ψ(x, t) = σ ˆp e )] } 2 2m c A(x, t) + σ0 [e A 0 (x, t) + V (x, t)] Ψ(x, t). How to implement reversal of time? We demand that σ 0 ˆp := σ 0 i x are both odd under reversal of time This is achieved by defining and σ (σ 1, σ 2, σ 3 ) t = t, x = +x. σ 0 ˆp = (σ 2 K) σ 0 ˆp (K σ 2 ) = σ 0 ˆp and σ = (σ 2 K) σ (K σ 2 ) = σ, where K represents the antilinear operation of complex conjugation. C. Mudry (PSI) Interactions in Topological Matter 34 / 108

35 We conclude that i t Ψ(x, t) = { 1 2m is invariant under the transformations t = t, x = +x, [ ( σ ˆp e )] } 2 c A(x, t) + σ0 [e A 0 (x, t) + V (x, t)] Ψ(x, t). σ 2 Ψ (+x, t ) (σ 2 K) Ψ(+x, t ) = Ψ(x, t ), (σ 2 K) σ (K σ 2 ) = σ, σ 0 ˆp (σ 2 K) σ 0 ˆp (K σ 2 ) = σ 0 ˆp, A(+x, t ) = A(x, t ), A 0 (+x, t ) = +A 0 (x, t ), V (+x, t ) = +V (x, t ). C. Mudry (PSI) Interactions in Topological Matter 35 / 108

36 Tutorial on one-dimensional Dirac Hamiltonians Let the Bloch Hamiltonian for one spinless fermion be H(k) := 2t cos k, π k < +π, (uniform nearest-neighbor hopping with t > 0). Linearization of this dispersion about the two Fermi points ±k F with π < k F < +π gives the rank-two Dirac Hamiltonian H D := τ 3 i x ( 1, v F = 2t sin k F 1). The lattice model uniquely specifies reversal of time for the rank-two Dirac Hamiltonian, namely conjugation by τ 1 K, for (τ 1 K) H D (K τ 1 ) = H D. C. Mudry (PSI) Interactions in Topological Matter 36 / 108

37 Let the Bloch Hamiltonian for one spin-1/2 fermion be H(k) := 2t σ 0 cos k, π k < +π, (uniform nearest-neighbor hopping t > 0). Linearization of this dispersion about the two Fermi points ±k F with π < k F < +π gives the rank two Dirac Hamiltonian H D := τ 3 σ 0 i x ( 1, v F = 2t sin k F 1). The lattice model uniquely specifies reversal of time for the rank-four Dirac Hamiltonian, namely conjugation by τ 1 σ 2 K, for (τ 1 σ 2 K) H D (K τ 1 σ 2 ) = H D. C. Mudry (PSI) Interactions in Topological Matter 37 / 108

38 Assume that you are given the rank-two Dirac Hamiltonian H D := τ 3 i x. In how many ways can you define reversal of time? Reversal of time involves complex conjuation since we must undo the sign change of the time derivative in i t Ψ(x, t) = H D Ψ(x, t) under t t. However, complex conjugation alone reverses the sign of the momentum operator ˆp := i x on the right-hand side of the Dirac equation. We can undo this change of sign by conjugation with either τ 1 or τ 2, { (τ H D = 1 K) H D (K τ 1 ), (τ 2 K) H D (K τ 2 ). With no reference to a microscopic model, there is no unique way to define reversal of time (This is not so for Weyl Hamiltonian in 3D!). C. Mudry (PSI) Interactions in Topological Matter 38 / 108

39 Tutorial on Anderson localization: Schrödinger equation with a delta-function double well d 2 H dw = 1 v [δ(x + d/2) + δ(x d/2)] 2 dx 2 in units with = m = 1. C. Mudry (PSI) Interactions in Topological Matter 39 / 108

40 Tutorial on Anderson localization: Schrödinger equation with short-range correlated potential disorder In units with = m = 1, consider the random Hamiltonian with the vanishing mean the second moment d 2 H = 1 2 dx 2 + V (x), V (x) = 0, V (x) V (y) = g 2 v e x y /ξ dis, and all higher moments vanishing. Locality means that ξ dis <. Typically, all states are exponentially localized for this one-dimensional random Hamiltonian. C. Mudry (PSI) Interactions in Topological Matter 40 / 108

41 Tutorial on Anderson localization: Tight-binding model with weak short-range correlated disorder Start from (a) (b) Break weakly translation invariance by adding uncorrelated on-site potentials µ i and uncorrelated nearest-neighbor hopping t i = t e iφ i of vanishing means. You then get H = iτ 3 x + a 0 (x) τ 0 + m 1 (x) τ 1 + m 2 (x) τ 2 + a 1 (x) τ 3 with a C., Mudry m,(psi) m, and a uncorrelated Interactions in Topological beyond Matter some length scale ξ. 41 / 108

42 Tutorial on Anderson localization: The quasi-one-dimensional case Define the quasi-d-dimensional Dirac Hamiltonian H(x) = i(α I) x + V(x), where α and β are a set of matrices that anticommute pairwise and square to the unit r min r min matrix, I is a unit N N matrix, and (1a) V(x) = m(x) β I + (1b) with representing all other masses, vector potentials, and scalar potentials allowed by the AZ symmetry class. C. Mudry (PSI) Interactions in Topological Matter 42 / 108

43 For one-dimensional space, the stationary eigenvalue problem H(x) Ψ(x; ε) = ε Ψ(x; ε) (2) with the given initial value Ψ(y; ε) is solved through the transfer matrix Ψ(x; ε) = M(x y; ε) Ψ(y; ε) where M(x y; ε) = P x exp x y dx i(α I) [ε V(x )]. (3a) (3b) The symbol P x represents path ordering. The limit N with all entries of V independently and identically distributed (iid) up to the AZ symmetry constraints, (averaging over the disorder is denoted by an overline) [ ] [Vkl ] V ij (x) v ij, V ij (x) v ij (y) v kl g 2 e x y /ξ dis, (4) for i, j, k, l = 1,, r min N defines the thick quantum wire limit. C. Mudry (PSI) Interactions in Topological Matter 43 / 108

44 The consequences of Eq. (3) are the following. First, the local symmetries defining the symmetry classes A, AII, and AI obeyed by ε V(x ) carry through to the transfer matrix at any single-particle energy ε. The local unitary spectral symmetries defining the symmetry classes AIII, CII, and BDI and the local anti-unitary spectral symmetries defining the symmetry classes D, DIII, C, and CI carry through to the transfer matrix at the single-particle energy ε = 0. C. Mudry (PSI) Interactions in Topological Matter 44 / 108

45 Second, the diagonal matrix entering the polar decomposition of the transfer matrix at the band center ε = 0 is related to the non-compact symmetric spaces from the column M in Table 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) Interactions in Topological Matter 45 / 108

46 Third, the composition law obeyed by the transfer matrix that encodes enlarging the length of a disordered wire coupled to perfect leads is matrix multiplication. It is then possible to derive a Fokker-Planck equation for the joint probability obeyed by the radial coordinates on the non-compact symmetric spaces from the column M in the Table as the length L of of a disordered wire coupled to perfect leads is increased. C. Mudry (PSI) Interactions in Topological Matter 46 / 108

47 In this way, the moments of the dimensionless Landauer conductance g in the columns δg and ln g can be computed. An infinitesimal increase in the length of the disordered region for one of the ten symmetry classes induces an infinitesimal Brownian motion of the Lyapunov exponents that is solely controlled by the multiplicities of the ordinary, long, and short roots of the corresponding classical semi-simple Lie algebra under suitable assumptions on the disorder (locality, weakness, and isotropy between all channels). 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) Interactions in Topological Matter 47 / 108

48 When the transfer matrix describes the stability of the metallic phase in the thick quantum wire limit of non-interacting fermions perturbed by static one-body random potentials with local correlations and of vanishing means in the bulk of a quasi-one-dimensional lattice model, the multiplicities of the short root entering the Brownian motion of the Lyapunov exponents always vanish. 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) Interactions in Topological Matter 48 / 108

49 However, when the transfer matrix describes the quasi-one-dimensional boundary of a two-dimensional topological band insulator moderately perturbed by static one-body random potentials with local correlations, the multiplicities of the short roots is nonvanishing in the Brownian motion of the Lyapunov exponents in the five AZ symmetry classes A, AII, D, DIII, and C. Correspondingly, the conductance is of order one along the infinitely long boundary, i.e., the insulating bulk supports extended edge states. These extended edge states can be thought of as realizing a quasi-one-dimensional ballistic phase of quantum matter robust to disorder. C. Mudry (PSI) Interactions in Topological Matter 49 / 108

50 Organization of the lectures Introduction with tutorials The tenfold way in quasi-one-dimensional space Fractionalization from Abelian bosonization Stability analysis for the edge theory in the symmetry class AII Construction of two-dimensional topological phases from coupled wires C. Mudry (PSI) Interactions in Topological Matter 50 / 108

51 Symmetry class A and r = 2: The Dirac Hamiltonian H = τ 3 k + A 0 τ 0 + M 1 τ 1 + M 2 τ 2 + A 1 τ 3 (5a) is said to belong to the AZ symmetry class A. The set V A d=1,r=2 := { β(θ) := τ 1 cos θ + τ 2 sin θ 0 θ < 2π} =: S1, (5b) a circle, is the topological space of normalized Dirac masses associated with the Dirac Hamiltonian. Note that Vd=1,r=2 A and U(1) are homeomorphic as topological spaces. Thus, they share the same homotopy groups. On the other hand, Vd=1,r=2 A is not a group under matrix multiplication, while U(1) is. C. Mudry (PSI) Interactions in Topological Matter 51 / 108

52 Symmetry class AII and r = 2: If charge conservation holds and TRS is imposed through H(k) = +τ 2 H ( k) τ 2, (6a) then H(k) = τ 3 k + τ 0 A 0. No mass matrix is permissible if TRS squares to minus the identity. The topological space of normalized Dirac masses in the symmetry class AII is the empty set (6b) V AII d=1,r=2 =. (6c) Because of the fermion-doubling problem, the only way to realize (6b) as the low-energy and long wavelength limit of a lattice model with short-range correlated disorder is on the boundary of a two-dimensional topological insulator in the symmetry class AII. C. Mudry (PSI) Interactions in Topological Matter 52 / 108

53 Symmetry class AI and r = 2: If charge conservation holds and TRS is imposed through H(k) = +τ 1 H ( k) τ 1, (7a) then H(k) = τ 3 k + τ 2 M 2 + τ 1 M 1 + τ 0 A 0. The same mass matrix as in the symmetry class A is permissible if TRS squares to the identity. The homeomorphy between the allowed masses in the symmetry classes A and AI is accidental. It does not hold for larger representations of the Dirac matrix. The topological space of normalized Dirac masses is (7b) V AI d=1,r=2 := {β(θ) 0 θ < 2π} =: S1. (7c) The topological spaces Vd=1,r=2 AI and U(1) are homeomorphic. [This homeomorphism is not a group homomorphism, for Vd=1,r=2 A is not a group while U(1) is.] Consequently, they share the same homotopy groups. C. Mudry (PSI) Interactions in Topological Matter 53 / 108

54 Symmetry class AIII and r = 2: If charge conservation holds together with the CHS H(k) = τ 1 H(k) τ 1, (8a) then H(k) = τ 3 k + τ 3 A 1 + τ 2 M 2. There is a unique mass matrix. The topological space of normalized Dirac masses obtained by adding to the Dirac kinetic contribution a mass matrix squaring to unity and obeying the CHS is (8b) V AIII d=1,r=2 = {±τ 2 }. (8c) C. Mudry (PSI) Interactions in Topological Matter 54 / 108

55 Symmetry class CII and r = 2: It is not possible to write down a 2 2 Dirac equation in the symmetry class CII. For example, imposing H(k) = τ 1 H(k) τ 1, H(k) = +τ 2 H ( k) τ 2, (9) enforces the symmetry class DIII, for composing the CHS with the TRS delivers a PHS that squares to the unity and not minus the unity. In order to implement the symmetry constraints of class CII, we need to consider a 4 4 Dirac equation. C. Mudry (PSI) Interactions in Topological Matter 55 / 108

56 Symmetry class BDI and r = 2: If charge conservation holds together with then H(k) = τ 1 H(k) τ 1, H(k) = +τ 1 H ( k) τ 1, (10a) H(k) = τ 3 k + τ 2 M 2. There is a unique mass matrix. The topological space of normalized Dirac masses obtained by adding to the Dirac kinetic contribution a mass matrix squaring to unity while preserving TRS and PHS (a product of TRS and CHS), both of which square to unity, is (10b) V BDI d=1,r=2 = {±τ 2 }. (10c) C. Mudry (PSI) Interactions in Topological Matter 56 / 108

57 Symmetry class D and r = 2: If we impose PHS through H(k) = H ( k), (11a) then H(k) = τ 3 k + τ 2 M 2. (11b) There is a unique mass matrix. The topological space of normalized Dirac masses obtained by adding to the Dirac kinetic contribution a mass matrix squaring to unity and preserving the PHS squaring to unity is Vd=1,r=2 D = {±τ 1 }. (11c) C. Mudry (PSI) Interactions in Topological Matter 57 / 108

58 Symmetry class DIII and r = 2: If we impose PHS and TRS through respectively, then H(k) = H ( k), H(k) = +τ 2 H ( k) τ 2, (12a) H(k) = τ 3 k. No mass matrix is permissible if TRS squares to minus the identity. The topological space of normalized Dirac masses in the symmetry class DIII is the empty set (12b) V DIII d=1,r=2 =. (12c) Because of the fermion-doubling problem,the only way to realize (12b) as the low-energy and long wavelength limit of a lattice model with short-range correlated disorder is on the boundary of a two-dimensional topological superconductor in the symmetry class DIII. C. Mudry (PSI) Interactions in Topological Matter 58 / 108

59 Symmetry class C and r = 2: If we impose PHS through H(k) = τ 2 H ( k) τ 2, (13a) then H(k) = τ 3 A 1 + τ 2 M 2 + τ 1 M 1. PHS squaring to minus unity prohibits a kinetic energy in any Dirac Hamiltonian of rank 2 in the symmetry class C. (13b) C. Mudry (PSI) Interactions in Topological Matter 59 / 108

60 Symmetry class CI and r = 2: If we impose PHS and TRS through H(k) = τ 2 H ( k) τ 2, H(k) = +τ 1 H ( k) τ 1, (14a) respectively, then H(k) = τ 2 M 2 + τ 1 M 1. (14b) PHS squaring to minus unity prohibits a kinetic energy in the symmetry class CI. C. Mudry (PSI) Interactions in Topological Matter 60 / 108

61 Symmetry class A and r = 4: The Dirac Hamiltonian in the symmetry class A is H(k) :=τ 3 σ 0 k + τ 3 σ ν A 1,ν + τ 2 σ ν M 2,ν + τ 1 σ ν M 1,ν + τ 0 σ ν A 0,ν. (15a) The topological space of normalized Dirac masses is { ( ) 0 U Vd=1,r=4 A := β = U U U(2)}. (15b) 0 As a topological space, it is thus homeomorphic to U(2) U(1) SU(2) S 1 S 3, an interpretation rendered plausible by the parameterization { } Vd=1,r=4 A = M X+N Y M 2 = cos 2 θ, N = tan θm = S 1 S 3, M := ( M 2,0, M 1,1, M 1,2, M 1,3 ), N := ( M1,0, M 2,1, M 2,2, M 2,3 ), X := (τ 2 σ 0, τ 1 σ 1, τ 1 σ 2, τ 1 σ 3 ), Y := ( τ 1 σ 0, τ 2 σ 1, τ 2 σ 2, τ 2 σ 3 ). (15c) C. Mudry (PSI) Interactions in Topological Matter 61 / 108

62 Symmetry class AII and r = 4: If charge conservation holds together with TRS through then H(k) = +τ 1 σ 2 H ( k) τ 1 σ 2, H(k) = τ 3 σ 0 k + τ 3 σ ν A 1,ν + τ 2 σ 0 M 2,0 ν=1,2,3 + τ 1 σ 0 M 1,0 + τ 0 σ 0 A 0,0. (16a) (16b) Observe that by doubling the Dirac Hamiltonian (6b), we went from no mass matrix to two anticommuting mass matrices. The topological space of normalized Dirac masses is V AII d=1,r=4 := {β = ( 0 U U 0 ) U = +σ 2 U T σ 2 U(2)}. (16c) C. Mudry (PSI) Interactions in Topological Matter 62 / 108

63 As a topological space, Vd=1,r=4 AII can be shown to be homeomorphic to U(2)/Sp(1) U(1) SU(2)/SU(2) U(1), an interpretation rendered plausible by the parameterization } Vd=1,r=4 {M AII = X M 2 = 1 =: S 1, M := ( M 2,0, M 1,0 ), X := (τ2 σ 0, τ 1 σ 0 ). C. Mudry (PSI) Interactions in Topological Matter 63 / 108

64 Symmetry class AI and r = 4: If charge conservation holds together with TRS through H(k) = +τ 1 σ 0 H ( k) τ 1 σ 0, (17a) then H(k) = τ 3 σ 0 k + τ 3 σ 2 A 1,2 + ( τ2 σ ν M 2,ν ν=0,1,3 (17b) ) + τ 1 σ ν M 1,ν + τ 0 σ ν A 0,ν. There are six mass matrices of rank r = 4 in the 1D symmetry class AI that can be arranged into the three pairs (M 1,ν, M 2,ν ) with ν = 0, 1, 3 of anticommuting masses. The topological space of normalized Dirac masses is V AI d=1,r=4 := {β = ( ) 0 U U U = +U T U(2)}. (17c) 0 C. Mudry (PSI) Interactions in Topological Matter 64 / 108

65 As a topological space, Vd=1,r=4 AI can be shown to be homeomorphic to U(2)/O(2) U(1)/{±1} SU(2)/U(1) S 1 S 2, an interpretation rendered plausible by the parameterization } Vd=1,r=4 {M AI = X +N Y M 2 = cos 2 θ, N = tan θm =: S 1 S 2, M := ( M 2,0, M 1,1, M 1,3 ), N := ( M1,0, M 2,1, M 2,3 ), X := (τ 2 σ 0, τ 1 σ 1, τ 1 σ 3 ), Y := ( τ 1 σ 0, τ 2 σ 1, τ 2 σ 3 ). C. Mudry (PSI) Interactions in Topological Matter 65 / 108

66 Symmetry class AIII and r = 4: If charge conservation holds together with the CHS then H(k) = τ 3 σ 0 k + H(k) = τ 1 σ 0 H(k) τ 1 σ 0, ν=0,1,2,3 (18a) ( τ3 σ ν A 1,ν + τ 2 σ ν M 2,ν ). (18b) The Dirac mass matrix τ 2 σ 0 M 2,0 that descends from Eq. (8b) commutes with the triplet of anticommuting mass matrices τ 2 σ 1 M 2,1, τ 2 σ 2 M 2,2, and τ 2 σ 3 M 2,3. The topological space of normalized Dirac masses is Vd=1,r=4 {β AIII := = τ 2 A A := U I m,n U, m, n = 0, 1, 2, m + n = 2, U U(2), m times n times } {}}{{}}{ I m,n := diag( 1,..., 1, +1,..., +1). C. Mudry (PSI) Interactions in Topological Matter 66 / 108

67 As a topological space, it is thus homeomorphic to U(2)/[U(2) U(0)] U(2)/[U(1) U(1)] U(2)/[U(0) U(2)], as is also apparent from the parameterization } Vd=1,r=4 AIII = {±τ 2 σ 0 {M } X M 2 = 1, M := ( M 2,1, M 2,2, M 2,3 ), X := (τ 2 σ 1, τ 2 σ 2, τ 2 σ 3 ), (recall that S 2 SU(2)/U(1) so that U(2)/[U(1) U(1)] S 2 ). C. Mudry (PSI) Interactions in Topological Matter 67 / 108

68 Symmetry class CII and r = 4: If charge conservation holds together with CHS and TRS respectively, then H(k) = τ 3 σ 0 k + H(k) = τ 1 σ 0 H(k) τ 1 σ 0, H(k) = +τ 1 σ 2 H ( k) τ 1 σ 2, ν=1,2,3 τ 3 σ ν A 1,ν + τ 2 σ 0 M 2,0. (19a) (19b) (19c) There is a unique mass matrix, as was the case in Eqs. (8b) and (10b). The topological space of normalized Dirac masses is { } Vd=1,r=4 CII := β Vd=1,r=4 AIII β = (τ 1 σ 2 ) β (τ 1 σ 2 ). (19d) As a topological space, Vd=1,r=4 CII can be shown to be homeomorphic to Sp(1)/Sp(1) Sp(0) Sp(1)/Sp(0) Sp(1), an interpretation rendered plausible by the parameterization V CII d=1,r=4 = {±τ 2 σ 0 }. (19e) C. Mudry (PSI) Interactions in Topological Matter 68 / 108

69 Symmetry class BDI and r = 4: If charge conservation holds together together with CHS and TRS H(k) = τ 1 σ 0 H(k) τ 1 σ 0, H(k) = +τ 1 σ 0 H ( k) τ 1 σ 0, respectively, then H(k) = τ 3 σ 0 k + τ 3 σ 2 A 1,2 + τ 2 σ ν M 2,ν. ν=0,1,3 (20a) (20b) (20c) The Dirac mass matrix τ 2 σ 0 M 2,0 that descends from Eq. (10b) commutes with the pair of anticommuting mass matrices τ 2 σ 1 M 2,1 and τ 2 σ 3 M 2,3. The topological space of normalized Dirac masses is V BDI d=1,r=4 := { β V AIII d=1,r=4 } β = (τ 1 σ 0 ) β (τ 1 σ 0 ). (20d) C. Mudry (PSI) Interactions in Topological Matter 69 / 108

70 As a topological space, Vd=1,r=4 BDI can be shown to be homeomorphic to O(2)/[O(2) O(0)] O(2)/[O(1) O(1)] O(2)/[O(0) O(2)], an interpretation rendered plausible from the parameterization } Vd=1,r=4 BDI = {±τ 2 σ 0 {M } X M 2 = 1, M := ( M 2,1, M 2,3 ), X := (τ 2 σ 1, τ 2 σ 3 ), (recall that S 1 O(2)/[O(1) O(1)]). C. Mudry (PSI) Interactions in Topological Matter 70 / 108

71 Symmetry class D and r = 4: If we impose PHS through then H(k) = H ( k), H(k) = τ 3 σ 0 k + τ 3 σ 2 A 1,2 + τ 2 σ ν M 2,ν ν=0,1,3 + τ 1 σ 2 M 1,2 + τ 0 σ 2 A 0,2. (21a) (21b) There are four Dirac mass matrices. None commutes with all remaining ones. However, each of them is antisymmetric and so is their sum. The topological space of normalized Dirac masses is V D d=1,r=4 = { M X M 2 = 1 } { } N Y N 2 = 1, M := ( M 2,1, M 2,3 ), N := ( M2,0, M 1,2 ), X := (τ 2 σ 1, τ 2 σ 3 ), Y := (τ 2 σ 0, τ 1 σ 2 ). As a topological space, Vd=1,r=4 D is homeomorphic to O(2), as Vd=1,r=4 D S1 S 1 U(1) Z 2 O(2). (22) C. Mudry (PSI) Interactions in Topological Matter 71 / 108

72 Symmetry class DIII and r = 4: If we impose PHS and TRS through then H(k) = H ( k), H(k) = +τ 2 σ 0 H ( k) τ 2 σ 0, (23a) H(k) = τ 3 σ 0 k + τ 3 σ 2 A 1,2 + τ 1 σ 2 M 1,2. Observe that there is only one Dirac mass matrix [there was none in Eq. (12b)]. Moreover, this Dirac mass matrix is Hermitian and antisymmetric. The topological space of normalized Dirac masses is (23b) V DIII d=1,r=4 = {±τ 1 σ 2 }. (23c) As a topological space, Vd=1,r=4 DIII is homeomorphic to O(2)/U(1). C. Mudry (PSI) Interactions in Topological Matter 72 / 108

73 Symmetry class C and r = 4: If we impose PHS through then H(k) = τ 0 σ 2 H ( k) τ 0 σ 2, H(k) = τ 3 σ 0 k + τ 3 σ ν A 1,ν + τ 2 σ 0 M 2,0 + ν=1,2,3 τ 1 σ ν M 1,ν + ν=1,2,3 ν=1,2,3 τ 0 σ ν A 0,ν. (24a) (24b) There are four mass matrices that anticommute pairwise. The topological space of normalized Dirac masses is } Vd=1,r=4 {M C = X M 2 = 1 =: S 3, M := ( ) M 2,0, M 1,1, M 1,2, M 1,3, (24c) X := (τ 2 σ 0, τ 1 σ 1, τ 1 σ 2, τ 1 σ 3 ). As a topological space, Vd=1,r=4 C is homeomorphic to Sp(1) since we have Sp(1) SU(2) S 3. C. Mudry (PSI) Interactions in Topological Matter 73 / 108

74 Symmetry class CI and r = 4: If we impose PHS and TRS through H(k) = τ 0 σ 2 H ( k) τ 0 σ 2, H(k) = +τ 1 σ 0 H ( k) τ 1 σ 0, (25a) respectively, then H(k) = τ 3 σ 0 k + τ 3 σ 2 A 1,2 + τ 2 σ 0 M 2,0 + ( ) τ1 σ ν M 1,ν + τ 0 σ ν A 0,ν. (25b) ν=1,3 There are three mass matrices that anticommute pairwise. The topological space of normalized Dirac masses is } Vd=1,r=4 {M CI = i X i M 2 i = 1 =: S 2, M := ( ) M 2,0, M 1,1, M 1,3, (26) X := (τ 2 σ 0, τ 1 σ 1, τ 1 σ 3 ). As a topological space, Vd=1,r=4 CI is homeomorphic to Sp(1)/U(1) since we have the homeomorphism Sp(1)/U(1) SU(2)/U(1) S 2. C. Mudry (PSI) Interactions in Topological Matter 74 / 108

75 Classifying spaces 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) U(N)/O(N) R 7 C. Mudry (PSI) Interactions in Topological Matter 75 / 108

76 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) Interactions in Topological Matter 76 / 108

77 B A B B A B B A B A B A C. Mudry (PSI) Interactions in Topological Matter 77 / 108

78 AZ symmetry class r min V d=1,r π 0 (V d=1,r ) Phase diagram from the Figure below 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) AIII, BDI, CII N odd N even (b) D, DIII (c) A, AI, AII, C, CI... ν= 1/2 ν=1/ ν= 1 ν=0 ν=1... Insulator Insulator ν=0 ν=1 Insulator ν= N/2 ν=n/2 ν= N/2 ν=n/ C. Mudry (PSI) Interactions in Topological Matter 78 / 108

79 Organization of the lectures Introduction with tutorials The tenfold way in quasi-one-dimensional space Fractionalization from Abelian bosonization Stability analysis for the edge theory in the symmetry class AII Construction of two-dimensional topological phases from coupled wires C. Mudry (PSI) Interactions in Topological Matter 79 / 108

80 Fractionalization from Abelian bosonization Universal data: K ij = K ji Z, with det K 0 and q i Z where ( 1) K ii = ( 1) q i,. Non-universal data: V ij = V ji Z, positive definite matrix. L [ 1 Ĥ := dx 4π V ( ) ( ) ij Dx û i D x û j 0 ( qi + A 0 2π K 1 ( ) ) ] D ij x û j (t, x), D x û i (t, x) := ( ) x û i + q i A 1 (t, x), [ ] ] û i (t, x), û j (t, y) = iπ [K ij sgn(x y) + L ij, 0, if i = j, L ij = L ji = ) sgn(i j) (K ij + q i q j, otherwise, û i (t, x + L) = û i (t, x) + 2πn i, n i Z. Chiral equations of motions: 0 = δ ik D 0 û k + K ij V jk D 1 û k Anomalous continuity equation: µ Ĵ µ A = σ 1 H t Ĵ 0 (t, x) = 1 2π q i K 1 ) (D ij 1 û j (t, x), Ĵ 1 (t, x) = 1 ) 2π q i V ij (D 1 û j (t, x) + σ H A 0 (t, x) with σ H 1 2π x ( q i K 1 ij z (a) q j ). B y (b) where C. Mudry (PSI) Interactions in Topological Matter 80 / 108

81 Application The conserved charge ( Q = dx ˆ ψ γ ˆψ) 0 (t, x) ɛ01 [ ˆφ(t, x = + ) ˆφ(t, x = )] 2π R for the static profile ϕ(x) is approximately given by (27) Q ɛ01 [ϕ(x = + ) ϕ(x = )]. (28) 2π On the other hand, the number of electrons per periode T = 2π/ω that flows across a point x Î = T 0 dt ( ˆ ψ γ 1 ˆψ) (t, x) ɛ10 2π [ ˆφ(T, x) ˆφ(0, x)] for the uniform profile ϕ(t) = ω t is approximately given (29) Î ɛ10 2π ω T = ɛ10. (30) C. Mudry (PSI) Interactions in Topological Matter 81 / 108

82 Tutorial on the anomalous continuity equation Define the quantum Hamiltonian (in units with the electric charge e, the speed of light c, and set to one) Ĥ = L 0 dx [ 1 ) ( 4π V qi ( )) ij (D x û i ) (D ] x û j + A 0 2π K 1 ij D x û j (t, x), D x û i (t, x) := ( x û i + q i A 1 ) (t, x). (31) The indices i, j = 1,, N label the bosonic modes. Summation is implied for repeated indices. The N real-valued quantum fields û i (t, x) obey the equal-time commutation relations [ ] ] û i (t, x), û j (t, y) = iπ [K ij sgn(x y) + L ij (32) for any pair i, j = 1,, N. The function sgn(x) = sgn( x) gives the sign of the real variable x and will be assumed to be periodic with periodicity L. C. Mudry (PSI) Interactions in Topological Matter 82 / 108

83 The N N matrix K is symmetric, invertible, and integer valued. Given the pair i, j = 1,, N, any of its matrix elements thus obey K ij = K ji Z, K 1 ij = K 1 ji Q. (33) The N N matrix L is anti-symmetric 0, if i = j, L ij = L ji = ) sgn(i j) (K ij + q i q j, otherwise, for i, j = 1,, N. The sign function sgn(i) of any integer i is here not made periodic and taken to vanish at the origin of Z. The N N matrix V is symmetric and positive definite (34) V ij = V ji R, v i V ij v j > 0, i, j = 1,, N, (35) for any nonvanishing vector v = (v i ) R N. The charges q i are integer valued and satisfy ( 1) K ii = ( 1) q i, i = 1,, N. (36) C. Mudry (PSI) Interactions in Topological Matter 83 / 108

84 The external scalar gauge potential A 0 (t, x) and vector gauge potential A 1 (t, x) are real-valued functions of time t and space x coordinates. They are also chosen to be periodic under x x + L. Finally, we shall impose the boundary conditions and for any i = 1,, N. û i (t, x + L) = û i (t, x) + 2πn i, n i Z, (37) ( x û i ) (t, x + L) = ( x û i ) (t, x), (38) First important result: The equations of motion are chiral. 0 = δ ik D 0 û k + K ij V jk D 1 û k, i = 1,, N, (39) C. Mudry (PSI) Interactions in Topological Matter 84 / 108

85 Proof of anomalous continuity equation Second important result: The anomalous continuity equation holds. Proof. µ Ĵ µ = +σ H A 1 t (40) With the help of [ ] D x û i (t, x), D y û j (t, y) = 2πi K ij δ (x y) (41) for i, j = 1,, N, one verifies that the total derivative of Ĵ0 (t, x) is Ĵ0 t [Ĵ0 ] = i, Ĥ A + σ 1 H t = Ĵ1 x + σ H A 1. (42) t C. Mudry (PSI) Interactions in Topological Matter 85 / 108

86 Proof. Alternatively, introduce the density and the current density ˆρ(t, x) = 1 2π q i K 1 ( ) ij x û j (t, x) (43a) ĵ(t, x) = 1 2π q i V ij ( x û j ) (t, x). (43b) First, verify that taking the divergence over Ĵµ gives µ Ĵ µ t Ĵ 0 + x Ĵ 1 = t ˆρ + σ H t A }{{} + 1 x ĵ + 1 ( ) qi V 2π ij q j x A 1 + σ H x A 0. (44) Second, verify with the help of the chiral equations of motion that t ˆρ + x ĵ = 1 2π ( qi V ij q j ) x A 1 σ H x A 0. (45) C. Mudry (PSI) Interactions in Topological Matter 86 / 108

87 Tutorial on quasi-particle (anyons) and Fermi-Bose excitations A first application of the Baker-Campbell-Hausdorff formula to any pair of quasi-particle vertex operator at equal time t but two distinct space coordinates x y gives Ψ q-p,i (t, x) Ψ Θq-p q-p,j (t, y) = e iπ ij Ψ q-p,j (t, y) Ψ q-p,i (t, x), (46a) where Θ q-p ij ( = K 1 ji sgn(x y) + K 1 ik K 1 jl Here and below, it is understood that ) K kl + q k K 1 ik K 1 jl q l sgn(k l). (46b) sgn(k l) = 0 (47) when k = l = 1,, N. Hence, the quasi-particle vertex operators obey neither bosonic nor fermionic statistics since K 1 ij Q. C. Mudry (PSI) Interactions in Topological Matter 87 / 108

88 The same exercise applied to the Fermi-Bose vertex operators yields Ψ f-b,i (t, x) Ψ f-b,j (t, y) = ( 1) K ii ( 1) q i q j Ψ f-b,i (t, y) Ψ f-b,i (t, x), if i = j, Ψ f-b,j (t, y) Ψ f-b,i (t, x), if i j, (48) when x y. The self statistics of the Fermi-Bose vertex operators is carried by the diagonal matrix elements K ii Z. The mutual statistics of any pair of Fermi-Bose vertex operators labeled by i j is carried by the product q i q j Z of the intger-valued charges q i and q j. Had we not assumed that K ij with i j are integers, the mutual statistics would not be Fermi-Bose because of the non-local term K ij sgn (x y). C. Mudry (PSI) Interactions in Topological Matter 88 / 108

89 A third application of the Baker-Campbell-Hausdorff formula allows to determine the boundary conditions Ψ q-p,i (t, x + L) = Ψ 1 K q-p,i (t, x) e 2πi ij N 1 i πi K e ii (49) and Ψ f-b,i (t, x + L) = Ψ f-b,i (t, x) e 2πi N i e πi K ii (50) obeyed by the quasi-particle and Fermi-Bose vertex operators, respectively. C. Mudry (PSI) Interactions in Topological Matter 89 / 108

90 Tutorial on the Abelian bosonization rules Table: Abelian bosonization rules in two-dimensional Minkowski space. The conventions of relevance to the scalar mass ˆ ψ ˆψ and the pseudo-scalar mass ˆ ψ γ 5 ˆψ are ˆ ψ = ˆψ γ 0 with ˆψ = ( ˆψ, ˆψ + ), whereby γ0 = τ 1 and γ 1 = iτ 2 so that γ 5 = γ 5 = γ 0 γ 1 = τ 3. Fermions Bosons Kinetic energy Current ˆ ψ iγ µ µ ˆψ 1 8π ( µ ˆφ)( µ ˆφ) ˆ ψ γ µ ˆψ 1 2π ɛ µν ν ˆφ Chiral currents 2 ˆψ ˆψ ± 1 2π x û Right and left movers ˆψ 1 4π a e iû Backward scattering ˆψ ˆψ 1 ˆφ + e i 4π a Cooper pairing ˆψ ˆψ + 1 4π a e i ˆθ Scalar mass ˆψ ˆψ + + ˆψ + ˆψ 1 2π a cos ˆφ Pseudo-scalar mass ˆψ ˆψ + ˆψ + ˆψ i sin ˆφ 2π a C. Mudry (PSI) Interactions in Topological Matter 90 / 108

91 Organization of the lectures Introduction with tutorials The tenfold way in quasi-one-dimensional space Fractionalization from Abelian bosonization Stability analysis for the edge theory in the symmetry class AII Construction of two-dimensional topological phases from coupled wires C. Mudry (PSI) Interactions in Topological Matter 91 / 108

92 Organization of the lectures Introduction with tutorials The tenfold way in quasi-one-dimensional space Fractionalization from Abelian bosonization Stability analysis for the edge theory in the symmetry class AII Construction of two-dimensional topological phases from coupled wires C. Mudry (PSI) Interactions in Topological Matter 92 / 108

93 Stability analysis for the edge theory in the symmetry class AII The edge of a two-dimensional insulator in the symmetry class AII is described by L Ĥ := Ĥ0 + Ĥint, Ĥ 0 := dx 1 4π x ˆΦ T V x ˆΦ, 0 L Ĥ int := dx ( h T (x) : cos T T ) K ˆΦ(x) + α T (x) :. (51a) T L 0 The real functions h T (x) 0 and 0 α T (x) 2π encode information about the disorder along the edge when position dependent. The set L := {T Z 2N } T T Q = 0, (51b) encodes all the possible charge neutral tunneling processes (i.e., those that just rearrange charge among the branches). The theory is quantized according to the equal-time commutators [ˆΦ i (t, x), ˆΦ j (t, x ( )] = iπ K 1 sgn(x x ) + Θ ij ij ), (51c) where K is a 2N 2N symmetric and invertible matrix with integer-valued matrix elements, and the Θ matrix accounts for Klein factors that ensure that charged excitations in the theory (vertex operators) satisfy the proper commutation relations. C. Mudry (PSI) Interactions in Topological Matter 93 / 108

94 Tutorial on quasi-particle/fermi-bose vertex operators ( ) The universal data are (K, Q). The non-universal data are V, h T (x), α T (x). The boundary conditions K ij ˆΦ j (t, x + L) = K ij ˆΦ j (t, x) + 2πn i (52) with n i Z for all i = 1,..., 2N are imposed. Together with the condition that the tunneling vectors T are restricted to have integer-valued components, this ensures that the Hamiltonian Ĥ is single-valued. The chiral nature of the bosonic quantum fields arises from demanding that the equal-time commutator [ˆΦi (t, x), ˆΦ j (t, x ( )] = iπ K 1 sgn(x x ) ) + Θ ij ij (53) holds for any pair i, j = 1,..., 2N. Here, Θ ij := K 1 L ik kl K 1 lj (54) and the antisymmetric 2N 2N matrix L is defined by ) L ij = sgn(i j) (K ij + Q i Q j, (55) where sgn(0) = 0 is understood. It then follows that the quadratic theory in Eq. (51) is endowed with chiral equations of motion. Finally, we need to impose the compatibility conditions ( 1) K ii = ( 1) Q i, i = 1,..., 2N, (56) in order to construct local excitations with well-defined statistics. C. Mudry (PSI) Interactions in Topological Matter 94 / 108