Quantum transport of 2D Dirac fermions: the case for a topological metal

Transcription

1 Quantum transport of 2D Dirac fermions: the case for a topological metal Christopher Mudry 1 Shinsei Ryu 2 Akira Furusaki 3 Hideaki Obuse 3,4 1 Paul Scherrer Institut, Switzerland 2 University of California at Berkeley, USA 3 RIKEN, JAPAN 4 Kyoto University, JAPAN Gregynog Hall, University of Wales, UK, November C. Mudry (PSI) Quantum transport of 2D Dirac fermions:... 1 / 28

2 Outline 1 Introduction 2 Two-dimensional Dirac fermions 3 Topological term as the sign of a Pfaffian 4 Conclusions 5 Appendix A: What are ripples? 6 Appendix B: Numerical results C. Mudry (PSI) Quantum transport of 2D Dirac fermions:... 2 / 28

3 1 Introduction 2 Two-dimensional Dirac fermions 3 Topological term as the sign of a Pfaffian 4 Conclusions 5 Appendix A: What are ripples? 6 Appendix B: Numerical results C. Mudry (PSI) Quantum transport of 2D Dirac fermions:... 3 / 28

4 Anderson localization Question: What is the return probability of a single quantum particle in a static random environment? Answer: dimensionality of space, weak versus strong disorder, short-range versus long-range disorder, symmetries. C. Mudry (PSI) Quantum transport of 2D Dirac fermions:... 3 / 28

5 Anderson localization and scaling Gang of Four (Abrahams, Anderson, Licciardello, and Ramakrishnan) for a noninteracting electron gas with TRS and SRS and with a finite density of state at the Fermi energy predicts: d ln g d ln L d= ln g d=2 d=1 C. Mudry (PSI) Quantum transport of 2D Dirac fermions:... 4 / 28

6 From symmetry to universality classes (say in Q1D) White-noise 10 symmetry classes Universality as N disorder in Q1D Hamiltonian Spectrum {}}{{}}{ (a random walk) dl dl<l<<l,nl TRS SRS CHS BdG {x n (0)} n=1,...,n* {x n (L)} n=1,...,n* {x n (L + δl)} n=1,...,n* (DMPK ) (Altland, Zirnbauer 1996) (Hüffmann 1990) C. Mudry (PSI) Quantum transport of 2D Dirac fermions:... 5 / 28

7 Universality Non-linear-Sigma Models (NLSM) For white-noise Gaussian disorder, the existence of a diffusive regime in any dimension, l L ξ, implies a NLSM description of Anderson localization. ξ l The fields in a NLSM are maps from base space to target space There is a 1-to-1 correspondence between the local data of the target space (metric and curvature) and the 10 symmetry classes. The global properties (topology) of the target manifold can also matter. C. Mudry (PSI) Quantum transport of 2D Dirac fermions:... 6 / 28

8 Symmetry classes with topological term in d = 2 TRS SRS CHS BdG Topology could matter π = θ term (Fendley) u(1) θ term (Pruisken) π = θ term (Fendley) WZW term (Guruswamy et al.) WZW term (Nersesyan et al.) WZW term (Fendley) u(1) θ term (Senthil et al.) u(1) θ term (Bocquet et al.) Warning: Symmetry alone does not guaranty the topological term, i.e., a microscopic derivation of the topological term is always needed. C. Mudry (PSI) Quantum transport of 2D Dirac fermions:... 7 / 28

9 Are topological terms of any importance? σ xx Khmelnitskii 1983 Pruisken 1985 d ln g d ln L σ xy φ=π ln g d=2 φ=0 C. Mudry (PSI) Quantum transport of 2D Dirac fermions:... 8 / 28

10 Questions and Answers Q: Is there a microscopic realization of the NLSM with π = θ topological term in the symmetry class with TRS and SRS? A: Yes, graphene with infinite long-range impurities and effective TRS. Q: What is the effect of the topological term? φ=π??? d ln g d ln L φ=π??? φ=π??? d ln g d ln L φ=π φ=0 d=2 g xx??? ln g??? g xy A: It is to make the two-dimensional metallic phase stable. d=2 ln g C. Mudry (PSI) Quantum transport of 2D Dirac fermions:... 9 / 28

11 1 Introduction 2 Two-dimensional Dirac fermions 3 Topological term as the sign of a Pfaffian 4 Conclusions 5 Appendix A: What are ripples? 6 Appendix B: Numerical results C. Mudry (PSI) Quantum transport of 2D Dirac fermions: / 28

12 Square lattice with π flux phase Direct space Reciprocal space k y k x K - π/a Py K + b 2 G π/a Px b 1 C. Mudry (PSI) Quantum transport of 2D Dirac fermions: / 28

13 Honeycomb lattice with graphene-like dispersion Direct space Reciprocal space y K + K - b1 K - K + s2 k x s3 s1 b2 K + K - K - k y K + b 1 G k x b 2 C. Mudry (PSI) Quantum transport of 2D Dirac fermions: / 28

14 Low-energy effective Hamiltonian without disorder H := d 2 r Ψ (r) H(r) Ψ(r) where H = 0 2i z 0 0 2i z i z 0 0 2i z 0, Ψ(r) = u b (r) u a (r) v a (r) v b (r). C. Mudry (PSI) Quantum transport of 2D Dirac fermions: / 28

15 Low-energy effective Hamiltonian with infinite-range disorder and with effective TRS For infinite-range disorder with effective TRS where H σ p + σ 0 V (r) V (r) = 0, V (r)v (r ) = gδ(r r ), g 0, the symplectic symmetry (Ludwig et al. 1994) ( +iσy ) H ( iσ y ) = H emerges. C. Mudry (PSI) Quantum transport of 2D Dirac fermions: / 28

16 Effective symplectic symmetry TRS SRS CHS BdG Topology could matter Is there a π = θ term? u(1) θ term (Pruisken) Is there a π = θ term? WZW term (Guruswamy et al.) WZW term (Nersesyan et al.) WZW term (Fendley) u(1) θ term (Senthil et al.) u(1) θ term (Bocquet et al.) Warning: The range of the disorder can control the presence or absence of a topological term, i.e., a microscopic derivation of the topological term is always needed. C. Mudry (PSI) Quantum transport of 2D Dirac fermions: / 28

17 1 Introduction 2 Two-dimensional Dirac fermions 3 Topological term as the sign of a Pfaffian 4 Conclusions 5 Appendix A: What are ripples? 6 Appendix B: Numerical results C. Mudry (PSI) Quantum transport of 2D Dirac fermions: / 28

18 Topological term as the sign of a Pfaffian: Step 1 Z := lim N r 0 = lim N r 0 D[V ]P[V ] R d 2 r 4N Pr ψ a iσ 2 [(σ p+σ 0 V) δ ab +iηλ ab ]ψ b a,b=1 D[ψ]e D[Q] e 2 R 4g d 2 r tr QQ T }{{} D[ψ]e R d 2 r ψ T iσ 2 (σ p+iηλ i Q)ψ (π/g) =: ln h1 + (a ) 2i }{{} =Pf e D[Q] (±) Det e D[Q] with Q O(4N r )/O(2N r ) O(2N r ) Q 2 = 1, Q = Q T, tr Q = 0. Observe that: The infinitesimal η lifts the degeneracy between the retarded and advanced sectors: G G/H, G = O(4N r ) H = O(2N r ) O(2N r ). D[Q] is antisymmetric. the imaginary part to the self-energy of plane waves in the self-consistent Born approximation opens a gap about 0 in the spectrum of D[Q] for Q close to Λ. C. Mudry (PSI) Quantum transport of 2D Dirac fermions: / 28

19 Topological term as the sign of a Pfaffian: Step 2 How to compute Pf D[Q]? 1 Use the identity Det D[Q] = Det D [Q] where D [Q] = σ p + σ 3 Q. 2 Take advantage of the fact that the spectrum of D [Q] comes in pairs of nonvanishing real-valued eigenvalues ±λ i. 3 For some reference Q, define Pf D[Q] := i λ i with λ i either one of λ i or +λ i. 4 Although the sign of Pf D[Q] is protected by the spectral gap under infinitesimal changes of Q, it is not protected globally as Z 2 if N r > 1, π 2 (G/H) = Z Z if N r = 1. C. Mudry (PSI) Quantum transport of 2D Dirac fermions: / 28

20 Topological term as the sign of a Pfaffian: Step 3 1 To get a countable index i for the spectrum of D[Q], we compactify base space to the surface of the sphere (x, y) R 2 (θ, φ) S 2, π 2 θ π 2, 0 φ 2π. 2 Following Weinberg, London, and Rosner (1984), we define I 2Nr Q k (θ, φ) := 0 q k (θ, φ) 0, 0 0 I 2Nr 2 ( ) cos θi2 sin θr q k (θ, φ) := k (φ) sin θrk T (φ) cos θi, 2 ( ) cos kφ sin kφ R k (φ) :=. sin kφ cos kφ 3 We have verified numerically that ) ) sgn Pf ( D[Qk ] = sgn Pf ( D[Qk+1 ], k Z. C. Mudry (PSI) Quantum transport of 2D Dirac fermions: / 28

21 Topological term as the sign of a Pfaffian: Final Step Z := lim N r 0 = lim N r 0 D[V ]P[V ] R d 2 r 4N Pr ψ a iσ 2 [(σ p+σ 0 V) δ ab +iηλ ab ]ψ b a,b=1 D[ψ]e D[Q] e 2 R 4g d 2 r tr QQ T }{{} D[ψ]e R d 2 r ψ T iσ 2 (σ p+iηλ i Q)ψ (π/g) =: ln h1 + (a ) 2i }{{} D[Q] ( 1) n[q] e S NLSM [Q] =Pf e D[Q] (±) Det e D[Q] Z topolo NLSM also derived by Ostrovsky, Gornyi, and Mirlin (2007) where S NLSM [Q] is the usual local action for the NLσM on G/H and n[q] = 0, 1 is the Z 2 topological quantum number of Q. The topological term has its origin in the Pfaffian arising from Majorana fermions, i.e., the global anomaly. C. Mudry (PSI) Quantum transport of 2D Dirac fermions: / 28

22 1 Introduction 2 Two-dimensional Dirac fermions 3 Topological term as the sign of a Pfaffian 4 Conclusions 5 Appendix A: What are ripples? 6 Appendix B: Numerical results C. Mudry (PSI) Quantum transport of 2D Dirac fermions: / 28

23 Conclusions The topological term in the NLSM corresponding to the symplectic symmetry class in 2D is realized by a single Majorana fermion with a random scalar potential. However, local tight-binding models with the translation symmetry of a regular lattice, conserved electric charge, and TRS support an even number of Majorana fermions. To evade this no-go theorem: Infinite range disorder [Nersesyan, Tsvelik, and Wenger (1994) + Ando and Suzuura (2002)]. Work on the 2D boundary of a 3D tight-binding model [Fradkin, Dagotto, and Boyanovsky(1986)]. The topological term cannot be treated perturbatively. Numerical simulations are consistent with a topological metal [Bardarson, Tworzydlo, Brouwer, and Beenakker + Nomura, Koshino, and Ryu + San-Jose, Prada, and Golubev (2007)]. C. Mudry (PSI) Quantum transport of 2D Dirac fermions: / 28

24 d ln g d ln L φ=π 0 ln g φ=0 d=2 C. Mudry (PSI) Quantum transport of 2D Dirac fermions: / 28

25 1 Introduction 2 Two-dimensional Dirac fermions 3 Topological term as the sign of a Pfaffian 4 Conclusions 5 Appendix A: What are ripples? 6 Appendix B: Numerical results C. Mudry (PSI) Quantum transport of 2D Dirac fermions: / 28

26 Appendix A: What are ripples? C. Mudry (PSI) Quantum transport of 2D Dirac fermions: / 28

27 1 Introduction 2 Two-dimensional Dirac fermions 3 Topological term as the sign of a Pfaffian 4 Conclusions 5 Appendix A: What are ripples? 6 Appendix B: Numerical results C. Mudry (PSI) Quantum transport of 2D Dirac fermions: / 28

28 Numerical set up Following the proposal of Witten (1982) to study the SU(2) global anomaly in four-dimensional space-time (π 4 [SU(2)] = Z 2 ), we study the spectral flow of D [Q t ] where Q t := (1 t)q i + tq f, 0 t 1 with Q i,f G/H. This is done by regularizing Det D [Q t ] with base space compactified to a torus T 2 whereby eigenvalues ±λ are labeled by the quantized momentum k µ = 2πn µ /L, n µ = (N 1)/2,..., (N 1)/2 } µ = 1, 2, N = L a an odd integer. ) ) The signature for a sign difference between Pf ( D[Qi ] and Pf ( D[Qf ] is an odd number of level crossings at zero eigenvalue. C. Mudry (PSI) Quantum transport of 2D Dirac fermions: / 28

29 Case of no level crossing Interpolating backround field Q t = (1 t)q i + tq f for 0 t 1 C. Mudry (PSI) Quantum transport of 2D Dirac fermions: / 28

30 Case of one level crossing Interpolating backround field Q t = (1 t)q i + tq f for 0 t 1 C. Mudry (PSI) Quantum transport of 2D Dirac fermions: / 28

31 Case of two (degenerate) level crossings Interpolating backround field Q t = (1 t)q i + tq f for 0 t 1 C. Mudry (PSI) Quantum transport of 2D Dirac fermions: / 28

32 Case of two (nondegenerate) level crossings Interpolating backround field Q t = (1 t)q i + tq f for 0 t 1 C. Mudry (PSI) Quantum transport of 2D Dirac fermions: / 28

33 Case of three (degenerate) level crossings Interpolating backround field Q t = (1 t)q i + tq f for 0 t 1 C. Mudry (PSI) Quantum transport of 2D Dirac fermions: / 28

34 Case of three (nondegenerate) level crossings Interpolating backround field Q t = (1 t)q i + tq f for 0 t 1 C. Mudry (PSI) Quantum transport of 2D Dirac fermions: / 28