Anderson localization, topology, and interaction

Transcription

1 Anderson localization, topology, and interaction Pavel Ostrovsky in collaboration with I. V. Gornyi, E. J. König, A. D. Mirlin, and I. V. Protopopov PRL 105, (2010), PRB 85, (2012) Cambridge, 20 September 2012

2 Outline 1 Introduction Weak (anti)localization Quantum Hall effect, quantum spin-hall effect Symmetry classification Topological insulators 2 Localization in chiral systems Absence of weak localization Non-perturbative localization by vortices Chiral topological insulators 3 Localization in symplectic systems Interaction effects QSHE phase diagram Effect of vortices 4 Summary and outlook Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

3 Introduction Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

4 Scaling theory of localization Abrahams, Anderson, Licciardello, Ramakrishnan 79 Dimensionless conductance [in units e 2 =h]: Metallic sample (Ohm s law): g L d 2 Insulating sample (tunneling): g e L= Universal scaling function: 8 d ln g < d ln L = (g) = d 2; g 1; (metal); : ln g; g 1; (insulator): 3D d ln g d ln L 0 2D 1D ln g Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

5 Quantum interference correction Metal with time-reversal symmetry =) Cooperon loop Z = 2e2 (d d q) >< = e2 l L ; 3D; h q 2 ln(l=l); 2D; h >: L l ; 1D Correction diverging in 2D and 1D Spin preserved (broken) =) weak (anti)localization No time-reversal symmetry =) two diffuson loops Very weak localization in 2D: = e4 2h 2 ln(l=l) Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

6 Weak localization correction in 2D Gor kov, Larkin, Khmelnitskii 79; Hikami, Larkin, Nagaoka 80 8 (g) d ln g >< 1 ; d ln L = (2g 2 ) 1 ; >: +(g) 1 ; orthogonal (TR preserved, spin preserved); unitary (TR broken); symplectic (TR preserved, spin broken) d lnσ d ln L 0 Σ Sp 1.4 Sp U O lnσ Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

7 Nonlinear sigma model Effective theory for interacting diffusons and Cooperons Z S[Q] = h i d 2 r tr D(rQ) 2 + 2i!ΛQ 8 Matrix field 8 U(2N)=U(N) U(N); >< Q 2 Sp(2N)=Sp(N) Sp(N); >: O(2N)=O(N) O(N); Replica limit N! 0 is assumed unitary (A); Renormalization of D =) weak (anti)localization orthogonal (AI); symplectic (AII): Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

8 Quantum Hall effect von Klitzing, Dorda, Pepper 80 Electron states localized extended E E F Quantized resistance Landau levels Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

9 Quantum Hall effect Pruisken 83; Khmelnitskii 83 Low-energy theory: Z unitary class sigma model with -term S[Q] = d 2 r tr xx 8 (rq)2 + xy 4 (Qr xqr y Q) Q : S 2 7! M = U(2N)=U(N) U(N) Z Top. invariant: 2(M) = Z d 2 r tr Qr x Qr y Q = 8in =) e S is periodic in xy! 2Π Two-parameter scaling: Θ 2ΠΣ xy Π 0 0 Σ U 0.6 Quantum-Hall transition at half-integer xy and xx = U 0:6 Σ xx Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

10 Quantum spin-hall effect Kane, Mele 05, Sheng et al 05 Bernevig, Zhang 06 No magnetic field but strong spin-orbit interaction =) Electrons with opposite spins feel opposite effective magnetic field Electric current leads to spin accumulation at the edges =) spin-hall effect Extreme spin-orbit coupling opens a band gap =) quantum spin-hall effect Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

11 Quantum (spin-)hall effect with disorder Impurities do not destroy the edge (spin) current: QHE QSHE due to chirality of carriers any disorder Z top. term in 2D due to time-inversion symmetry no magnetic impurities Z 2 top. term in 1D Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

12 Quantum spin-hall effect: experiment Molenkamp group 07 HgTe/CdTe quantum well I d = 5:5nm: normal insulator II, III, IV d = 7:3nm: inverted band gap topological insulator Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

13 Symmetry Classification Wigner 51, Dyson 62, Altland and Zirnbauer 97, Schnyder et al 08 Always present: diffuson Time-reversal symmetry (T): H = UH T U 1 ; T 2 = UU = 1 Cooperon Chiral symmetry (C): H = UHU 1 RR diffuson Particle-hole symmetry (CT): H = UH T U 1 ; CT 2 = UU = 1 RR Cooperon R A R R R A R R Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

14 Symmetry Classification Wigner 51, Dyson 62, Altland and Zirnbauer 97, Schnyder et al 08 T 2 C CT 2 NLM WL A U(2N)=U(N) U(N) 0 Z 0 0(-) AI Sp(2N)=Sp(N) Sp(N) AII O(2N)=O(N) O(N) Z 2 Z AIII U(N) Z 0 Z 0 BDI U(2N)=Sp(2N) Z CII U(N)=O(N) Z Z 2 Z 2 0 D O(2N)=U(N) 0 Z 0 + C Sp(2N)=U(N) 0 Z Z 2 - DIII O(N) Z 2 0 Z + CI Sp(2N) 0 0 Z - Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

15 Classification: Bott periodicity Kitaev 08, Schnyder et al 08 T 2 C CT 2 NLM A U(2N)=U(N) U(N) 0 Z 0 Z AIII U(N) Z 0 Z 0 AI Sp(2N)=Sp(N) Sp(N) Z BDI U(2N)=Sp(2N) Z D O(2N)=U(N) 0 Z 0 0 DIII O(N) Z 2 0 Z 0 AII O(2N)=O(N) O(N) Z 2 Z 2 0 Z CII U(N)=O(N) Z Z 2 Z 2 0 C Sp(2N)=U(N) 0 Z Z 2 Z 2 CI Sp(2N) 0 0 Z Z 2 Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

16 Classification of topological insulators Kitaev 08, Schnyder et al 08 Z topology Let d = Z (as in QHE at d = 2) =) topological term may appear in d dimensions =) (d 1)-dimensional surface states delocalized =) d-dimensional Z topological insulator Z 2 topology Let d = Z 2 (as on the d = 2 surface of 3D BiSb) =) topological term with = =) absence of localization (as in graphene with potential disorder) =) (d + 1)-dimentional Z 2 topological insulator Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

17 Classification of topological insulators Kitaev 08, Schnyder et al 08 T 2 C CT 2 d = 1 d = 2 d = 3 d = 4 A Z (QHE) 0 Z AIII Z 0 Z 0 AI Z BDI Z D Z 2 (InSb) Z (TQHE) 0 0 DIII Z 2 Z 2 Z 0 AII Z 2 (QSHE) Z 2 (BiSb et al) Z CII Z 0 Z 2 Z 2 C Z (SQHE) 0 Z 2 CI Z 0 Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

18 Localization in chiral systems Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

19 From metal to insulator Gradually remove sites from graphene Metal-insulator transition expected! Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

20 From metal to insulator Gradually remove sites from graphene Metal-insulator transition expected! Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

21 From metal to insulator Gradually remove sites from graphene Metal-insulator transition expected! Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

22 From metal to insulator Gradually remove sites from graphene Metal-insulator transition expected! Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

23 Numerics: Chiral Network Models Bocquet and Chalker 03 Chiral unitary (AIII) network model Both critical (Gade) and localized phase observed Similar results for dimerized lattice model [Motrunich et al 02] Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

24 Localization from metallic perspective Gade and Wegner 91, Gade 93 2D nonlinear sigma model for a chiral system Z h i S [Q] = d 2 x tr rq 1 c rq 8 8 h i 2 tr Q 1 rq Matrix field 8 U(N); >< Q 2 U(N)=Sp(N); >: U(N)=O(N); unitary (AIII); orthogonal (BDI); symplectic (CII): Replica limit N! 0 is assumed conductivity per square Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

25 Localization from metallic perspective Gade and Wegner 91, Gade 93 Rewrite Q = e i U (det U = 1) Z h i S [U; ] = d 2 x tr ru 1 + Nc ru + N (r) 2 8 Decoupled Gaussian theory in : d ( + Nc) = 0 dlnl 8 Replica limit d dlnl = N dc dlnl N!0! 0 Absence of localization to all orders in perturbation theory! Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

26 Status quo Apparent controversy Strong disorder induces localization in a chiral system (intuition + numerics) No traces of localization in the perturbation theory in the metallic limit (Gade and Wegner) No room for topological protection against localization How to resolve? Take into account non-perturbative effects Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

27 Bypass Gade and Wegner argument Loophole to escape Gade and Wegner argument: det Q = e i 2 U(1) ' S 1 ) vortex excitations allowed! Recalls Berezinskii-Kosterlitz-Thouless transition! Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

28 BKT Transition Berezinskii 70, Kosterlitz and Thouless 73 Z Continuum limit of xy-model: S[] = J 2 d 2 x (r) 2 Vortex excitations with core energy S core Large J (low temperature): =) vortices strongly bound in tiny dipoles =) ordered phase (quasi long-range order) Small J (high temperature): =) vortex plasma, disordered phase Renormalization group (fugacity y = L 2 e Score ) dj d ln L = y 2 J 2 dy = (2 J) y d ln L Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

29 BKT Transition Berezinskii 70, Kosterlitz and Thouless 73 RG-flow in the vicinity of the critical end point. Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

30 Renormalization group Expand the fast field Q near 1 =) One-loop perturbative RG: d d ln L = dc 0; d ln L = 1 Exact in AIII class [Guruswamy et al 00] Include one vortex-antivortex dipole in Q (lowest order in fugacity y = e Score ) d d ln L = y 2 ; dc d ln L = 1 ( + 2c)y 2 ; dy d ln L = 2 + c 4 y Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

31 Flow diagram In terms of stiffness parameter K = ( + c)=4 y d d ln L = y 2 ; dk d ln L = 1 4 2Ky 2 ; dy d ln L = (2 K ) y Fixed points: metal (Gade) critical insulator K No minimal metallic conductivity Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

32 Vortices vs. topology Chiral symplectic class CII admits Z 2 -term ) Vortices attract instantons ) Vortex-instanton fusion changes S core 7! S core + i ) Internal Z 2 degree of freedom in each vortex Chiral unitary class AIII admits Wess-Zumino term ) Vortices break global gauge symmetry ) Internal Goldstone degree of freedom in each vortex ) Random Im S core Presence of topological terms in sigma-model action prevents the theory from vortices! Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

33 Localization in symplectic systems Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

34 Interaction in disordered system Altshuler, Aronov 79; Finkelstein 83 Coulomb interaction =) new soft mode: plasmon Corrections to diffusion: + Logarithmic in 2D: g = 2e2 hn log(l T =l) To be added to weak (anti)localization N = number of independent flavours (valleys, spin etc.) Finkelstein sigma model Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

35 Scaling with interaction Coulomb interaction kills metallic phase in symplectic class Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

36 QSHE: Coulomb interaction Edge modes are protected w.r.t. Coulomb interaction =) Distinction between normal and QSH insulator is robust Coulomb interaction kills supermetal phase (a) no interaction supermetal (b) with interaction critical 0disorder QSH insulator inverted 0 band gap normal insulator normal 0disorder QSH insulator inverted 0 band gap normal insulator normal Interaction restores direct quantum spin-hall transition via a novel critical state Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

37 Z 2 vortices in symplectic system Fu, Kane, arxiv: Fu&Kane: Localization in symplectic class is due to Z 2 vortices!!! no votices =) no localization Combine everything No vortices + Coulomb interaction = critical state for QSHE transition Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33

38 Summary Results 1 Renormalization of sigma model due to Z vortices 2 Metal-insulator transition in chiral systems 3 Topological prevention from vortex-induced localization 4 Interaction-induced critical state in QSHE 5 Z 2 vortices + Coulomb interaction = critical state for QSHE transition Outlook 1 Numerical study of localization transition in chiral systems 2 Higher-dimensional effects (vortex loops) 3 Z 2 vortices in DIII PRL 105, (2010), PRB 85, (2012) Pavel Ostrovsky Anderson localization, topology, and interaction 20 September /33