disordered topological matter time line

Transcription

1

2 disordered topological matter time line

3 disordered topological matter time line 80s quantum Hall SSH

4 quantum Hall effect (class A)

5 quantum Hall effect (class A) 1998 Nobel prize press release

6 quantum Hall effect (class A) 1998 Nobel prize press release

7 quantum Hall continued universal interplay disorder/topology described by two parameters : average longitudinal transport coefficient : average topological index two parameter criticality Khmelnitskii, 1983

8 quantum Hall continued universal interplay disorder/topology described by two parameters : average longitudinal transport coefficient : average topological index two parameter criticality Khmelnitskii, 1983

9 disordered topological matter time line 80s quantum Hall

10 disordered topological matter time line 80s quantum Hall 90s symmetry classes

11 disordered topological matter time line 80s 90s quantum free fermion Hall systems classified by unitary and anti-unitary symmetries pre-nineties: 3 Wigner-Dyson classes there exist 10=(3+7) symmetry classes, (Zirnbauer, AA, 96) symmetry classes labeled A, AI, AII, AIII, BDI, C, CI, CII, D, DIII (courtesy Cartan)

12 disordered topological matter time line 80s quantum Hall 90s symmetry classes

13 disordered topological matter time line 80s quantum Hall 90s 1d delocalization symmetry classes novel quantum Hall effects 00s 2d delocalization

14 1d delocalization phenomena delocalization in quasi-one dimensional geometries 1998 AIII quantum wire (Brouwer, Mudry, Simons, AA) 1999 D quantum wire (Brouwer, Mudry, Furusaki) 2004 AIII, D, BDI, DIII, CII (Read, Gruzberg, Vishveshwara)

15 1d delocalization phenomena delocalization in quasi-one dimensional geometries 1998 AIII quantum wire (Brouwer, Mudry, Simons, AA) 1999 D quantum wire (Brouwer, Mudry, Furusaki) 2004 AIII, D, BDI, DIII, CII (Read, Gruzberg, Vishveshwara) Unconventional criticality in 2d 2001 Class C spin quantum Hall effect (Chalker et al.) 2001 Class D quantum criticality (Fisher et al., Read et al.) universal interpretation: topological insulators at quantum critical point

16 disordered topological matter time line 80s quantum Hall 90s 1d delocalization symmetry classes novel quantum Hall effects 00s 2d delocalization

17 disordered topological matter time line 80s quantum Hall 90s 1d delocalization symmetry classes novel quantum Hall effects 00s 2d delocalization topological matter

18 disordered topological matter time line 80s quantum Hall 90s 1d delocalization symmetry classes novel quantum Hall effects 00s 2d delocalization topological matter 10s topological Anderson insulator

19 topological Anderson insulator

20 topological Anderson insulator

21 topological Anderson insulator

22 Addition of disorder to a clean band insulator invalidate k-theory of band gaps/topological indices

23 Addition of disorder to a clean band insulator invalidate k-theory of band gaps/topological indices destroys band gaps and

24 Addition of disorder to a clean band insulator invalidate k-theory of band gaps/topological indices destroys band gaps and turns insulator into nominal metal renders topological indices statistically distributed

25 Addition of disorder to a clean band insulator invalidate k-theory of band gaps/topological indices destroys band gaps and turns insulator into nominal metal renders topological indices statistically distributed clean disordered k-theory + - band gap + - index + (+)

26 topological Anderson insulator cf. Motrunich, Damle and Huse 2001, Groth et al. 2010

27 topological Anderson insulator cf. Motrunich, Damle and Huse 2001, Groth et al. 2010

28 topological Anderson insulator cf. Motrunich, Damle and Huse 2001, Groth et al. 2010

29 topological Anderson insulator cf. Motrunich, Damle and Huse 2001, Groth et al early works

30 topological Anderson insulator cf. Motrunich, Damle and Huse 2001, Groth et al early works Brouwer et al., 2012; Groth et al. 2010;

31 topological Anderson insulator cf. Motrunich, Damle and Huse 2001, Groth et al critical theory?

32 Universal theory of disordered TI checklist describe topological invariants without reference to k-space describe criticality in terms of two observables: : transport coefficient : configurational average of index state bare values generically: critical: describe edge state formation similar architecture in all symmetry classes, d=1,2

33 Universal theory of disordered TI checklist describe topological invariants without reference to k-space describe criticality in terms of two observables: : transport coefficient : configurational average of index state bare values generically: critical: describe edge state formation similar architecture in all symmetry classes, d=1,2

34 Universal theory of disordered TI checklist describe topological invariants without reference to k-space describe criticality in terms of two observables: : transport coefficient : configurational average of index state bare values generically: critical: IQH SSH Kitaev chain Majorana quantum wire topolgical insulator (HgTe) spin QH describe edge state formation similar architecture in all symmetry classes, d=1,2

35 Universal theory of disordered TI checklist describe topological invariants without reference to k-space describe criticality in terms of two observables: : transport coefficient : configurational average of index state bare values generically: critical: describe edge state formation similar architecture in all symmetry classes, d=1,2

36 quantum criticality of the 1d topological Anderson insulator Benasque, Alexander Altland, Dmitry Bagrets (Cologne) Alex Kamenev (Minnesota) Lars Fritz (Utrecht)! the 1d AIII insulator topological quantum criticality generalization to 1d BDI, CII comparison to 2d A, C, D generalization to Z2 classes, 1d D,DIII, 2d AII, DIII arxiv ; PRL 112, (2014)

37 the 1d AIII system (aka SSH)

38 AIII quantum wire

39 AIII quantum wire

40 AIII quantum wire

41 AIII quantum wire

42 AIII quantum wire winding number (clean) single channel: channels:

43 generalized definition of winding number

44 generalized definition of winding number generalized definition

45 generalized definition of winding number generalized definition cf. Nazarov, 94

46 topological quantum criticality

47 field integral AA & Merkt, 2001 matrix fields boundary conditions bare (SCBA) values of loc. length and topological parameter, resp. good picture: path integral of quantum point particle in time L.

48 diffusive systems for diffusive conductance non-integer average invariant

49 Understanding the path integral

50 Understanding the path integral

51 Understanding the path integral focus on compact sector: theory reduces to QM on a ring with twisted boundary conditions and subject to magnetic flux

52 Understanding the path integral focus on compact sector: theory reduces to QM on a ring with twisted boundary conditions and subject to magnetic flux

53 Understanding the path integral focus on compact sector: theory reduces to QM on a ring with twisted boundary conditions and subject to magnetic flux

54 transfer matrix solution of full problem

55 transfer matrix solution of full problem solvable problem: eigenfunctions: eigenvalues: solution by spectral sum:

56 results where lnhgl c

57 phase diagram

58 phase diagram phase boundaries: lines of half integer bare topological index flow describes stabilization of self-averaging topological phase/boundary state generation

59 phase diagram mêt bi AI ti tai n=3 2 n= wêt phase boundaries: lines of half integer bare topological index flow describes stabilization of self-averaging topological phase/boundary state generation

60 physics at boundary deep localization regime

61 physics at boundary deep localization regime

62 physics at boundary deep localization regime boundary action (generalized for finite excitation energies) S[T ]= s 4 str(t + T 1 )+n str ln(t ), s =

63 physics at boundary deep localization regime boundary action (generalized for finite excitation energies) S[T ]= s 4 str(t + T 1 )+n str ln(t ), s = Mourik et al, 12

64 universality

65 1d universality classes AIII, BDI, CII described by unified approach: physical response probed by insertion of gauge flux structurally identical low energy action and flow of system parameters. lnhgl c

66 2d universality

67 2d universality classes A, C, D described by unified approach: probed by insertion of gauge flux (Pruisken s background field, cf. Laughlin gauge argument) structurally identical low energy action S[Q] = g 2 Z d 2 x str(@ µ Q@ µ Q)+ 2 Z d 2 x µ str(q@ µ Q@ Q) σ 11 and flow of system parameters. σ* 11 n n + 1 σ 12

68 Z-universality 1d AIII, BDI, CII 2d A, C, (D) φ S[M] = gs di [M]+ S top [M] σ 11 σ* 11 n n + 1 σ 12 generic flow (g, ) L!1! (0,n) ns top [M]! ns boundary [T ]

69 Z2

70 generalization to Z2 1 2 A AIII AI BDI D DIII AII CII C CI

71 generalization to Z2 case study: AII, d=2 (Kane & Fu, 2012) system probed by topological point defects field theory admits point-like excitations theta-term > fugactiy term 2 parameter criticality

72 generalization to Z2 case study: AII, d=2 (Kane & Fu, 2012) system probed by topological point defects field theory admits point-like excitations theta-term > fugactiy term 2 parameter criticality

73 generalization to Z2 case study: AII, d=2 (Kane & Fu, 2012) system probed by topological point defects field theory admits point-like excitations theta-term > fugactiy term 2 parameter criticality

74 1d Z2 D and DIII field theory

75 1d Z2 D and DIII field theory

76 1d Z2 D and DIII field theory

77 1d Z2 D and DIII field theory generic two parameter flow

78 Summary real space approach to translationally non-invariant topological insulators 2-parameter field theory probed by continuous (Z) or point-like (Z2) topological sources universal scaling stabilization of topology by localization Thouless topology 2d class A Khmelnitskii/Pruisken criticality 2d class A Chalker et al. 2d class C Ludwig et al. exact solution 2d class C Fisher et al., Read et al., Zirnbauer/Serban 2d class D Mirlin, Mudry, Gruzberg et al. disordered topological matter Beenakker et al., Brouwer et al. scattering theory of topological matter Read, Gruzberg, Vishveshwara 1d topological quantum criticality Ludwig et al. disorder vs. bulk-boundary correspondence Kane/Fu, 2d class AII Zirnbauer 1d super-fourier analysis

79 previous work Thouless topology 2d class A Khmelnitskii/Pruisken criticality 2d class A Chalker et al. 2d class C Ludwig et al. exact solution 2d class C Fisher et al., Read et al., Zirnbauer/Serban 2d class D Zirnbauer 1d super-fourier analysis Mirlin, Mudry, Gruzberg et al. disordered topological matter Beenakker et al., Brouwer et al. scattering theory of disordered topological matter Read, Gruzberg, Vishveshwara, 1d topological quantum criticality Ludwig et al., disorder vs. bulk-boundary correspondence Kane/Fu, 2d AII