Interpolating between Wishart and inverse-wishart distributions

Transcription

1 Interpolating between Wishart and inverse-wishart distributions Topological phase transitions in 1D multichannel disordered wires with a chiral symmetry Christophe Texier December 11, 2015 with Aurélien Grabsch, cond-mat arxiv:

2 Wishart distribution (Laguerre ensemble) real (β = 1) M : N N Hermitian matrix with complex (β = 2) quaternionic (β = 4) elements and positive eigenvalues DM P(M) Haar {}}{ DM (det M) η 1 β 2 (N 1) exp [ 12 ] Tr {M} for η > β 2 (N 1) Inverse-Wishart DM P(M) DM (det M) η 1 β 2 (N 1) exp [ 12 Tr { M 1}]

3 Wishart distribution (Laguerre ensemble) real (β = 1) M : N N Hermitian matrix with complex (β = 2) quaternionic (β = 4) elements and positive eigenvalues DM P(M) Haar {}}{ DM (det M) η 1 β 2 (N 1) exp [ 12 ] Tr {M} for η > β 2 (N 1) Inverse-Wishart DM P(M) DM (det M) η 1 β 2 (N 1) exp [ 12 Tr { M 1}]

4 Interpolation between Wishart and inverse-wishart f (Z ) (det Z ) µ β(n 1) exp [ 12 Tr { G 1 (Z + k 2 Z 1 ) }] for Z > 0, with G = G and µ

5 Interpolation between Wishart and inverse-wishart f (Z ) (det Z ) µ β(n 1) exp [ 12 Tr { G 1 (Z + k 2 Z 1 ) }] for Z > 0, with G = G and µ Consider G = 1 N : k 2 = 0 : Wishart with η = µ > β 2 (N 1)

6 Interpolation between Wishart and inverse-wishart f (Z ) (det Z ) µ β(n 1) exp [ 12 Tr { G 1 (Z + k 2 Z 1 ) }] for Z > 0, with G = G and µ Consider G = 1 N : k 2 = 0 : Wishart with η = µ > β 2 (N 1) k 2 : inverse-wishart with η = +µ > β 2 (N 1)

7 Physical context : Multichannel disordered wires with chiral symmetry 1D Dirac Hamiltonian with a random mass : H D = i σ 2 1 N x + σ 1 M(x) acting on a spinor with 2N components chiral symmetry : σ 3 H D σ 3 = H D Disorder : real (β = 1) Gaussian white noise (N N matrix) complex (β = 2) quaternion (β = 4) [ P [M(x)] exp 1 2 dx Tr { (M(x) µ G) G 1 (M(x) µ G) }] where µ R and G = G.

8 Physical context : Multichannel disordered wires with chiral symmetry 1D Dirac Hamiltonian with a random mass : H D = i σ 2 1 N x + σ 1 M(x) acting on a spinor with 2N components chiral symmetry : σ 3 H D σ 3 = H D Disorder : real (β = 1) Gaussian white noise (N N matrix) complex (β = 2) quaternion (β = 4) [ P [M(x)] exp 1 2 dx Tr { (M(x) µ G) G 1 (M(x) µ G) }] where µ R and G = G.

9 Motivations

10 The strictly 1D Dirac equation with random mass [ ] i σ 2 x + σ 1 m(x) Ψ(x) = ε Ψ(x) }{{} =H D In condensed matter and statistical physics : 1D disordered metal at half filling H D = iσ 3 x + V 0 (x) + σ 1 V π (x) Kappus-Wegner (band center) anomaly Disordered spin chains and spin ladders π V 0 V π ε k V 0 k π (Fisher 94, 95 ; Steiner, Fabrizio, Gogolin, Mélin, Steiner, 97 ;...) Classical diffusion in a random environment H 2 D H ± = 2 x + m 2 ± m FPE t P = x ( x 2m)P Sinai diffusion ( m = 0), etc. Bouchaud, Comtet, Georges & Le Doussal, 1990 ;...

11 The strictly 1D Dirac equation with random mass [ ] i σ 2 x + σ 1 m(x) Ψ(x) = ε Ψ(x) }{{} =H D In condensed matter and statistical physics : 1D disordered metal at half filling H D = iσ 3 x + V 0 (x) + σ 1 V π (x) Kappus-Wegner (band center) anomaly Disordered spin chains and spin ladders π V 0 V π ε k V 0 k π (Fisher 94, 95 ; Steiner, Fabrizio, Gogolin, Mélin, Steiner, 97 ;...) Classical diffusion in a random environment H 2 D H ± = 2 x + m 2 ± m FPE t P = x ( x 2m)P Sinai diffusion ( m = 0), etc. Bouchaud, Comtet, Georges & Le Doussal, 1990 ;...

12 The strictly 1D Dirac equation with random mass [ ] i σ 2 x + σ 1 m(x) Ψ(x) = ε Ψ(x) }{{} =H D In condensed matter and statistical physics : 1D disordered metal at half filling H D = iσ 3 x + V 0 (x) + σ 1 V π (x) Kappus-Wegner (band center) anomaly Disordered spin chains and spin ladders π V 0 V π ε k V 0 k π (Fisher 94, 95 ; Steiner, Fabrizio, Gogolin, Mélin, Steiner, 97 ;...) Classical diffusion in a random environment H 2 D H ± = 2 x + m 2 ± m FPE t P = x ( x 2m)P Sinai diffusion ( m = 0), etc. Bouchaud, Comtet, Georges & Le Doussal, 1990 ;...

13 Multichannel disordered wires Standard classes (orthogonal, unitary, symplectic) Dorokhov, 1982 ; 1988 ;... Mello, Pereyra, Kumar, 1988 : «DMPK» Beenakker, Rev. Mod. Phys. (1997) Additional discrete symmetries : chiral & BdG classes spectral properties, localisation Brouwer, Mudry, Simons, Altland, PRL (1998) Brouwer, Mudry, Furusaki, PRL (2000) ;.... Ludwig & Schulz-Baldes, J. Stat. Phys. (2013) (localisation) Topological phase transitions driven by strong disorder Potter, Lee, PRL (2010)... Rieder, Brouwer, Adagideli, PRL (2013)...

14 Multichannel disordered wires Standard classes (orthogonal, unitary, symplectic) Dorokhov, 1982 ; 1988 ;... Mello, Pereyra, Kumar, 1988 : «DMPK» Beenakker, Rev. Mod. Phys. (1997) Additional discrete symmetries : chiral & BdG classes spectral properties, localisation Brouwer, Mudry, Simons, Altland, PRL (1998) Brouwer, Mudry, Furusaki, PRL (2000) ;.... Ludwig & Schulz-Baldes, J. Stat. Phys. (2013) (localisation) Topological phase transitions driven by strong disorder Potter, Lee, PRL (2010)... Rieder, Brouwer, Adagideli, PRL (2013)...

15 Multichannel disordered wires Standard classes (orthogonal, unitary, symplectic) Dorokhov, 1982 ; 1988 ;... Mello, Pereyra, Kumar, 1988 : «DMPK» Beenakker, Rev. Mod. Phys. (1997) Additional discrete symmetries : chiral & BdG classes spectral properties, localisation Brouwer, Mudry, Simons, Altland, PRL (1998) Brouwer, Mudry, Furusaki, PRL (2000) ;.... Ludwig & Schulz-Baldes, J. Stat. Phys. (2013) (localisation) Topological phase transitions driven by strong disorder Potter, Lee, PRL (2010)... Rieder, Brouwer, Adagideli, PRL (2013)...

16 Topological insulators and quantum phase transitions Example : Integer Quantum Hall Effect (class A) boundaries: 1D chiral metal «bulk-edge correspondence» trivial insulator bulk: 2D topological insulator trivial insulator

17 Topological insulators and quantum phase transitions Example : Integer Quantum Hall Effect (class A) boundaries: 1D chiral metal trivial insulator bulk: 2D topological insulator trivial insulator «bulk-edge correspondence» Quantum phase transition σ xy = n n σ xy σ xx ν 1/B

18 Topological insulators and quantum phase transitions Example : Integer Quantum Hall Effect (class A) boundaries: 1D chiral metal trivial insulator bulk: 2D topological insulator trivial insulator «bulk-edge correspondence» Quantum phase transition without symmetry breaking 2 1 σ xy σ xx ν 1/B «topological phase transition»

19 What are the possible topological insulators? Symmetries of RMT and disordered systems : (Altland & Zirnbauer, 96 ; 97) Two discrete symmetries in condensed matter : Chiral (sublattice) & particle-hole (superconductors) Periodic table of topological insulators : topological index TRS p-h S ch. S SRS 0D 1D 2D Wigner- AI (orthogonal) +1 no no yes Z no no Dyson A (unitary) no no no indiff. Z no Z AII (symplectic) 1 no no no Z no Z 2 Chiral BDI (chiral orth.) yes yes Z 2 Z no AIII (chiral unit.) no no yes indiff. no Z no CII (chiral sympl.) 1 1 yes no no Z no BdG D no +1 no no Z 2 Z 2 Z DIII 1 +1 yes no no Z 2 Z 2 C no 1 no yes no no Z CI +1 1 yes yes no no no Kitaev (2009) ; Ryu, Schnyder, Furusaki, Ludwig, New J. Phys. (2010)

20 What are the possible topological insulators? Symmetries of RMT and disordered systems : (Altland & Zirnbauer, 96 ; 97) Two discrete symmetries in condensed matter : Chiral (sublattice) & particle-hole (superconductors) Periodic table of topological insulators : topological index TRS p-h S ch. S SRS 0D 1D 2D Wigner- AI (orthogonal) +1 no no yes Z no no Dyson A (unitary) no no no indiff. Z no Z AII (symplectic) 1 no no no Z no Z 2 Chiral BDI (chiral orth.) yes yes Z 2 Z no AIII (chiral unit.) no no yes indiff. no Z no CII (chiral sympl.) 1 1 yes no no Z no BdG D no +1 no no Z 2 Z 2 Z DIII 1 +1 yes no no Z 2 Z 2 C no 1 no yes no no Z CI +1 1 yes yes no no no Kitaev (2009) ; Ryu, Schnyder, Furusaki, Ludwig, New J. Phys. (2010)

21 Question 1D multichannel Dirac equation with random mass : Q : How does disorder drive topological phase transitions? Relation with the matrix model

22 Table of contents 1 The scattering problem and the Riccati matrix 2 Distribution of the Riccati matrix 3 Spectral density of the disordered wires 4 Topological phase transitions 5 Conclusion

23 Plan 1 The scattering problem and the Riccati matrix 2 Distribution of the Riccati matrix 3 Spectral density of the disordered wires 4 Topological phase transitions 5 Conclusion

24 The scattering problem M(x) M(x)= 0 L 1 x S matrix Weyl (chiral) representation : H D = i σ 3 1 N x + σ 1 M(x) Scattering state : 2N N «spinor» : (for ε > 0) ( ) ( ) 0 Ψ ε (x) = e iε(x L) S(ε) + e +iε(x L) 1 N 0 for x > L S(ε) : N N scattering matrix

25 The scattering problem M(x) M(x)= 0 L 1 x S matrix Weyl (chiral) representation : H D = i σ 3 1 N x + σ 1 M(x) Scattering state : 2N N «spinor» : (for ε > 0) ( ) ( ) 0 Ψ ε (x) = e iε(x L) S(ε) + e +iε(x L) 1 N 0 for x > L S(ε) : N N scattering matrix

26 Riccati matrix Z and scattering matrix S Majorana representation : H D Ψ = ε Ψ with H D = i σ 2 1 N x + σ 1 M(x) 2N N «spinor» : obeys Ψ = ( ) ϕ χ Z = def ε χ ϕ 1 Z (x) = ε 2 Z (x) 2 M(x) Z (x) Z (x) M(x) Relation to the S-matrix : S(ε) = i [ ε i Z (L) ][ ε + i Z (L) ] 1

27 Riccati matrix Z and scattering matrix S Majorana representation : H D Ψ = ε Ψ with H D = i σ 2 1 N x + σ 1 M(x) 2N N «spinor» : obeys Ψ = ( ) ϕ χ Z = def ε χ ϕ 1 Z (x) = ε 2 Z (x) 2 M(x) Z (x) Z (x) M(x) Relation to the S-matrix : S(ε) = i [ ε i Z (L) ][ ε + i Z (L) ] 1

28 Riccati matrix Z and scattering matrix S Majorana representation : H D Ψ = ε Ψ with H D = i σ 2 1 N x + σ 1 M(x) 2N N «spinor» : obeys Ψ = ( ) ϕ χ Z = def ε χ ϕ 1 Z (x) = ε 2 Z (x) 2 M(x) Z (x) Z (x) M(x) Relation to the S-matrix : S(ε) = i [ ε i Z (L) ][ ε + i Z (L) ] 1

29 Plan 1 The scattering problem and the Riccati matrix 2 Distribution of the Riccati matrix 3 Spectral density of the disordered wires 4 Topological phase transitions 5 Conclusion

30 Random matricial process M(x) a Gaussian white noise (N N matrix) Stochastic matricial differential equation Z (x) = ε 2 Z (x) 2 M(x) Z (x) Z (x) M(x) (Stratonovich) Matricial Fokker-Planck equation ε = ik ir if M(x) = 0, fixed point Z (x) = k 1 N if M(x) 0 : stationary (equil.) distribution (proven for β = 1 & 2) : f (Z ) = C 1 N,β (det Z 1 ) µ 1 2 β(n 1) exp [ 12 Tr { G 1 (Z + k 2 Z 1 ) }] for Z > 0

31 Random matricial process M(x) a Gaussian white noise (N N matrix) Stochastic matricial differential equation Z (x) = ε 2 Z (x) 2 M(x) Z (x) Z (x) M(x) (Stratonovich) Matricial Fokker-Planck equation ε = ik ir if M(x) = 0, fixed point Z (x) = k 1 N if M(x) 0 : stationary (equil.) distribution (proven for β = 1 & 2) : f (Z ) = C 1 N,β (det Z 1 ) µ 1 2 β(n 1) exp [ 12 Tr { G 1 (Z + k 2 Z 1 ) }] for Z > 0

32 Random matricial process M(x) a Gaussian white noise (N N matrix) Stochastic matricial differential equation Z (x) = ε 2 Z (x) 2 M(x) Z (x) Z (x) M(x) (Stratonovich) Matricial Fokker-Planck equation ε = ik ir if M(x) = 0, fixed point Z (x) = k 1 N if M(x) 0 : stationary (equil.) distribution (proven for β = 1 & 2) : f (Z ) = C 1 N,β (det Z 1 ) µ 1 2 β(n 1) exp [ 12 Tr { G 1 (Z + k 2 Z 1 ) }] for Z > 0

33 Plan 1 The scattering problem and the Riccati matrix 2 Distribution of the Riccati matrix 3 Spectral density of the disordered wires 4 Topological phase transitions 5 Conclusion

34 Density of states (DoS) Krein-Friedel relation : DoS Scattering S(ε) = i [ε iz (L)] [ε + iz (L)] 1 integrated DoS N (ε) Z (L) = DZ f (Z ) Z (+ analytic continuation, ε = ik ir)

35 Density of states (DoS) Krein-Friedel relation : DoS Scattering S(ε) = i [ε iz (L)] [ε + iz (L)] 1 integrated DoS N (ε) Z (L) = DZ f (Z ) Z (+ analytic continuation, ε = ik ir)

36 Normalisation constant C N,β = DZ (det Z ) µ β(n 1) exp [ 12 Tr { G 1 (Z + k 2 Z 1 ) }] Z >0 Integrated DoS N (ε) : G = diag(g 1,, g N ) Ω = Tr {Z } = N i=1 g 2 i ln C N,β g i ( G=g1 N Ω = g N µ + k ln C ) N,β k N (ε) = 1 π Im [Ω k= iε]

37 Normalisation constant C N,β = DZ (det Z ) µ β(n 1) exp [ 12 Tr { G 1 (Z + k 2 Z 1 ) }] Z >0 Integrated DoS N (ε) : G = diag(g 1,, g N ) Ω = Tr {Z } = N i=1 g 2 i ln C N,β g i ( G=g1 N Ω = g N µ + k ln C ) N,β k N (ε) = 1 π Im [Ω k= iε]

38 Determinantal representations (G = g1 N ) Chiral orthogonal (β = 1, BDI) Pfaffian [ ] C N,1 pf Chiral unitary (β = 2, AIII) Hankel determinant C N,2 = N! 2 N k Nµ det [ ] K µ+1+n i j (k/g) 1 n, m N Chiral symplectic (β = 4, CII) [ ] C N,4 = N! 2 N k Nµ pf (m n) K µ+1+2n n m (k/g) 1 n, m 2N similar result in : Titov, Brouwer, Furusaki & Mudry, PRB (2001)

39 Density of states ρ(ε) = N (ε) µ = mass disorder N (ε) ε α as ε 0 µ = 0 : parity effect (Brouwer, Furusaki, Mudry, PRL 00 ; Physica E 01)

40 Exponent of the DoS ρ(ε) = N (ε) ε α 1 µ = mass disorder : N (ε) ε β ln ε : N (ε) 1/ ln 2 ε (Dyson singularity β) superuniversality (Gruzberg, Read, Vishveshwara, PRB 2005) α = 0 topological phase transition

41 Plan 1 The scattering problem and the Riccati matrix 2 Distribution of the Riccati matrix 3 Spectral density of the disordered wires 4 Topological phase transitions 5 Conclusion

42 Topological quantum number (N = 1 channel case) m(x)= 0 m(x) L 1 x e iδ set m(x) = ±µg = m 0 Phase-shift δ ± (ε) Bulk-edge correspondence case with no disorder subgap state : IDoS N ± L (ε) = δ±(ε) 2π

43 Topological quantum number (N = 1 channel case) m(x)= 0 m(x) L 1 x e iδ set m(x) = ±µg = m 0 Phase-shift δ ± (ε) Bulk-edge correspondence case with no disorder subgap state : m(x)= m(x)=m 0<0 gap x 2 m no zero mode 0... m(x)= closure & reopening of the gap IDoS N ± L Ψ 0 (x) m(x)=m 0>0 gap 1/ m 2m x (ε) = δ±(ε) 2π

44 Topological quantum number (N = 1 channel case) m(x)= 0 m(x) L 1 x e iδ set m(x) = ±µg = m 0 Phase-shift δ ± (ε) Bulk-edge correspondence case with no disorder subgap state : m(x)= m(x)=m 0<0 gap x 2 m no zero mode 0... m(x)= closure & reopening of the gap IDoS N ± L Ψ 0 (x) m(x)=m 0>0 gap 1/ m 2m x (ε) = δ±(ε) 2π ( ) = 1 2 Witten index ( 1 2 # of zero modes) : N + L (0) N L (0) = δ +(0) δ (0) 2π = ( ) β ( β) { } = Tr σ 3 e βh 2 D

45 Witten index and Riccati matrix M(x) 1 M(x)= 0 L x S matrix S(ε) = i [ ε i Z (L) ][ ε + i Z (L) ] 1 Eigenvalues : z n = ε tan δ n (phase shift) Witten index ( 1 2 # of zero modes) δ ± (ε) = i ln det S(ε) for M(x) = ±µg 1 N 1 2 n sign(z n) n sign(z n k) δ + (ε) δ (ε) ( ) = ε 0 2π 1 2 for ε R for ε = ik ir

46 Witten index and Riccati matrix M(x) 1 M(x)= 0 L x S matrix S(ε) = i [ ε i Z (L) ][ ε + i Z (L) ] 1 Eigenvalues : z n = ε tan δ n (phase shift) Witten index ( 1 2 # of zero modes) δ ± (ε) = i ln det S(ε) for M(x) = ±µg 1 N 1 2 n sign(z n) n sign(z n k) δ + (ε) δ (ε) ( ) = ε 0 2π 1 2 for ε R for ε = ik ir

47 k = iε 0 limit of the distribution f (Z ) P(z 1,, z N ) = C 1 N,β µ β(n 1)/2 > 0 : z i z j β i<j l P(z 1,, z N ) k 0 Wishart z µ β(n 1)/2 1 l e 1 2g (z l +k 2 /z l ) µ β(n 1)/2 < 0 ( P(z 1,, z N ) Wishart ({z i }) inverse Wishart { z ) j k 0 k 2 } Condensation of n charges to z = 0

48 k = iε 0 limit of the distribution f (Z ) P(z 1,, z N ) = C 1 N,β µ β(n 1)/2 > 0 : z i z j β i<j l P(z 1,, z N ) k 0 Wishart z µ β(n 1)/2 1 l e 1 2g (z l +k 2 /z l ) µ β(n 1)/2 < 0 ( P(z 1,, z N ) Wishart ({z i }) inverse Wishart { z ) j k 0 k 2 } Condensation of n charges to z = 0

49 k = iε 0 limit of the distribution f (Z ) P(z 1,, z N ) = C 1 N,β µ β(n 1)/2 > 0 : z i z j β i<j l P(z 1,, z N ) k 0 Wishart z µ β(n 1)/2 1 l e 1 2g (z l +k 2 /z l ) µ β(n 1)/2 < 0? ( P(z 1,, z N ) Wishart ({z i }) inverse Wishart { z ) j k 0 k 2 } Condensation of n charges to z = 0

50 k = iε 0 limit of the distribution f (Z ) P(z 1,, z N ) = C 1 N,β µ β(n 1)/2 > 0 : z i z j β i<j l P(z 1,, z N ) k 0 Wishart z µ β(n 1)/2 1 l e 1 2g (z l +k 2 /z l ) µ β(n 1)/2 < 0 ( P(z 1,, z N ) Wishart ({z i }) inverse Wishart { z ) j k 0 k 2 } k 2 inverse Wishart n charges 0 k N n charges Wishart z Condensation of n charges to z = 0

51 k 0 : two uncorrelated sets of charges 1 ( ) = lim k 0 2 Case θ = µ β(n 1)/2 > 0 sign(z n k) n Case θ = µ β(n 1)/2 ] n β, (n 1)β[ Case θ = µ β(n 1)/2 = n β ( ) = N 2n+1 2

52 k 0 : two uncorrelated sets of charges 1 ( ) = lim k 0 2 Case θ = µ β(n 1)/2 > 0 sign(z n k) n Wishart k 0 ( ) = N 2 z Case θ = µ β(n 1)/2 ] n β, (n 1)β[ Case θ = µ β(n 1)/2 = n β ( ) = N 2n+1 2

53 k 0 : two uncorrelated sets of charges 1 ( ) = lim k 0 2 Case θ = µ β(n 1)/2 > 0 sign(z n k) n Wishart k 0 ( ) = N 2 z Case θ = µ β(n 1)/2 ] n β, (n 1)β[ k 2 inverse Wishart n charges N n charges 0 k Wishart z ( ) = N 2n 2 Case θ = µ β(n 1)/2 = n β ( ) = N 2n+1 2

54 k 0 : two uncorrelated sets of charges 1 ( ) = lim k 0 2 Case θ = µ β(n 1)/2 > 0 sign(z n k) n Wishart k 0 ( ) = N 2 z Case θ = µ β(n 1)/2 ] n β, (n 1)β[ k 2 inverse Wishart n charges N n charges 0 k Wishart z ( ) = N 2n 2 Case θ = µ β(n 1)/2 = n β ( ) = N 2n+1 2

55 Witten index and phase diagram Witten index Phase diagram : plane (m 0, g)

56 Witten index and phase diagram Witten index Phase diagram : plane (m 0, g) g 1/2 g +1/2 g 1/2 g +1/2 g 0 1/2 +1/ m /2 +1/2 3/2 +3/2 +1 3/2 m 0 m 0 m 0 m 0 N =1 N =2 N =3 N =4 N = /2 1/2 +1/2 3/2 +3/2 5/2 +5/2...

57 Plan 1 The scattering problem and the Riccati matrix 2 Distribution of the Riccati matrix 3 Spectral density of the disordered wires 4 Topological phase transitions 5 Conclusion

58 Conclusions H D = i σ 2 1 N x + σ 1 M(x) Relation to Random Matrix Products Π N = M N M 1 ( ) ( ) 1N cos θ 1 M n = N sin θ e W 0 1 N 1 N sin θ 1 N cos θ 0 1 N e W Sp(2N) with W = W Stationary distribution for the matricial process Z (x) : f (Z ) = C 1 N,β (det Z 1 ) µ 1 2 β(n 1) exp [ 12 Tr { G 1 (Z + k 2 Z 1 ) }] if G 1 N : no isotropy assumption DoS : Z (x) = DZ f (Z ) Z C N,β N (ε) Topological quantum number (Witten index ( )) : from f (Z ) (i.e. of the S-matrix) Topological phase transitions driven by µ = mass /disorder

59 Conclusions H D = i σ 2 1 N x + σ 1 M(x) Relation to Random Matrix Products Π N = M N M 1 ( ) ( ) 1N cos θ 1 M n = N sin θ e W 0 1 N 1 N sin θ 1 N cos θ 0 1 N e W Sp(2N) with W = W Stationary distribution for the matricial process Z (x) : f (Z ) = C 1 N,β (det Z 1 ) µ 1 2 β(n 1) exp [ 12 Tr { G 1 (Z + k 2 Z 1 ) }] if G 1 N : no isotropy assumption DoS : Z (x) = DZ f (Z ) Z C N,β N (ε) Topological quantum number (Witten index ( )) : from f (Z ) (i.e. of the S-matrix) Topological phase transitions driven by µ = mass /disorder

60 Conclusions H D = i σ 2 1 N x + σ 1 M(x) Relation to Random Matrix Products Π N = M N M 1 ( ) ( ) 1N cos θ 1 M n = N sin θ e W 0 1 N 1 N sin θ 1 N cos θ 0 1 N e W Sp(2N) with W = W Stationary distribution for the matricial process Z (x) : f (Z ) = C 1 N,β (det Z 1 ) µ 1 2 β(n 1) exp [ 12 Tr { G 1 (Z + k 2 Z 1 ) }] if G 1 N : no isotropy assumption DoS : Z (x) = DZ f (Z ) Z C N,β N (ε) Topological quantum number (Witten index ( )) : from f (Z ) (i.e. of the S-matrix) Topological phase transitions driven by µ = mass /disorder

61 Conclusions H D = i σ 2 1 N x + σ 1 M(x) Relation to Random Matrix Products Π N = M N M 1 ( ) ( ) 1N cos θ 1 M n = N sin θ e W 0 1 N 1 N sin θ 1 N cos θ 0 1 N e W Sp(2N) with W = W Stationary distribution for the matricial process Z (x) : f (Z ) = C 1 N,β (det Z 1 ) µ 1 2 β(n 1) exp [ 12 Tr { G 1 (Z + k 2 Z 1 ) }] if G 1 N : no isotropy assumption DoS : Z (x) = DZ f (Z ) Z C N,β N (ε) Topological quantum number (Witten index ( )) : from f (Z ) (i.e. of the S-matrix) Topological phase transitions driven by µ = mass /disorder

62 Conclusions H D = i σ 2 1 N x + σ 1 M(x) Relation to Random Matrix Products Π N = M N M 1 ( ) ( ) 1N cos θ 1 M n = N sin θ e W 0 1 N 1 N sin θ 1 N cos θ 0 1 N e W Sp(2N) with W = W Stationary distribution for the matricial process Z (x) : f (Z ) = C 1 N,β (det Z 1 ) µ 1 2 β(n 1) exp [ 12 Tr { G 1 (Z + k 2 Z 1 ) }] if G 1 N : no isotropy assumption DoS : Z (x) = DZ f (Z ) Z C N,β N (ε) Topological quantum number (Witten index ( )) : from f (Z ) (i.e. of the S-matrix) Topological phase transitions driven by µ = mass /disorder

63 At the transition : delocalisation in one channel Lyapunov spectrum (ε = 0) : γ n = [µ β ] 2 (N 2n + 1) g m(x) Ψ 0 (x) x with n {1,, N} 1D : two symmetry classes (D, DIII) Z 2 -insulators Morimoto, Furusaki, Mudry, PRB (june 2015) application of our method? No isotropy among channels (G 1 N ). Most general : P[M(x)] exp [ dx Tr { A 1 M(x)B 1 M(x) } ]

64 At the transition : delocalisation in one channel Lyapunov spectrum (ε = 0) : γ n = [µ β ] 2 (N 2n + 1) g m(x) Ψ 0 (x) x with n {1,, N} 1D : two symmetry classes (D, DIII) Z 2 -insulators Morimoto, Furusaki, Mudry, PRB (june 2015) application of our method? No isotropy among channels (G 1 N ). Most general : P[M(x)] exp [ dx Tr { A 1 M(x)B 1 M(x) } ]

65 Thank you. A. Grabsch & C. Texier, preprint cond-mat arxiv:

66 Appendices

67 Appendix A : Matricial Fokker-Planck equation ( ) Z = Z 2 E µgz µzg Z M MZ with E = ε 2 Case β = 1 (class BDI) M ab (x) = σ ab ζ ab (x) }{{} normalised { def ga if a = b, with σ ab = g ag b g a+g b if a b. ( ) Z mn = [ Z 2 E µgz µzg ] mn + k l B mn,kl (Z ) ζ kl, where B mn,kl (Z ) = def 2 δ kl σkl (Z mk δ nl + Z ml δ nk + Z ln δ km + Z kn δ lm ) 2

68 Forward generator G Define and ( ) def = 1 + δ mn, Z mn 2 Z mn σ ab def = 2 δ ab σ ab. 2 { [ [ G = 2 Tr Z Z σ + Tr ( ) ]]} T Z Z Z Z + { ( Z 2 + E + µgz + µzg )}, Z [A B] mn = A mnb mn : Hadamard product Stationary solution G f (Z ) = 0