Harish-Chandra Spherical Functions, Topology & Mesoscopics

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1 Harish-Chandra Spherical Functions, Topology & Mesoscopics Dmitry Bagrets A. Altland, DB,, A. Kamenev,, L. Fritz, H. Schmiedt,, PRL 11, 0660 A. Altland, DB,, A. Kamenev, arxiv: : , to appear in PRB School on Topological Quantum Matter, February 9 th -1 st, 015

2 Acknowledgment Alexander Altland Univ. zu Köln Alex Kamenev Univ. of Minnesota Lars Fritz Utrecht University Hanno Schmiedt Univ. zu Köln

3 Scope Topological quantum wires: experimental motivation Disordered quantum wires: Anderson local. revisited - Class Z wires: low-energy field theory, phase diagram - Class Z wires: kinks, emergent SUSY Harish-Chandra spherical functions - Laplace operator on a symmetric space (G/H) - Example #1: sphere S = SU()/U(1), spectrum is l(l+1) - Example #: class D disordered wire at criticality

4 Altland-Zirnbauer classification Symmetries of mesoscopic system * Time reversal: ˆ T U ˆ T H UT = H, UTUT = ± 1 * Particle-hole: ˆ T U ˆ CH UC = H, UCUC = ± 1 Chiral (parity): PHP ˆ = Hˆ 90s Altland & Zirnbauer 97

5 Classification table of topological 00s insulators & superconductors Kitaev 009 Ludwig, Ryu, Schnyder, Furusaki 009 this talk IQHE SQHE

6 1d BDI system

7 Topological spin-orbit nanowire 10s 1D spin-orbit orbit-coupled wire in proximity to s-wave s superconductor Class BDI { ph H σ } 1 σ ˆ, = 0, Hˆ σ ph T ph 1 1 = Hˆ H ± iu x i( ± h) = i( h) iu ± x Lutchyn, Sau, Das Sarma 010 Oreg, Refael, von Oppen 010 mass

8 Topological spin-orbit nanowire 10s Mourik, Zuo, Frolov, Plissard, Bakkers, Kouwenhoven 01

9 1d AIII system

10 Polyacetylene chain 80s Su, Schrieffer, and Heeger 79 Model Hamiltonian (e-ph coupling) L k H = t ' (1 + ) + h.c. + u u l= = L c + ( ) 1 l ul cl + 1 l 1 l+ 1 l Peierls instability u l l ( 1) δ, = µ t ' δ, t t ' + δ End state t t µ µ µ t µ End state

11 µ Topological Z-number bi n=0 n + π dk = i Q Q π π 1 tr( k k k ) t ti -t bi 0 n=0 n=1 n = 1 π i H k Qk { ˆ AB =, H, σ } 3 = 0 T Q k ik tde µ + te ik Chiral symmetry Im(e ik ) Re(e ik )

12 How to define Z-index of a random chain? Winding number in k-space does not work N-channel disordered AIII wire Disorder strength L / s= 1 + kk ' kk ' k ' * s, k s s+ 1, k ' h.c., s s ss ' ( k ) ( ) ' H R = a R b + R R = w N δ

13 Winding number with disorder Axial gauge transform the same sign a a e, a a e + + iϕ s / L iϕ s / L s, k s, k s, k s, k b b e, b b e + + iϕ s / L iϕ s / L s, k s, k s, k s, k SUSY partition function Axial gauge flux Z ( ϕ ) = + det i0 H F ( ϕ1) + det i0 H B( iϕ 0) dis fermions (φ->j 1 ) bosons (φ->-ij 0 )

14 Winding number with disorder Topological number Conductance n = i Z ϕ ( ) ϕ ϕ = ( ) ( ) ϕ = 0 g = ϕ + Z 0 ϕ1 ϕ SUSY partition function Axial gauge flux Z ( ϕ ) = + det i0 H F ( ϕ1) + det i0 H B( iϕ 0) dis fermions (φ->j 1 ) bosons (φ->-ij 0 )

15 Q: How to find disorder averaged Z? A: Derive low-energy field theory and solve it.

16 Non-linear sigma-model (NLσM) Classical Heisenberg ferromagnet D D S[ m] = d x λ i = 1 ( m) order parameter below T c Coordinate representation - take explicit parametrization parametrization,, e.g. i ( sin cos,sin sin, cos ) m = θ ϕ θ ϕ θ - then action becomes (exercise) S[ m] = d x θ + sin ϕ D D ( i ) θ ( i ) λ i = 1 i.e., just a metric on a sphere m unit sphere

17 Non-linear sigma-model (NLσM) Geometric approach Q = gτ g 1, g SU() 3 Q spans the coset space: SU()/U(1) m Action iτ 3 ψ g g ' = ge, Q = Q ' D 1 D S[ Q] = d x tr i λ i = 1 ( Q) i / / g = e e Sphere S ~SU()/U(1) ϕτ 3 iθτ take explicit parametrization,,, and check equivalence with the coordinate approach (exercise)

18 Field theory (NLσM) SUSY partition function twisted boundary conditions Supermatrix field T lives in a symmetric space Class AIII wire: BDI p-wave p wire: Action S T ( L) ( ) T [ T ] 1 L [ ] ( 1) ( 1 T = dx str str ) xt xt + χ T xt 0 ξ 4 ϕ0 iϕ1 ( ) T L ( ) Z ϕ = D exp S, ( ) = diag e, e L T GL(1 1) - group manifold T GL( ) / OSp( ) - coset space BF π 1 (T F )=Z bare localization length SCBA topological index

19 Flow diagram L ξ = 1 1 4,,1,,...,3 Universal description of the Z-topological chiral wires: two parameter scaling χ g ( L) = n erf l ± ( n χ ) ( L) = 1 4 ξ π L l Z+ 1/, ± l Z+ 1/ e ( l χ ) L / ξ L ξ Class AIII - running topological index - (thermal) conductance

20 Phase diagram and criticality Critical conductance g ( L) ~ ξ L Localization length ξ ( ) χ = ξ χ n 1/ - - critical exponent ν=

21 Phase diagram (BDI wire) χ=3/ SCBA topological index ( i0 H ) w Σ+ = Σ Half-integer index defines critical lines in (µ,w)-plane χ + ( G P H ) k i = tr ˆ M.-T. Rieder, P.Brouwer, I. Adagideli 013 n ( n 1) ( ) 1/ w = t N +

22 von Klitzing, Dorda, Pepper 80 Integer QHE 80s

23 IQHE (Class A in d) 80s Two parameter flow Khmelnitskii 84 & Pruisken 85 Fixed points (g*, n+1/) with g*~1 (strong coupling limit)? σ xx - average longitudinal conductivity σ xy - average topological index quantum critical point

24 Scattering matrix Landauer-Büttiker approach to e/h transport Sˆ r t ' = t r ' DMPK decomposition Lyapunov exponents Sˆ U th( λ / ) 1/ ch( λ / ) U ' = V 1/ ch( λ / ) th( λ / ) V ' λ=diag(λ 1,... λ 4N ), λ s can have either sign! If any of λ k =0, then one has delocalized state!

25 NLσM vs. T-matrix χ=3/ T-matrix approach ψ l+ 1 H1, l H0, l H1, l H ψ 1, l l ψ = l 1 0 ψ l 1 L T = T exp( Lλ ) l= 0 l Z-index = # (negative Lyapunov exponents) j

26 1d Delocalization 90s Disorder-induced induced localization D : all electron states are localized (Anderson 58) Delocalization in quasi-1d geometries (DMPK) 98: AIII quantum wire (Brouwer, Mudry, Simons, Altland) 99: D quantum wire (Gruzberg, Brouwer, Mudry, Furusaki) 04: AIII, BDI, CII, D, DIII (Gruzberg, Read, Vishveshwara) Universal interpretation Topological insulators at quantum critical point

27 How to solve AIII action?

28 Particle on a ring χ = Φ Φ 0 - Euclidean action (à( la Feynman) : β 1 SE [ ϕ] = dτ ( τϕ ) iχ τϕ 0 - Hamiltonian (à( la Heisenberg): Q: What is the partition sum? A: ˆ 1 H = ( i ϕ χ) 1 Z D e n n Z S [ ] ( ) exp ( ) E ϕ β = ϕ = β χ ϕ ( 0) = ϕ ( β ) + πz (discrete) spectrum of H

29 Q. M. on the GL(1 1), class AIII Grassmann field Parametrization y1 1 e 0 ρ T = UTzU, Tz =, U exp iy = 0 e σ 0 bf Action metric on the group gauge field L L ξ y1 iy0 S [ z] = dx ( y ) + ( y ) + 4sinh σ ρ iχ dy + idy 4 0 x 0 x 1 x x Berezinian depends only on radial variables det g 1 J ( y) sdet sinh ( g ) iy 1/ B 1 0 = = = det gf y

30 Schrödinger equation Imaginary time Schrödinger eq. vector poitential 1 ξ xψ ( y, x) = y Ψ,, = 1, J ( ) ( ia ) J ( )( ia ) ( y x ) A ( i ) α α α α χ y length becomes imaginary time! Spectral decomposition dl Ψ = + PΨ e = + l ( y L) 0 ( π ) εll / ξ ( y) P π ( l i ) 1, 1 l l, l 4 l = Z+ 1/ Eigenfunctions & spectrum initial conditions y + iy Ψ l = = il α yα ( y) sinh e, ε ( l χ ) ( l iχ ) l 0 1 Solution can be found via Sutherland transformation 1

31 1d class D system

32 Topological spin-orbit nanowire 10s Spin-orbit interaction breaks spurious chiral symmtery in the multi-channel wire ˆ T ˆ, ph UCH UC = H U C = σ - p/h is the only (true) symmetry 1

33 Class D wires Field theory of granular array { t } n Qx Q x + 1 chaotic quantum dots Class D coset space is disconnected (!) Q yτ 1 yτ 1 1 e τ e 0 Ω = UQzU Q = U = 0 Ω 3, z, exp ± τ 3 bf two configurtions (±) can t be smoothly deformed into each other SpO( ) / GL(1 1)

34 Class D wires Field theory of granular array { t } n Qx Q x + 1 chaotic quantum dots S L N 1 t Q Q x 4 x= 1 n=1 4 n [ ] = str ln 1 ( Q ) + ln σ ( Q, Q ) Kink s s fugacity x+ 1 x x+ 1 topological term S N e k r χ n= 1 k SpO( ) / GL(1 1)

35 Hidden super-symmetry Transfer matrix with kinks ladder operators ξ d ( ) x + 1 ˆ ˆ, ˆ d Φ = D + i χd Φ D = d + = i y ± A( y), A( y) = 1/ sinh y Spectrum of SUSY Schrödinger eq. d + + d y Super-potential λ( λ 1) λ( λ 1) 1 1 = +, = + cosh y sinh y λ = ±, εl l iχl

36 Flow diagram Class D L ξ = 1 1 4,,1,,...,3 Z -topological chiral wires: two parameter scaling 1 πl χll χ ( L) = dl coth sin e ξ 1 πl χll g ( L) = dl coth cos e 8 ξ l l L ξ L ξ - running topological index - (thermal) conductance

37 Harish-Chandra Harish-Chandra was an Indian American mathematician and physicist who did fundamental work in representation theory, especially harmonic analysis on semisimple Lie groups Known for Harish-Chandra's c-function Harish-Chandra's character formula Harish-Chandra homomorphism Harish-Chandra isomorphism Harish-Chandra module Harish-Chandra's regularity theorem Harish-Chandra's Schwartz space Harish-Chandra transform Harish-Chandra's Ξ functions From Wikipedia (or spherical functions) Born Died 11 October 193 Kanpur, British India 16 October 1983 (aged 60) Princeton, New Jersey, United States

38 Action NLσM on a symmetric space ξ [ ] = tr + [ Q] Action S Q dx ( ) xq Stop L 0 Q Coset space: Q spans the coset manifold M~G/K Q Invariant subgroup: 1 = gλg, Λ = 1, g G g is Lie group! M ~ G/K generalized sphere K { 1 k G kλk 1} = = g g ' = gk, Q = Q '

39 Zonal spherical functions - take (some) explicit parametrization { } Q = Q ( z k ), k = 1,...,dim M dim M=dim G dim K Q - evaluate the metric on M 1 tr kl ( dq) = g ( z) - construct Laplace-Beltrami operator 1 dz kl # = g g = + g z k z l k dz radial part l... Zonal functions are eigenfunctions of the operator M ~ G/K generalized sphere #

40 Legendre polynomials - Laplace-Beltrami operator 1 1 # = sinθ + sinθ θ θ sin θ ϕ - Eigenfunctions: # P l ( cosθ ) = l( l + 1) P ( cosθ ) - P l (cosθ) can be used for Fourier transform l S ~SU()/U(1) Harish-Chandra Chandra s s contribution: Integral representation for zonal spherical functions on the general symmetric space M=G/K, their asymptotic properties, Fourier transform on M, functional relations, etc.