Various Facets of Chalker- Coddington network model

Various Facets of Chalker- Coddington network model V. Kagalovsky Sami Shamoon College of Engineering Beer-Sheva Israel

Context Integer quantum Hall effect Semiclassical picture Chalker-Coddington Coddington network model Various applications Inter-plateaux transitions Floating of extended states New symmetry classes in dirty superconductors Effect of nuclear magnetization on QHE

Inter-plateaux transition is a critical phenomenon

Semiclassical picture: strong magnetic field B and slowly varying random potential V(r) Criterion: l V << ħω c or l<<ξcor Where magnetic length l= c eb ħ cyclotron frequency eb ω c = mc correlation length of potential ξ cor

Hamiltonian of a 2D electron in a perpendicular magnetic filed and random potential H 1 ˆ e = p+ ˆ ( ) V ( ) 2m c A r + r 2 Standard change of variables ˆ π = pˆ e ˆ, Xˆ x c ˆ, ˆ c c A = π y Y = y+ ˆ π eb eb Equations of motion for the guiding center coordinates Xɺ i H, X c V, Y i = H, Y c V = ɺ = = ħ eb y ħ eb x x

π ˆ, π ˆ = ħeb ic δπ δπ x X = c ˆ eb π y l x y x y ħ eb c In the limit of strong magnetic field X x, Y y dv V Xɺ + V Yɺ c V V c V V 0 dt X = Y eb = X Y eb Y X Therefore electron moves along lines of constant potential Scattering in the vicinity of the saddle point potential T Transmission probability = 1 1 + exp(- πε ) Percolation + tunneling MS

The network model of Chalker and Coddington. Each node represents a saddle point and each link an equipotential line of the random potential (Chalker and Coddington; 1988) z 1 z 2 z 3 z 4 Z Z 1 3 = M Z Z 4 2 M = e iϕ 0 1 e 0 iϕ 2 cosh θ sinh θ i sinh θ e cosh θ 0 ϕ 3 e 0 iϕ 4 Crit. value argument

Fertig and Halperin, PRB 36, 7969 (1987) Exact transmission probability through the saddle-point potential V = U( x2+ y2) + V SP 0 T = 1 1 + exp(- πε ) E ħω 2 c ε ( E ( n+ 1/2) E V )/ E E 1 2U mω c 2 0 1 ħ for strong magnetic fields For the network model T = 1 cosh 2 θ ε = 2 π ln(sinh θ )

Total transfer matrix T of the system is a result of N iterations. Real parts of the eigenvalues are produced by diagonalization of the product M system width Lyapunov exponents λ 1 >λ 2 > >λ M/2 >0 Localization length for the system of width M ξ M is related to the smallest positive Lyapunov exponent: ξ M ~ 1/λ M/2 2 2 2 1 1/2 2 1 ( ) M M N N N N N N e e e e e e T T λ λ λ λ λ λ = Loc. Length explanation

ξ M /M ξ Μ /Μ Renormalized localization length as function of energy and system width 0.9 ξ M 0.8 = const M 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.00 0.0 0 0.2 2 0.4 4 0.6 6 0.8 8 10 1.0 εμ 1/ν ε One-parameter scaling fits data for different M on one curve ξ M ( ε ) M M=16 M=32 M=64 M=128 M = f ξ ( ε )

The thermodynamic localization length is then defined as function of energy and diverges as energy approaches zero ν = 2.5 ± 0.5 ξ ( ε ) ~ ε ν Main result in agreement with experiment and other numerical simulations Is that it?

Generalization: each link carries two channels. Mixing on the links is unitary 2x2 matrix U i e e cos φ α iγ iδ = e iγ iα sin φ e e sin φ cos φ Lee and Chalker, PRL 72, 1510 (1994) Main result two different critical energies even for the spin degenerate case

energies ε and ε-, respectively T = U U 1 C S 3 U S C U 2 4 C = 1+ exp[ π ( ε )] 0 0 1+ exp[ πε ] S = exp[ π ( ε )/2] 0 0 exp[ πε /2] = E / E = f ( B) 2 1

One of the results: Floating of extended states ε Landau level (B) PRB 52, R17044 (1996) V.K., B. Horovitz and Y. Avishai

General Classification: Altland, Zirnbauer, PRB 55 1142 (1997) S N S

Compact form of the Hamiltonian Ĥ = 1 h h c c c 2 T c The 4N states are arranged as (p,p,h,h ) Four additional symmetry classes: combination of time-reversal and spin-rotational symmetries Class C TR is broken but SROT is preserved corresponds to SU(2) symmetry on the link in CC model (PRL 82 3516 (1999)) with ν 1.12, µ 1.45 Renormalized localization length ξ ( ε, ) M = 1 f εm ν, M M 1 µ Unidir. Motion argument

At the critical energy ξ = f ενm, µ M M Energies of extended states ( ) =± c ε = const and is independent of M, meaning the ratio between two variables is constant! c µ ν σ = 0 xy σ = 2 xy σ = 1 xy Spin transport PRL 82 3516 (1999) V.K., B. Horovitz, Y. Avishai, and J. T. Chalker

Class D TR and SROT are broken Can be realized in superconductors with a p-wave spin-triplet pairing, e.g. Sr 2 RuO 4 (Strontium Ruthenate) The A state (mixing of two different representations) total angular momentum J z =1 broken time-reversal symmetry Triplet broken spin-rotational symmetry

y k x ik y θ = θ k1 k2 p-wave x θ θ cos( ) sin( ) k1 = θ i θ 0 k1 k1 2 0 = cos( θ ) + isin( θ ) k k2 k2 = cos( θ ) + isin( θ ) = only for θ = 90 k1 k2 0 k1 k1 SNS with phase shift π there is a bound state S N S Chiral edge states imply QHE (but neither charge nor spin) heat transport with Hall coefficient 2π 2k B 2 Ratio Kxy = Kxy / T is quantized 3h

Class D TR and SROT are broken corresponds to O(1) symmetry on the link one-channel CC model with phases on the links (the diagonal matrix element ) ϕ = l 0 with probability W π with probability 1-W λ =!!! The result: 0 M M=2 exercise coshθ sinhθ 1 0 1 e 1 sinh cosh 0 A 1 A θ θ θ A = After many iterations... = e [( + A+ AB+ ABC+...) θ ] 1 ABC...

coshθ sinhθ 1 0 1 = e Aθ 1 sinhθ coshθ 0 A 1 A... = e After many iterations [ ( + A+ AB+ ABC+...) θ] 1 ABC... After many iterations there is a constant probability α for ABC =+1, and correspondingly 1- α for the value -1. Then: αw+(1- α)(1-w)= α α=1/2 except for W=0,1 Both eigenvectors have EQUAL probability, and their contributions therefore cancel each other leading to λ =0

Change the model Node matrix cosh Aθ sinh Aθ = sinh Aθ cosh Aθ Cho, M. Fisher PRB 55, 1025 (1997) Random variable A=±1 with probabilities W and 1-W respectively cosh Aθ sinh Aθ 1 0 coshθ sinhθ 1 0 sinh Aθ cosh Aθ 0 A sinhθ coshθ 0 A Disorder in the node is equivalent to correlated disorder on the links correlated O(1) model M=2 exercise cosh Aθ sinh Aθ 1 e 1 sinh A cosh A 1 A θ θ θ 1 = λ=0 only for <A>=0, i.e. for W=1/2 Sensitivity to the disorder realization!

0.6 0.4 σ xy =1 Heat transport ε 0.2 0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50-0.2-0.4 σ xy =0 W METAL -0.6 Ilya Gruzberg et. al. PRB 65, 012506 (2001) J. T. Chalker, N. Read, V. K., B. Horovitz, Y. Avishai, A. W. W. Ludwig: Another approach to the same problem

Another approach to the same problem ξ M /M ξ M /M 2 M =16 M = 32 M = 64 M = 128 M = 256 2 M =16 M = 32 M = 64 M = 128 M = 256 (a) (b) 1 1 0 0 0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 25 30 35 ε εm 1/ν W=0.1 is fixed ν=1.4

ξ M /M ξ M /M 2 M = 16 M = 32 M = 64 M = 128 2 M = 16 M = 32 M = 64 M = 128 (a) (b) 1 1 0.00 0.05 0.10 0.15 W ε=0.1 is fixed 0 1 2 3 4 5 6 W-0.2 M 1/ν ν=1.4

ξ M /M 7 6 5 M=16 M=32 M=64 M=128 4 3 2 1 0 0 1 2 3 ε W-0.19 M 1/ν ν=1.4

ε σxy = 0 0.19 0.2 0.2 METALLIC W σxy = 1 cond-mat 0806.2744 V.K. D. Nemirovsky

Back to the original network model Height of the barriers fluctuate - percolation

Random hyperfine fields H = γ ħi H int n i e Nuclear spin Magnetic filed produced by electrons H 8 e g π β seδ = re R 3 i e Additional potential V hf = µ B B hf

Nuclear spin relaxation Spin-flip in the vicinity of long-range impurity S.V. Iordanskii et. al., Phys. Rev. B 44, 6554 (1991) Yu.A. Bychkov et. al., Sov. Phys-JETP Lett. 33, 143 (1981),

First approximation infinite barrier with probability p If p=1 then 2d system is broken into M 1d chains All states are extended independent on energy Lyapunov exponent λ=0 for any system size as in D-class superconductor

Naive argument a fraction p of nodes is missing, therefore a particle should travel a larger distance (times 1/(1-p)) to experience the same number of scattering events, then the effective system width is M(1-p) -1 and the scaling is ξ M M ( 1 p) 1 = M f ξ( ε) But missing node does not allow particle to propagate in the transverse direction. Usually ξ M ~M, we, therefore, can expect power ν>1

ξ M /M 20 18 16 14 M=16 M=32 M=64 12 10 8 6 4 2 0 0.0 0.2 0.4 0.6 0.8 1.0 p Renormalized localization length at critical energy ε=0 as function of the fraction of missing nodes p for different system widths. Solid line is the best fit 1.24(1-p) -1.3. Dashed line is the fit with "naive" exponent ν=1

(1-p) 1.3 ξ Μ /M 1.4 1.2 1.0 0.8 0.6 0.4 0.2 M=16 p=0.5 M=32 p=0.5 M=64 p=0.5 M=16 p=0 M=32 p=0 M=64 p=0 M=16 ε=0.3 M=32 ε=0.3 M=64 ε=0.3 M=16 ε=0.5 M=32 ε=0.5 M=64 ε=0.5 M=16 ε=0 M=32 ε=0 M=64 ε=0 0.0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 Mε 2.5 Data collapse for all energies ε, system widths M and all fractions p 1 of missing nodes

Ferdinand Evers The effect of directed percolation can be responsible for the appearance of the value ν 1.3. By making a horizontal direction preferential, we have introduced an anisotropy into the system. Our result practically coincides with the value of critical exponent for the divergent temporal correlation length in 2d critical nonequilibrium systems, described by directed percolation models H. Hinrichsen, Adv. Phys. 49, 815 (2000) G. Odor, Rev. Mod. Phys. 76, 663 (2004) S. Luebeck, Int. J. Mod. Phys. B 18, 3977 (2004) It probably should not come as a surprise if we recollect that each link in the network model can be associated with a unit of time C. M. Ho and J. T. Chalker, Phys. Rev. B 54, 8708 (1996).

Summary Scaling ξ ν cl = 1 p f M M ε νq ν cl 4 / 3 ν q 2.5 The fraction of polarized nuclei p is a relevant parameter PRB 75, 113304 (2007) V.K. and Israel Vagner

Applications of Chalker-Coddington Network Model Link Matrix Physical System Results Level Statistics U(1) QHS: Single lowest Landau level ν 2.5 GUE U(2) QHS: Two levels (Landau or spin) with mixing Floating of extended states SU(2) Singlet SC with SROT invariance and TR symmetry broken Coalescence of extended states ν 4/3 O(1) Correlated O(1) - CF model Triplet SC SROT and TR symmetries broken 1D metal 3 phase diagram: 2D metal & two insulators (σxy=0,1) GUE GUE for all phases?

Summary Applications of CC network model QHE one level critical exponents QHE two levels two critical energies floating QHE current calculations QHE generalization to 3d QHE - level statistics SC spin and thermal QHE novel symmetry classes SC level statistics SC 3d model for layered SC Chiral ensembles RG QHE and spin QHE in graphene