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Metal-Insulator TransitionMetal-Insulator Transition& Energy Level Statistics& Energy Level Statistics

S.M. NishigakiS.M. NishigakiDept. Physics, Univ. ConnecticutDept. Physics, Univ. Connecticut

SMN PRE58, R6915 (1998); E59, 2853 (1999)SMN PRE58, R6915 (1998); E59, 2853 (1999)A.Garcia, SMN, J.Verbaarschot PRE66, 16132 (2002)A.Garcia, SMN, J.Verbaarschot PRE66, 16132 (2002)

**.**.2002 location

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•• LocalizationLocalization　　　　 Anderson Model / ScalingAnderson Model / Scaling　　

•• Spectral Statistics Spectral Statistics Many-Body/ Quantum Chaos/ DisorderMany-Body/ Quantum Chaos/ Disorder

•• Random Matrices Random Matrices UniversalityUniversality

•• Mobility EdgeMobility Edge　　　　Multifractality / Critical StatisticsMultifractality / Critical Statistics

GOALGOAL Derive Energy Level Fluctuation Derive Energy Level Fluctuation from Universality of M-I Transition from Universality of M-I Transition

•• Critical Statistics & Random MatricesCritical Statistics & Random Matrices

•• Other SystemsOther Systems　　　　Gauge Theory / Non-HermitianGauge Theory / Non-Hermitian

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LOCALIZATIONLOCALIZATION

Vx

x

tamplitude t

random #of range W

•• ee---e-e-- Coulomb force Coulomb force ××　→　　→　One-Body HamiltonianOne-Body Hamiltonian•• Lattice oscillation Lattice oscillation ×ו• Internal freedom Internal freedom ××　→　　→　1 state/site, hopping 1 state/site, hopping tt•• ImpurityImpurity　　　　　　　　　　 →→　　Random potential Random potential WW

Anderson Model of Impure CrystalAnderson Model of Impure Crystal

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t >> |V0-V1|

ψx

t << |V0-V1|

t > W　⇒　Prob(|ψ0|~|ψn|) = 1　　　　　　　　 WF ExtendedWF Extended

t < W　⇒   Prob(|ψ0|~|ψn|) = (t/W)n → 0

ψx

WF LocalizedWF Localized

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ScalingScaling

LL+dL

Scale dependence ofScale dependence ofdimdim’’ lessless conductance conductance g(L) g(L)

g(L+dL) = F(g(L) , (L+dL)/L)　⇒ dlog g/dlog L = β(g)β(g)

Insulatorg(L) ~ exp(-L/ξ)β(g) ~ log g

Metalg(L) ~ LD-1/Lβ(g) ~ D-2

D>2D=2D<2

g*D>2 ⇒ ヨ Fixed Pt M-I Transition

Flow at L→∞

g0 .

6Exact computation impossible Exact computation impossible →→ Treat Statistically Treat Statistically

Highly Excited Levels of Heavy Nuclei

SPECTRAL STATISTICSSPECTRAL STATISTICS

Complex Many-Body SystemComplex Many-Body System

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UniversalityUniversalityStatistics independent of systemsStatistics independent of systemsas long as they are as long as they are ““ complicated enoughcomplicated enough””

Distribution of neighboring energy levelsDistribution of neighboring energy levels’’ spacing spacing

Excited levels of heavy nuclei Elastic levels of irregular quartz

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Quantum ChaosQuantum ChaosNon-separable 2D QMNon-separable 2D QM

Irregular Billiards 　 Free motion on Curved surface　 Quantum Dots

Classical Orbit exhausts Phase Space Classical Orbit exhausts Phase Space ＝＝ ErgodicErgodic

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Energy LevelsEnergy Levels

→→treattreat

statisticallystatistically

Level Spacing DistributionLevel Spacing Distribution

as long as Classical Orbit is Ergodic,as long as Classical Orbit is Ergodic,Statistical fluctuaton of energy levels isStatistical fluctuaton of energy levels is UniversalUniversal

Sinai BilliardSinai Billiard

PoissonPoisson

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Anderson ModelAnderson Model Lattice Gauge TheoryLattice Gauge Theory

N states/site (color of quarks=3)Hopping: U SU(N) matrix: random

Hopping amplitude : constantPotential : random

chiral chiral == bipartite lattice o ,x bipartite lattice o ,xnonchiral = simple latticenonchiral = simple lattice

Disordered SystemDisordered System

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Level spacing distr.Level spacing distr. Distribution of smallest Dirac EV Distribution of smallest Dirac EV chiralchiral

(different from left)

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Statistics of energy levelsStatistics of energy levelsof such extended states isof such extended states is UniversalUniversal

What is Ergodicity in Disordered SystemsWhat is Ergodicity in Disordered Systems？？

δx2 = D δτ

scattering by impurityscattering by impuritycreation/absorption of gluoncreation/absorption of gluon

Brownian motion : 2D FractalBrownian motion : 2D Fractal

covers whole region in covers whole region in ttimeime > >>> L2/D

typical time scale for level statistics :typical time scale for level statistics :　　　　((mean level spacingmean level spacing) ) −1 = h/∆

∆ << Ec ≡ hD / L2　

⇔⇔ ErgodicErgodic

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Overlap of Overlap of ExtendedstatesExtendedstates ⇒⇒ avoid degeneracy avoid degeneracy

Level repulsionLevel repulsionProb(E1 ,E2) ~|E1 -E2|β

β=1,2,4 due to symmetry

Universal behavior should be

derived from a simple model Random Matrix TheoryRandom Matrix Theorywith this property　　

H+ atom in magnetic field

Origin of UniversalityOrigin of Universality

　

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H = H+ = (Hij) 　　indep, normal distribution indep, normal distribution R, C, HR, C, H

dµ(Hij) = exp(-tr H2) ΠdβHij

••InvarianceInvariance　 dµ(H) = dµ(UHU+)　 invariant under change of basis　→　Eigenvector is delocalized

••Eigenvalue distributionEigenvalue distribution　　　　　dµ(Ei) = Πi dEi exp(-tr Ei

2) Πi>j|Ei - Ej|β

β = # real comp = 1, 2, 4

RANDOM MATRCESRANDOM MATRCES

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Level spacing distributionLevel spacing distribution Two-level correlationTwo-level correlation

β=0 no correlationβ=1β=2β=4

~ sβ ~ exp(-cβ s2)

exp(-s)

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Spectral rigidity (Level number variance)Spectral rigidity (Level number variance)

~ log L

Σ2 (L) = 〈 ( N −〈N〉) 2 〉 (within width L )

L

β=0 no correlationβ=1β=2β=4

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Distribution of Distribution of 11stst~4~4thth Diral levels Diral levels chiralchiral

β=2

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MOBILITY EDGEMOBILITY EDGEEnergy band near criticality W~WEnergy band near criticality W~Wcc

E

1/ξ

|ψ |2

ξ

|ψ |2

1/V

V

ξ >> L ξ << L

MultifractalMultifractal〈 Σ |ψ |2p〉~ Vdp(1-p)

〈 Σ |ψ |2p〉~ V1-p 〈 Σ |ψ |2p〉 indep of V

Fractal dim.Fractal dim.0 < {dp}p=2,3,... < 1

ξ ~ L

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　 ⇒　⇒　sparce overlap between WFssparce overlap between WFs　　 　 　Z(E-E‘) =〈 Σ |ψΕ

|2 |ψΕ‘ |2 〉~ |E − E‘|-(1-d2)

MultifractalityMultifractality〈 Σ |ψ |2p〉~ V-dp(p-1)

3D Anderson model

β=2 (B≠0)β=1

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Critical StatisticsCritical Statistics

sparce overlap sparce overlap 〈 Σ |ψΕ |2 |ψΕ‘

|2 〉~ |E − E‘|-(1-d2)

　 ⇒⇒ weak correlation between remote levels weak correlation between remote levels

Numbe variance〈 δN2 〉~ χ L , χ = (1-d2)/2Level spacing　 p(s) ~ exp(-κ s) , κ = 1/(1-d2)

Nearby levelsNearby levels：： Random Matrix (Wigner-Dyson)Random Matrix (Wigner-Dyson)　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　 hybrid statisticshybrid statisticsRemote levelsRemote levels：： No Correlation (Poisson)No Correlation (Poisson)

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Critical 3D Anderson model　(β=1: no B)

Level spacing distr.Level spacing distr.　　　　　　 Number variance Number variance

~ s1 ~ log L ~ χ L~ exp(-s/2χ)

•• indep of system sizeindep of system size　　　　　　　　 　　Infrared fixed pointInfrared fixed point•• indep of type of randomness indep of type of randomness Quasi-UniversalityQuasi-Universality•• dep on dimension, boundary dep on dimension, boundary ConductanceConductance gg＊＊

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CRITICAL STATISTICS & RMCRITICAL STATISTICS & RM

Nonlinear Sigma ModelNonlinear Sigma Model

Anderson modelAnderson model　　　　　　　　　　　　 Quantum Field Theory Quantum Field Theoryreplicareplica

supersymm.supersymm.

••Ergdic Ergdic Ec → ∞ 　　 0 0 dim, coincides with RM dim, coincides with RM••DiffusiveDiffusive Ec >> ∆　　 perturbationperturbation ○○••Mobility edge Mobility edge Ec ~ ∆　　 perturbationperturbation ××

L(Q) = 1/(V∆) [D tr (▽Q)2 + δE tr ΛQ] , Q(x) : NGNG boson boson

appeal to Universality ansatzappeal to Universality ansatz

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H = (Hij) normal distribution, indep.normal distribution, indep.　　　　variance variance = 1/(1+a2(i-j)2) ⇒⇒ invariance invariance H→UHU+

Multifractal eigenvectorMultifractal eigenvector　　　　　　　　　　dp ~ 1 - (a/2π2)p

Weak Multifractal WF of Anderson modelWeak Multifractal WF of Anderson model(g*>>1)　　　　　　　　　　dp ~ 1 - (1/g*)p

Banded Random MatricesBanded Random Matrices

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derive Level statisticsderive Level statistics

Level spacing distr.Level spacing distr.

~ sβ ~ e-s/2χ

β=1

β=2

β=4

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1-parameter fitting to1-parameter fitting tocritical statistics data ofcritical statistics data of3D Anderson models3D Anderson models

β=1

β=2

β=4

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fitting tofitting tocritical 3D Anderson modelcritical 3D Anderson model

β=1Number varianceNumber variance Level spacing distr.Level spacing distr.

fitting tofitting tocritical 2D Anderson modelcritical 2D Anderson model

β=4

~ χ L~ log L

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DisorderedDisorderedMetalMetal

RandomRandomMatrixMatrix

QuantumQuantumChaosChaos

Extended WFExtended WF

nearby levels: nearby levels: replusionreplusion　　

remote levels: correlationremote levels: correlation

WignerWigner-Dyson statistics-Dyson statistics

DisorderedDisorderedInsulatorInsulator

IntegrableIntegrableSystemSystem

Localized WFLocalized WF

no level correlationno level correlation

Poisson statisticsPoisson statistics

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MobilityMobilityEdgeEdge

BandedBandedRandom MatrixRandom Matrix

Multifractal Multifractal WFWFconductance g*conductance g*

nearby nearby lebels lebels repulsion repulsion ○○remote levels correlation remote levels correlation ××

Critical StatisticsCritical Statistics

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Other SystemsOther Systems

Gauge Field TheoryGauge Field Theory

Instanton Liquid ModelInstanton Liquid Modelof 4D QCDof 4D QCD

vsvsChiral Random MatricesChiral Random Matrices

β=2

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Non-Non-HermitianHermitian

Open / Dissipative system Open / Dissipative system 　＋　　＋　RandomnessRandomness

3D Anderson modelwith leads

Critical statistics of nonhermitian systemsCritical statistics of nonhermitian systemsbe described by nonhermitian RM withbe described by nonhermitian RM withMultifractal eigenvectorsMultifractal eigenvectors

conjectureconjecture

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Number variance ofNumber variance ofcomplex energy levelscomplex energy levels

same g* , vary nonhermiticity L

Σ2 (L) ~ χ L , χ indep of nonhermiticityconjecturconjecture e ??Multifractal dim of weakly nonhermitianMultifractal dim of weakly nonhermitiandisordered systems be indep of nonhermiticitydisordered systems be indep of nonhermiticity

β=2

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•• Energy levels at Mobility Edge areEnergy levels at Mobility Edge are associated with Multifractal WFsassociated with Multifractal WFs

•• Quasi-Universal Level StatisticsQuasi-Universal Level Statistics depending only on g* depending only on g* Energy levels partially uncorrelated Energy levels partially uncorrelated

•• If If g*>>1, dg*>>1, describedescribed by Random Matrices by Random Matrices sharing the set of sharing the set of multifractal multifractal dimsdims

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For those who want to study moreFor those who want to study more……

T.Guhr, A.M.-Groeling, H.Weidenmuller cond-mat/9707301

C.Beenakker cond-mat/9612179