Wavefunction extreme values statistics in Anderson localization P. R. N. Falc ao Instituto de F sica Universidade Federal de Alagoas 57072-900 Macei o AL Brazil and

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Wavefunction extreme values statistics in Anderson localization

P. R. N. Falc˜ao∗

Instituto de F´ısica, Universidade Federal de Alagoas, 57072-900 Macei´o, AL, Brazil and

Institute of Theoretical Physics, Jagiellonian University in Krakow, ul. Lojasiewicza 11, 30-348 Krak´ow, Poland

M. L. Lyra†

Instituto de F´ısica, Universidade Federal de Alagoas, 57072-900 Macei´o, AL, Brazil

We consider a disordered one-dimensional tight-binding model with power-law decaying hopping

amplitudes to disclose wavefunction maximum distributions related to the Anderson localization

phenomenon. Deeply in the regime of extended states, the wavefunction intensities follow the

Porter-Thomas distribution while their maxima assume the Gumbel distribution. At the critical

point, distinct scaling laws govern the regimes of small and large wavefunction intensities with a

multifractal singularity spectrum. The distribution of maxima deviates from the usual Gumbel

form and some characteristic ﬁnite-size scaling exponents are reported. Well within the localization

regime, the wavefunction intensity distribution is shown to develop a sequence of pre-power-law,

power-law, exponential and anomalous localized regimes. Their values are strongly correlated,

which signiﬁcantly aﬀects the emerging extreme values distribution.

I. INTRODUCTION

The Anderson localization transition is a key physical

phenomenon related to a drastic change in the spatial

distribution of Hamiltonian eigenfunctions promoted by

disorder [1–3]. In general lines, the eigenfunctions be-

come exponentially localized in the regime of strong dis-

order while remaining spatially extended for weak disor-

der. Anderson transition is a quite general phenomenon

aﬀecting the transport of electronic, acoustic, magnetic

and optical waves [4–8].

The critical behavior of the Anderson transition has

been extensively studied over the past decades and shown

to strongly depend on the system’s dimensionality, sym-

metries and the short or long-range character of the un-

derlying couplings [3]. In particular, all eigenstates be-

come exponentially localized in one-dimensional systems

with short-range couplings for any ﬁnite amount of un-

correlated disorder. In contrast, one-dimensional systems

with power-law decaying couplings support an Anderson

transition and have been frequently used as a prototype

model to investigate its universality classes and critical

behavior [9–24].

Field theoretical renormalization group and random

matrix (RM) theories yield several relevant aspects of the

Anderson localization transition. In particular, random

matrix theory unveiled universal characteristics of the

eigenvalues statistics in the localized and extended phases

[25]. Exponentially localized states are uncorrelated and

randomly distributed along the energy band, resulting

in a Poissonian probability distribution function (PDF)

of the level spacements. On the other hand, the level

repulsion typical of spatially extended states leads to a

new spacement PDF that depends of the nature of the

∗Electronic address: pedro.falcao@ﬁs.ufal.br

†Electronic address: marcelo@ﬁs.ufal.br

random matrix ensemble (Gaussian orthogonal, unitary

or symplectic). RM theory has also been used to explore

the statistical properties of the extreme eigenvalues [26].

Extreme events play a fundamental role in the study

of disordered physical systems [27,28]. These are events

with magnitude much larger than the average value

of a physical stochastic process. In condensed matter

physics, they have a key impact on transport phenomena

where the largest energy barriers and maximally localized

modes hinder the transmission of physical excitations

[29,30]. The extreme values statistics of uncorrelated

and identically distributed (IID) random variables is well

understood [31]. Unbounded random variables having a

PDF with a faster than power-law tail have a Gumbel dis-

tribution of the extreme values in long sub-sequences. A

Fr´echet extreme values PDF is in order for random vari-

ables with power-law decaying PDF tails. In the case of

bounded stochastic variables, the extreme values are dis-

tributed according to an asymptotic Weilbull PDF. For

sequences of strongly correlated random variables, very

little is known regarding the extreme values statistics (for

a recent review see [28]). In the context of Anderson lo-

calization, although RM theory discloses the PDF of the

extreme values for the correlated eigenvalues of Gaussian

ensembles [26], studies of the extreme values statistics of

the own eigenfunction intensities are missing. Although

the structureless nature of Gaussian extended eigenfunc-

tions with an exponentially decaying Porter-Thomas in-

tensity distribution[32] allows us to anticipate that the

maximum distribution of extended states shall fall in the

Gumbel class, multifractal correlations present in critical

states as well as the structured aspect of exponentially

localized states point to new distributions of the wave-

function intensity maxima that are still unexplored.

In this work, we address the above relevant open ques-

tion aiming to unveil how the extended or localized na-

ture of single-particle eigenstates in a disordered sys-

tem is reﬂected in the extreme values statistics of their

eigenfunctions. We will consider the prototype one-

arXiv:2210.06365v1 [cond-mat.dis-nn] 12 Oct 2022

dimensional Anderson model with random power-law de-

caying hopping amplitudes that exhibits a well-known

localization-delocalization transition. We will focus in

unveiling the scaling behavior of the average maximum

value of the eigenfunctions as well as its probability dis-

tribution in the extended, critical and localized regimes.

Further, we will unfold the role played by intrinsic eigen-

function correlations in the extreme values distributions.

II. MODEL AND NUMERICAL METHODS

We consider the following tight-binding Hermitian

Hamiltonian model with power-law decaying hopping

amplitudes on a linear chain with Nsites and periodic

boundary conditions:

H=X

i|iihi|+X

i>j

tij (|iihj|+|jihi|),(1)

with irepresenting the on-site potentials and hopping

amplitudes given by tij =Wij /rσ

ij .rij is the distance

between the chain sites, restricted to the interval 1 ≤

rij ≤N/2 (rij =i−jfor i−j < N/2 and rij =N−

(i−j) for i−j > N/2). σis a characteristic power-law

exponent that controls the eﬀective range of the hopping

amplitudes. We consider a large ensemble of the above

Hamiltonian with iand Wij being random real numbers

distributed uniformly in the interval [−1,1].

The above Hamiltonian belongs to the class of power-

law random band models. A perturbation analysis based

on a ﬁeld-theoretical model of interacting supermatrices,

supported by exact diagonalization results, has settled

that a well-deﬁned Anderson transition occurs as a func-

tion of the control exponent σ[9–11]. All states are ex-

tended for σ < 1. In this regime, the statistical prop-

erties are those of the Gaussian orthogonal ensemble of

RM for σ < 1/2, with stronger ﬂuctuations developing

for 1/2< σ < 1. All states are critical at σ= 1, exhibit-

ing a multifractal character [33]. For σ > 1 all states are

localized with integrable power-law tails |φ(r)|2∝r−2σ.

A superdiﬀusive short-time spreading of wavepackets sets

up for 1 < σ < 3/2.

Here, we will perform a statistical analysis of the ex-

treme values of the above Hamiltonian eigenfunctions.

Exact numerical diagonalization will be employed to

compute the full spectrum of eigenvalues and the cor-

responding eigenfunctions of a large ensemble of disor-

der conﬁgurations. After a preliminary analysis of the

density of states and level spacing distribution of the dif-

ferent regimes, we will focus on the eigenfunction statis-

tics. One can associate a sequence of random variables to

each eigenfunction {φ2

1, φ2

2, ..., φ2

N}, from which the dis-

tribution of eigenfunction intensities P(φ2) can be ex-

tracted. We further identify the maximum φ2

mof each

eigenfunction and its distribution F(φ2

m) over a spectral

range around the band center.

In the regime where the eigenfunction intensities are

uncorrelated, the above two distribution functions are

closely related. The cumulative distribution of maxima

of IID eigenfunction intensities can be expressed as

QN(φ2

m) = "1−Z1

φ2

P(φ2)dφ2#N

,(2)

from which the distribution of maxima can be derived

as F(φ2

m) = dQN/dφ2

m. Three limiting distributions of

maxima can be anticipated for IID variables [31]. In par-

ticular, when the tail of the parent distribution P(φ2)

decays faster than a power-law and the upper cutoﬀ can

be ignored, the distribution of maxima converges to the

Gumbel form

F(φ2

m)∝exp−aφ2

m−bexp−aφ2

m,(3)

where aand bare distribution parameters related to the

average value and dispersion. Parent distributions with

power-law tails lead to a distinct Fr´echet distribution of

maxima. On the other hand, when the parent distri-

bution vanishes at and above a cutoﬀ value, an asymp-

totic Weibull distribution of maxima of IID sequences of

random variables is predicted. In what follows, we di-

rectly compute both the parent distribution P(φ2) and

the maxima distribution F(φ2

m) of extended, critical and

strongly localized states, with particular emphasis on

possible deviations from the prediction based on the as-

sumption of IID variables.

−5 0 5

0.0

0.1

0.2

0.3

DOS

σ= 0.50

σ= 1.00

σ= 2.00

FIG. 1: Density of states for three values of the hopping de-

cay exponent σrepresenting the extended (σ= 0.5), critical

(σ= 1) and localized (σ= 2) regimes. The smooth proﬁle of

extended states evolves to a rough proﬁle typical of localized

states. The shaded region represents the band central region

|E| ≤ 0.5, where the DOS has a ﬂat proﬁle. Data are from

103disorder conﬁgurations on closed chains with N= 103

sites.

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WavefunctionextremevaluesstatisticsinAndersonlocalizationP.R.N.Falc~aoInstitutodeFsica,UniversidadeFederaldeAlagoas,57072-900Maceio,AL,BrazilandInstituteofTheoreticalPhysics,JagiellonianUniversityinKrakow,ul.Lojasiewicza11,30-348Krakow,PolandM.L.LyrayInstitutodeFsica,UniversidadeFederaldeAlag...

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Wavefunction extreme values statistics in Anderson localization P. R. N. Falc ao Instituto de F sica Universidade Federal de Alagoas 57072-900 Macei o AL Brazil and

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