Coherent forward scattering as a robust probe of multifractality in critical disordered media Maxime Martinez1Gabriel Lemarié123Bertrand Georgeot1Christian Miniatura23456and Olivier Giraud7

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Coherent forward scattering as a robust probe of multifractality in critical disordered

media

Maxime Martinez,

Gabriel Lemarié,

1, 2, 3

Bertrand Georgeot,

Christian Miniatura,

2, 3, 4, 5, 6

and Olivier Giraud

Laboratoire de Physique Théorique, Université de Toulouse, CNRS, UPS, France

MajuLab, CNRS-UCA-SU-NUS-NTU International Joint Research Unit,Singapore

Centre for Quantum Technologies, National University of Singapore, Singapore

Université Côte d’Azur, CNRS, INPHYNI, Nice, France

Department of Physics, National University of Singapore, Singapore

School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore

Université Paris Saclay, CNRS, LPTMS, 91405 Orsay, France

(Dated: October 10, 2022)

We study coherent forward scattering (CFS) in critical disordered systems, whose eigenstates are

multifractals. We give general and simple arguments that make it possible to fully characterize

the dynamics of the shape and height of the CFS peak. We show that the dynamics is governed

by multifractal dimensions

and

, which suggests that CFS could be used as an experimental

probe for quantum multifractality. Our predictions are universal and numerically veriﬁed in three

paradigmatic models of quantum multifractality: Power-law Random Banded Matrices (PRBM),

the Ruijsenaars-Schneider ensembles (RS), and the three-dimensional kicked-rotor (3DKR). In the

strong multifractal regime, we show analytically that these universal predictions exactly coincide

with results from standard perturbation theory applied to the PRBM and RS models.

PACS numbers: 05.45.Df, 05.45.Mt, 71.30.+h, 05.40.-a

I. INTRODUCTION

Wave transport in disordered systems is a long-standing

topic of interest in mesoscopic physics. In particular, wave

interference can have dramatic consequences on quantum

transport properties. The most celebrated example is

probably Anderson localization (AL) [

], that is, the

suppression of quantum diﬀusion and the exponential

localization of quantum states. AL is ubiquitous in wave

physics and has been observed in many experimental

situations: with acoustic waves [

], light [

–

], matter

waves [9–15].

Appearance of AL depends on several characteristics,

in particular dimensionality, disorder strength and corre-

lations. For instance, it is well established that 3d disor-

dered lattices undergo a genuine disorder-driven metal-

insulator transition (MIT), associated with a mobility

edge in the spectrum, separating the insulating phase

with localized eigenstates from the conducting phase with

extended eigenstates. Near the critical point of such

disorder driven transitions, eigenstates

φα

(with energy

ωα

) can display multifractal behavior, for instance at the

MIT in Anderson model [

–

] and graphs [

–

], but

also for Weyl-semimetal–diﬀusive transition [

]. They

are extended but non-ergodic, and characterized by the

anomalous scaling of their moments Iq(E):

Iq(E) = hPn,α |φα(n)|2qδ(E−ωα)i

hPαδ(E−ωα)i∼N−Dq(q−1),(1)

where

are the multifractal dimensions, forming a con-

tinuous set with

real (

h. . .i

represents an average over

disorder conﬁgurations). Extreme cases

= 0 and

(the dimension of the system) for all

, cor-

respond respectively to localized and extended ergodic

eigenstates.

While Anderson MIT has been observed directly in

atomic matter waves [

], experimental observation of

multifractality remains challenging [

–

]. In particular,

there exists to our knowledge no direct experimental

observation of dynamical multifractality, i.e. manifestation

of multifractality through transport properties (e.g. power-

law decay of the return probability [27,28]).

Another celebrated wave interference eﬀect is the co-

herent backscattering (CBS). It describes the doubling

of the scattering probability (with respect to incoherent

classical contribution) of an incident plane wave with

wave vector

, in the backward direction

−k0

. Coherent

backscattering has been observed in many experimental

situations: with light [

–

], acoustic waves [

], seis-

mic waves [

] and cold atoms [

]. Recently, it was

demonstrated that in the presence of AL a new robust scat-

tering eﬀect emerges [

–

], namely the doubling of the

scattering probability in the forward direction +

. This

phenomenon, which appears at long times, was dubbed

coherent forward scattering (CFS). CBS and CFS actually

have a distinct origin: CBS comes from pair interference

of time-reversed paths (and thus requires time-reversal

symmetry), while CFS is present even in the absence of

time-reversal symmetry [

]. From an experimental

point of view, CFS has recently been observed with cold

atoms [38].

In this work, we discuss the fate of CFS at the critical

point of a disorder-driven transition with multifractal

eigenstates. This problem was ﬁrst addressed for a bulk

3d Anderson lattice [

], for which it was shown that CFS

survives at the transition, with however a scattering prob-

ability smaller than in the localized phase. More precisely,

arXiv:2210.04796v1 [cond-mat.dis-nn] 10 Oct 2022

Figure 1. CFS contrast Λ

(

k, t

;

)deﬁned by

(40)

in critical

disordered systems.

is the wave vector of the incident plane

wave and

are the multifractal dimensions of the eigenstates.

(a) In systems of inﬁnite size

N→ ∞

, the emergence of the

CFS peak as a function of time is governed by the nonergodic

properties of multifractal eigenstates. The CFS wings decay

asymptotically like (

|k−k0|t1/d

)

−D2

, see Eqs.

(62)

and

(63)

while the CFS peak height grows algebraically in time like

t−D2/d

and ﬁnally reaches the compressibility value

= 1

−D1

in the long-time limit

t→ ∞

, see Eq.

(58)

. (b) For systems

of ﬁnite size

, the long-time dynamics of the CFS peak

is governed by the box boundaries. The CFS peak height

reaches 1

−αN−D2

for

t→ ∞

with

some numerical factor,

see Eq.

(56)

. The wings of the CFS peak are then described

by Eq. (54).

it was conjectured from numerical evidence that, instead

of a doubling of the classical incoherent contribution, the

forward scattering probability corresponds to a multipli-

cation by a factor (2

−D1/d

), with

the dimension of

the system and

the information dimension. In our

previous study [

], we gave scaling arguments that cor-

roborate this conjecture, backed by numerical simulations

on the Ruijsenaars-Schneider ensemble, a Floquet system

with critical disorder and tunable multifractal dimensions.

We also studied CFS at the transition in ﬁnite-size sys-

tems, unveiling a new regime, where CFS properties have

ﬁnite-size scaling related to the multifractal dimension

D2[46].

This article is based on the approach developed in our

previous work [

] and, somehow, in the spirit of the

random matrix theory point of view discussed in [

]. In

particular, we give a complete description of the dynam-

ics of CFS peak in critical disordered systems, including

height and shape of the scattering probability, in two

distinct dynamical regimes. Our ﬁndings are summa-

rized in the sketch in Fig. 1. In particular, we present

new links between CFS dynamics and the multifractal

dimension

, that are relevant for most experimental

situations. Our analytical predictions are veriﬁed on three

diﬀerent critical disordered models with multifractal eigen-

states: Power-Law Random Banded Matrices (PRBM),

Ruijsenaars-Schneider ensemble (RS) and unitary three-

dimensional random Kicked Rotor (3DKR). Our predic-

tions are also corroborated by perturbative expansions

for RS and PRBM models in the strong multifractality

regime. These results pave the way to a direct observa-

tion of a dynamical manifestation of multifractality in a

critical disordered system.

II. CRITICAL DISORDERED MODELS

As explained, in the following, our predictions will

be compared to numerical simulations on three diﬀerent

models. All of them can be mapped onto the generalized

-dimensional Anderson model, deﬁned by the following

tight-binding Hamiltonian

H=X

εn|nihn|+X

n6=m

tnm |nihm|,(2)

where

|ni

are the lattice site states,

εn

the on-site ener-

gies and

tnm

the hopping between two sites at distance

|n−m|

. Both

εn

and

tnm

can be considered arbitrary

random variables, whose exact properties will depend on

the system considered (see Table I). We will be interested

in ﬁnite-size eﬀects, and will consider a system with linear

size N, i.e. with a total number of sites equal to Nd.

MIT in the generalized Anderson model

(2)

has been

intensively studied (see [

] and references therein).

The three relevant parameters are the spatial dimension

of the lattice, the range of the hopping

tnm

, and existence

of correlations in the random entries of the Hamiltonian.

We recall here some well established facts: (i) in the

absence of disorder correlations and if

h|tnm|i

decay faster

than 1

/|n−m|d

, Anderson transition only occurs for

d >

2; (ii) in the absence of disorder correlations, critical

eigenstates can appear if

h|tnm|i

decay as fast as 1

/|n−

m|d

; (iii) correlations in diagonal disorder

εn

weaken

localization while correlations in oﬀ-diagonal disorder

tnm can favour localization.

We now discuss the characteristics and properties of

the diﬀerent models we used, as well as their link with

the Anderson model (2). A summary is given in Table I.

A. Power-Law Random Banded Matrices (PRBM)

Power-law random banded matrices were ﬁrst intro-

duced in [

]. They were inspired from earlier random

banded matrix ensembles with exponential decay describ-

ing the transition from integrability to chaos [

]. The

PRBM model is deﬁned by symmetric or Hermitian matri-

ces whose elements are identical independently distributed

(i.i.d.) Gaussian random variables with zero mean and

variance decreasing as a power law with the distance

from the diagonal. The critical PRBM model corresponds

to an Anderson model

(2)

with random long-range hop-

ping whose variance decays as the inverse of the distance

between sites.

More precisely, let

(

µ, σ

)be a Gaussian distribution

of mean

and standard deviation

. In the following

we use the version of PRBM considered in [

], with

periodic boundary conditions, where for

N×N

matrices

Model PRBM RS 3DKR

Tunable multifractal dimensions DqYes with b∈[0,∞[Yes with a∈[0,∞[No

Type Hamiltonian Floquet Floquet

Energy dependent properties Yes No No

Hopping range tnLong-range ∼1/n Long-range ∼1/n Short-range (exponential decay)

Dimension d= 1 d= 1 d= 3

Direct (disorder) space Position Momentum Momentum

Table I. Summary of some of the main properties of three models considered in this article (see text pour more details).

diagonal entries

εn

are i.i.d. with distribution

1),

and real and imaginary parts of the oﬀ-diagonal entries

tmn are i.i.d. with distribution N0, σnm/√2,

σ2

nm =1 + sin2(π|n−m|/N )

(bπ/N)2−1

.(3)

In particular we have

ph|tnm|2i

σnm

, which scales as

∼1/|n−m|for b |n−m|  N.

The density of states is deﬁned as

ρ(E) = h1

NdX

δ(E−ωα)i,(4)

which for this model gives

ρPRBM(E) = (1

√2πexp−E2

2b1,

2bπ2√4bπ −E2b1.(5)

Eigenvectors are multifractal, and their multifractal di-

mensions

, which depend on both

and parameter

, can be analytically computed [

]. Parameter

makes it possible to explore the whole range of multifrac-

tality regime: the weak multifractality regime

Dq→

1is

reached for

b→ ∞

and the strong multifractality regime

Dq→

0is reached for

b→

0. All numerical data pre-

sented in this work are performed at the center of the

band E= 0.

B. Ruijsenaars-Schneider model

Let us consider the following deterministic kicked rotor

model [51,52]

H=τˆp2

2+V(ˆx)X

δ(t−n),(6)

with a 2

-periodic sawtooth potential

(

) =

for

−π < x < π

, and where

is a constant parameter. As

a direct consequence of spatial periodicity of

(

), mo-

menta only take quantized values

= 0

,±

, . . .

(here

= 1). Additionally, we consider a truncated basis

space, with periodic boundary conditions, so that

the total number of momenta states

|pni

accessible is

This implies that position basis is also discretized (

are

separated by intervals 2π/N, with kan integer).

It is well-known that kicked Hamiltonians such as

(6)

can be mapped onto the Anderson models

(2)

[

The

quantized plane waves

|pni

then play the role

of lattice site states

|ni

. The mapping is given (for an

eigenvector of the Floquet operator with eigenphase e

iω

)

εn= tanω/2−τn2/4,(7)

tnm =−Zπ

−π

2πtan[V(x)/2]e−ix(m−n),(8)

where the on-site energy

εn

takes evenly distributed

pseudo-random value, provided

is suﬃciently irrational.

As a consequence of the Fourier transform relation in

Eq.

(8)

, discontinuity of the sawtooth potential

(

)cre-

ates a long-range decay of the couplings

tnm ∼

/|n−m|

and actually induces multifractal eigenstates.

The Ruijsenaars-Schneider (RS) model was introduced

in the context of classical mechanics [

–

]. Its quantum

properties were studied in [

–

]. It is deﬁned (for an

arbitrary real parameter

) by the Floquet operator of

the Hamiltonian (6) (with truncated basis in pspace)

U= e−iϕˆpe−iaˆx,(9)

where the deterministic kinetic phase has been replaced by

random phases

ϕˆp

(consequently the on-site energies

εn

in Eq.

(8)

are truly uncorrelated), and

is taken modulo

2π[61].

Importantly, unlike for PRBM, eigenstate properties

of the RS matrix ensemble do not depend on their quasi-

energy. In particular it has a ﬂat density of states

ρRS(E) = 1

2π.(10)

Eigenvectors are multifractal; the multifractal dimensions

can be derived in certain perturbation regimes, and only

depend on the parameter

[

–

]. This parameter

allows us to explore the whole range of multifractality

regimes : the weak multifractality regime

Dq→

1is

reached for

a→

1and the strong multifractality regime

Dq→0is reached for a→0.

C. 3d Random Kicked Rotor (3DKR)

Our three-dimensional (3d) model is the deterministic

kicked rotor, deﬁned by the following Hamiltonian [66]

H=τxp2

2+τyp2

2+τzp2

2+V(q)X

δ(t−n),(11)

where

τi

are constant parameters and the spatial potential

writes

(

) =

(

)

(

)

(

the kick strength, with

V(x) = √2

2cos x+1

2sin 2x,(12)

so that the system breaks the time-reversal symmetry

[45].

As previously stated, the Hamiltonian

(11)

can be

mapped onto the 3d Anderson model

(2)

. For a given

eigenstate of the system with eigenphase e

iω

, this mapping

writes

εn= tanω/2−τxn2

x/4−τyn2

y/4−τzn2

z/4,(13)

tnm =−ZZ π

−π

(2π)3tan[V(q)/2]e−iq·(m−n),(14)

where energies

εn

take pseudo-random values (provided

that (

τx, τy, τz

)are incommensurate numbers), and where

hopping terms

tnm

decay exponentially fast with distance

between sites |n−m|[66].

The 3d random Kicked Rotor (3DKR) that we consider

in the following corresponds to the Floquet operator of

Hamiltonian (11)

U= e−iφˆ

pe−iV (ˆ

q),(15)

where deterministic kinetic phases are replaced by uni-

formly distributed random phases

φp

(this implies in

particular that energies εnin Eq. (13) are uncorrelated).

The 3DKR can be seen as the Floquet counterpart

of the usual 3d unitary Anderson Model. In particular

it undergoes an Anderson transition monitored by the

parameter

(that is related to the hopping intensity).

Using techniques inspired by [

], we found that the

critical value is

Kc≈

58 (see Appendix A). However,

unlike the 3d Anderson model, this unitary counterpart

has a ﬂat density of states

ρ3DKR(E) = 1

2π,(16)

and no mobility edge.

Furthermore, we assumed that 3DKR has the same mul-

tifractal dimensions than the corresponding unitary 3d

Anderson model, because it belongs to the same universal-

ity class. The values that were determined in [

] (using

the same techniques as in [

]) are

= 1

912

007

and D2= 1.165 ±0.015.

III. GENERAL FRAMEWORK FOR THE

STUDY OF CFS IN CRITICALLY DISORDERED

SYSTEMS

A. Eigenstates and time propagator

In the following, we will analytically and numerically

address CFS in critical disordered systems within a very

general framework, including both Floquet and Hamil-

tonian cases. The numerical methods are presented in

Appendix B.

For the sake of clarity, we use a common notation:

|φαi

refer to eigenstates (or Floquet modes) with energy (or

quasienergy)

ωα

. The time propagator of the system then

writes

U(t) = X

e−iωαt|φαihφα|,(17)

where time will be considered a continuous variable. In

particular, we use the following convention and notation

for the temporal Fourier transform:

f(ω) = Z∞

−∞

dt f(t)eiωt, f(t) = Z∞

−∞

dω

2πf(ω)e−iωt.

(18)

B. Direct and reciprocal spaces

As illustrated by the models introduced above, in

generic critical disordered systems disorder can be present

either in position space (e.g. PRBM, Anderson model)

or momentum space (e.g. 3DKR, RS). From now on, we

refer to the basis where disorder is present (labeled with

kets

|ni

) as the direct space and to its Fourier-conjugated

basis (labeled with kets

|ki

) as the reciprocal space. This

distinction is particularly important because multifractal-

ity of eigenstates is a basis-dependent property that only

appears in direct space, where disorder is present, while

CFS is an interference eﬀect taking place in reciprocal

space.

Importantly, we choose to use standard notations

of spatially disordered lattice systems, as in Eq.

(2)

For a

-dimensional system, direct space is spanned

by discrete lattice sites states

|ni

|n1, . . . , ndi

(

−N/

2 + 1

, . . . N/

2) (

will be considered even). The

dimension of the associated Hilbert space is

. Con-

sequently, the reciprocal space is spanned by a basis

|ki

|k1, . . . , kdi

(where

±π

N,±3π

N··· ± (N−1)π

We also choose the following convention for the change of

basis (see Appendix Cfor details)

φα(k) = X

n∈]−N/2,N/2]d

φα(n)e−ik·n,(19)

φα(n) = 1

NdX

k∈]−π,π]d

φα(k)eik·n,(20)

so that in the limit

N→ ∞

the system tends to a inﬁnite-

size discrete lattice, that is,

φα(k)−→

N→∞

∞

n1=−∞ ··· ∞

nd=−∞

φα(n)e−ik·n,(21)

φα(n)−→

N→∞ ZZ π

−π

ddk

(2π)dφα(k)eik·n.(22)

We insist that for 3DKR and RS models, direct space

is the momentum space. For instance for the RS model

the basis

|ni

corresponds to plane waves with discrete

momenta

(with

= 1) because of spatial 2

periodicity of kicked Hamiltonians. Consequently, the

reciprocal space corresponds to position space, so that

|ki

corresponds to discrete positions

±π

N,±3π

N···±

(N−1)π

. Spatial discretization comes from the imposed

periodic boundary conditions in the truncated momentum

basis, so that the linear system size in direct space is N.

C. Form factor and level compressibility

Previous studies [

–

] found that CFS dynamics

could be related to the form factor. We will show that it

is the same in critical disordered systems. We recall some

deﬁnitions that will be useful in forthcoming calculations.

1. Form factor

The form factor is the Fourier transform of the two-

point energy correlator; it is usually deﬁned as

KN(t) = 1

NdhX

α,β

e−iωαβ ti,(23)

with ωα,β =ωβ−ωα. It can be rewritten as

KN(t) = ZdE ρ(E)KN(t;E)(24)

with

KN(t;E) = 1

Ndρ(E)*X

αβ

e−iωαβ tδE−ωα+ωβ

2+.

(25)

The component

(

;

)of the form factor can be in-

terpreted as coming from contributions of all interfering

pairs of states whose average energy is

. In order to

lighten forthcoming calculations, we introduce the follow-

ing implicit notation

α,β

. . .iE≡*1

ρ(E)X

α,β

δE−ωα+ωβ

2. . .+,(26)

hf(ωα)iE≡*1

ρ(E)X

δ(E−ωα)f(ωα)+,(27)

so that KN(t;E)writes

KN(t;E) = 1

NdhX

α,β

e−iωαβ tiE.(28)

2. Compressibility and link to multifractal dimensions

The level compressibility χis deﬁned as

χ= lim

t/N d→0KN(t;E).(29)

It is a measure of long-range correlations in the spectrum.

It estimates how much the variance of the number of

states in a given energy window scales with the size of

the window. For usual random matrices (GOE, GUE. . . )

χ= 0, while for Poisson statistics χ= 1.

For critical systems that have intermediate statistics,

the level compressibility lies in between 0

<χ<

It was proposed that

could actually be related to mul-

tifractal dimension

via

= 1

−D2/2d

[

], but

it was later observed that this relation fails in the weak

multifractal regime. Another relation was then conjec-

tured [

], relating

to the information dimension

χ= 1 −D1

d,(30)

and has since been veriﬁed in many diﬀerent systems [

72–74] (see also Appendix B).

The information dimension

appearing in Eq.

(30)

is deﬁned trough the asymptotic expansion of Eq.

(1)

the limit q→1

hPn,α δ(E−ωα)|φα(n)|2ln |φα(n)|2i

hPαδ(E−ωα)i∼D1ln N, (31)

and can be seen as the Shannon entropy of eigenstates

|φα(n)|2.

D. Energy decomposition and contrast deﬁnition

CFS is an interference eﬀect that appears when the

system is initially prepared in a state localized in recip-

rocal space,

|ψ(t= 0)i

Nd/2|k0i

(our Fourier trans-

form and normalization conventions are listed in Ap-

pendix C). The observable of interest is the disorder

averaged scattering probability in direction

, deﬁned

(

k, t

) =

Ndh|hk|ˆ

U(t)|k0i|2i

. Using

(17)

, it can be

expanded over eigenstates as

n(k, t) = 1

NdhX

α,β

e−iωαβ tφα(k)φ?

α(k0)φβ(k0)φ?

β(k)i.

(32)

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CoherentforwardscatteringasarobustprobeofmultifractalityincriticaldisorderedmediaMaximeMartinez,1GabrielLemarié,1,2,3BertrandGeorgeot,1ChristianMiniatura,2,3,4,5,6andOlivierGiraud71LaboratoiredePhysiqueThéorique,UniversitédeToulouse,CNRS,UPS,France2MajuLab,CNRS-UCA-SU-NUS-NTUInternationalJointResear...

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Coherent forward scattering as a robust probe of multifractality in critical disordered media Maxime Martinez1Gabriel Lemarié123Bertrand Georgeot1Christian Miniatura23456and Olivier Giraud7

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