A smooth transition towards a Tracy-Widom distribution for the largest eigenvalue of interacting k-body fermionic Embedded Gaussian Ensembles

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Citation: Carro E.; Benet L.; Pérez

Castillo I. A smooth transition

towards a Tracy-Widom distribution

for the largest eigenvalue of

interacting k-body fermionic

Embedded Gaussian Ensembles.

Journal Not Speciﬁed 2022,1, 0.

https://doi.org/

Received:

Accepted:

Published:

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iations.

Submitted to Journal Not Speciﬁed

for possible open access publication

under the terms and conditions

of the Creative Commons Attri-

bution (CC BY) license (https://

creativecommons.org/licenses/by/

4.0/).

Article

A smooth transition towards a Tracy-Widom distribution for the

largest eigenvalue of interacting k-body fermionic Embedded

Gaussian Ensembles

Ernesto Carro 1,∗, Luis Benet 1and Isaac Pérez Castillo 2

Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Av. Universidad s/n, Col. Chamilpa, CP

62210 Cuernavaca, Mor., Mexico; jecarro@icf.unam.mx; benet@icf.unam.mx

2Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, Ciudad de

México 09340, Mexico; iperez@izt.uam.mx

*Correspondence: jecarro@icf.unam.mx

Abstract:

In spite of its simplicity, the central limit theorem captures one of the most outstanding

phenomena in mathematical physics, that of universality. It states that, while navigating the set of all

possible distributions by their convolution there exists a ﬁxed point, the family of Gaussian distributions,

with a fairly large basin of attraction comprising a wide family of weakly correlated distributions. More

colloquially, this means roughly speaking that the sum of independent and identically distributed random

variables follows a Gaussian distribution irrespective of the details of the original distributions from

which these random variables are drawn. While this classical result is well understood, and is the reason

behind many physical phenomena, it is still not very clear what happens to this universal behaviour

when the random variables become correlated. Not a general overarching theorem exists on a possible

emergence of new universal behaviour and, so far as we are aware of, this area of research must be

explored in a case-by-case basis. A fruitful mathematically laboratory to investigate the rising of new

universal properties is offered by the set of eigenvalues of random matrices. In this regard a lot of work

has been done using the standard random matrix ensembles and focusing on the distribution of extreme

eigenvalues. In this case, the distribution of the largest —or smallest— eigenvalue departs from the

Fisher-Tippett-Gnedenko theorem yielding the celebrated Tracy-Widom distribution. One may wonder,

yet again, how robust is this new universal behaviour captured by the Tracy-Widom distribution when

the correlation among eigenvalues changes. Few answers have been provided to this poignant question

and our intention in the present work is to contribute to this interesting unexplored territory. Thus, we

study numerically the probability distribution for the normalized largest eigenvalue of the interacting

-body fermionic orthogonal and unitary Embedded Gaussian Ensembles in the diluted limit. We ﬁnd

a smooth transition from a slightly asymmetric Gaussian-like distribution, for small

k/m

, to the Tracy-

Widom distribution as

k/m→

1, where

is the rank of the interaction and

is the number of fermions.

Correlations at the edge of the spectrum are stronger for small values of

k/m

, and are independent of

the number of particles considered. Our results indicate that subtle correlations towards the edge of the

spectrum distinguish the statistical properties of the spectrum of interacting many-body systems in the

dilute limit, from those expected for the standard random matrix ensembles.

Keywords: Extreme value theorem; Tracy-Widom distribution; Embedded random matrix ensembles

1. Introduction

An important goal of basic sciences is to boldly search for universal behaviour, a common

denominator that underlies the description of phenomena based on simple ingredients, either

Version October 13, 2022 submitted to Journal Not Speciﬁed https://www.mdpi.com/journal/notspeciﬁed

arXiv:2210.05730v1 [cond-mat.dis-nn] 11 Oct 2022

Version October 13, 2022 submitted to Journal Not Speciﬁed 2 of 11

in the physical or mathematical reality. When this occurs they tend to be a paradigm-shifting

moment: Newton’s, Maxwell’s, Einstein’s, Boltzmann’s seminal works in physics come running

to mind as well as those of Descartes, Fermat, Klein, Hamilton, Riemann, Langlands, and many

other seminal works in the realm of mathematics. Focusing on the physical reality, whatever

that means, we all would agree that many, if not all, physical systems can be modelled as a set

of interacting random variables. This is apparent when describing the macroscopic behaviour

of systems composed of a large number of constituents. Here the main goal of statistical

mechanics is, by relating the macrostate of a system with all the microstates compatible to it, to

explain all phases of matter and transitions between them. While descriptions of pure phases

rely on the central limit theorem, close to a phase transition, when the microscopic constituents

of the system become more and more correlated, the law of large numbers miserably fails and

other techniques, such a renormalization group, must be used in its stead. Obviously, studying

one physical system after another, identifying their basic constituents and their interactions,

and seeking suitable mathematical and physical techniques to analyse them can be a tall order

and, at times, quite frankly, tiring. In the last couple of decades, a more mundane and basic

approach has been used: look for those mathematical models in which correlations of random

variables can be easily modelled and manipulated, and study in these systems the emergence

of new universal laws. Due to this, Random Matrix Theory (RMT) has been at the forefront

in the study of emergent behaviour of correlated random variables. Originally introduced to

deal with the complexities of heavy nuclei Hamiltonian systems [

] —and historically also to

deal with noisy linear systems of equations [

]— RMT has grown to be an extremely successful

theory with a surprisingly wide range of applications [

–

]. Its three main symmetry classes,

the so-called canonical ensembles, were originally unveiled in [

] and several others [

]

—like the Wishart [

], circular, or non-Hermitian ensembles— took more relevance over the

years.

It was ﬁrst Tracy and Widom [

–

] who dared to look at how the probability distribution

of the largest eigenvalue typically behaves, and whether its distribution departed from the

one described by the extreme value theorem of Fisher-Tippett-Gnedenko for independent and

identically distributed random variables [

]. It was noticed that a new emergent distribu-

tion appears, the now celebrated Tracy-Widom distribution. After this, a ﬂurry of research

followed suit, originated by the seminal work of Dean and Majumdar [

], focusing primarily

in understanding the large deviation properties of extreme eigenvalues in standard ensembles

of random matrices. By exploiting Dyson’s log-gas analogy and, shrewdly using saddle-point

techniques, they managed to obtain the left and right rate functions of extreme eigenvalues.

Importantly, they noticed that the deviations to the left of, say, the largest eigenvalues are

markedly different to the ones on its right, scaling differently with the system size. More

research was done along these lines for other standard random matrix ensembles [

–

], in

diluted ensembles of random matrices [

–

], and generalizations based on Dyson’s log-gas

analogy [

–

], among many others, to ascertain the robustness of this new emergent law on

the statistics of extreme values.

While RMT has been a fruitful mathematical laboratory in this particular endeavour, it

was also recognized that its ensembles are somewhat unrealistic in the sense that they assume

interactions to involve all Hilbert-space states, whereas typical forces in nature involve two-,

three-, or a few-body interactions. This criticism led to the introduction of the two-body

ensembles [

–

], which eventually was generalized to the

-body embedded ensembles

by French and Mon [

]. Many results are known for the

-body embedded ensembles of

Gaussian random matrices, which mostly focus on the properties of the mean-level density [

–

]. Regarding the ﬂuctuation properties of the spectrum, while it has not been proven that

they are of RMT type for the fermionic two-body embedded ensembles in the limit of large

matrices, there is vast numerical evidence that points on that direction, at least for the central

Version October 13, 2022 submitted to Journal Not Speciﬁed 3 of 11

part of the spectrum; see e.g. [

–

]. As far as we are aware of, much less is known about

the behavior at the edge of the spectrum for these ensembles. The main goal of the present

paper is to address the ﬂuctuation properties of the largest eigenvalue of the fermionic

-body

embedded ensembles of Gaussian random matrices in terms of its parameters. As, in contrast

to the standard matrix ensembles, the number of mathematical techniques to tackle statistical

problems in embedded ensembles is rather limited, our analysis is solely based in numerical

methods.

2. Matrix model and deﬁnitions

Recall that the

-body fermionic Embedded Gaussian Ensemble (fEGE) is the family of

matrix representations of quantum Hamiltonian systems consisting of

interacting spinless

fermions, which can occupy any of

possible degenerate single-particle states. The interaction

among them is taken to be a

-body operator. More concretely, following [

], we introduce

the operator

Ψ†

k;ρ≡Ψ†

j1...jk=∏k

s=1a†

that creates

k≤m

particles. Here,

a†

is the fermionic

creation operator of the single-particle state

, while

is the set of indices

(j1

. . .

jk)

, with

the ordering 1

≤j1<· · · <jk≤`

. In the number operator representation, the

-body

interaction is then given by

V(β)

k=∑

ρ,σ

v(β)

k;ρ,σΨ†

k;ρΨk;σ, (1)

where the coefﬁcients

v(β)

k;ρ,σ

, while obeying that

v(β)

k;ρ,σ= [v(β)

k;σ,ρ]?

, are independently distributed

Gaussian random variables with zero mean and a constant variance, which henceforth is

ﬁxed to one. Finally, depending on Dyson’s

parameter, the set of coefﬁcients

v(β)

k;ρ,σ

is either

real (for

β=

1) or complex (for

β=

2). We refer to

in Eq. (1) as the rank of the interac-

tion. The

-particle Hilbert space is spanned by a basis created by ﬁlling up

states out

, that is

|µi=Ψ†

m;µ|

, which readily implies that this space has dimension

N=(`

Using this basis, the matrix representation of the

-body interaction has entries given by

hµ|V(β)

k|νi=h

|Ψm;µV(β)

kΨ†

m;ν|

. The particular case of

k=m

coincides with the classical

Gaussian ensembles of RMT, while for

m>k

the matrix elements

hµ|V(β)

k|νi

display correla-

tions or may even be identically zero [45].

The limit

N→∞

for the fEGEs can be attained either as

`→∞

for ﬁxed

, or by ﬁxing

the ﬁlling factor

m/`

and taking the limits

`→∞

. These limits have different properties in

terms of the moments of the density of states [

]. In the present study, we shall focus on

the limit where

remains constant, which is more advantageous from the numerical point of

view. This is the so-called dilute limit.

To characterize the probability distribution of the largest eigenvalue for the fEGEs, that

we denote as

λN

, we assume that its mean value can be written as

hλNi=p2βNα

, while its

standard deviation behaves as

σλN∼Nγ

. Here

h(· · · )i

corresponds the the ensemble average

in the embedded ensemble. The normalized largest eigenvalue, denoted as

λN

, is then given by

λN=λN−p2βNα

Nγ. (2)

As it is well known, for the standard Gaussian ensembles, which correspond in our case

k=m

the celebrated Tracy-Widom distribution corresponds to having scaling exponents

α=

2 and

γ=−1/6 [14–16].

3. Results

For the fEGEs the scaling exponents remain unknown though. As an exact mathematical

analysis is rather difﬁcult, we henceforth rely on numerical methods in order to infer

and

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Citation:CarroE.;BenetL.;PérezCastilloI.AsmoothtransitiontowardsaTracy-Widomdistributionforthelargesteigenvalueofinteractingk-bodyfermionicEmbeddedGaussianEnsembles.JournalNotSpecied2022,1,0.https://doi.org/Received:Accepted:Published:Publisher'sNote:MDPIstaysneutralwithregardtojurisdictionalclaims...

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A smooth transition towards a Tracy-Widom distribution for the largest eigenvalue of interacting k-body fermionic Embedded Gaussian Ensembles

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