Level Spacing Statistics of Bidimensional Fermi Liquids. II. Landau Fixed Point and Quantum Chaos

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Level Spacing Statistics of Bidimensional Fermi Liquids.

II. Landau Fixed Point and Quantum Chaos

R. Mélin

To cite this version:

R. Mélin. Level Spacing Statistics of Bidimensional Fermi Liquids. II. Landau Fixed Point and Quantum Chaos. Journal de Physique I, EDP Sciences, 1995, 5 (7), pp.787-804. �10.1051/jp1:1995169�.

�jpa-00247103�

Classification Physics Abstracts 05.20-y 05.30Fk

Level Spacing Statistics of Bidimensional Fermi Liquids.

Il. Landau Fixed Point and Quantum Chaos

R. Mélin

CRTBT-CNRS, BP 166X, 38042 Grenoble Cedex 09, France

(Received

10 November 1994, revised 27 February 1995, accepted 29 March

1995)

Abstract. We investigate the _presence of quantum ^chaos in the spectrum of trie bidimen- sional Fermi hquid by _means of analytical and numencal methods. This model is integrable _{m a}

certain lirait by bosomzation of the Fermi surface. We study the effect _on trie level statistics of

the momentum cut-off A present in the bidimensional bosonization procedure. We first analyse

the level spacing statistics in trie.A-restricted Hilbert space in _one dimension. With _g2 and g4 interactions, the level statistics _are found to be Poissonian at low energies, and G-O-E- at

higher energies, for _a given cut-off A. In order to study this cross-over, a finite temperature _is introduced _{as a way} of focussing, for _a large _inverse temperature fl, _on the low energy many-

body states, ^and driving the statistics from G-O-E- to Poissonian. As for _as two dimensions

are

concerned, _we diagonalize trie Fermi liquid Hamiltonian with _a small number of orbitals. Trie level spacing statistics _are found to be Poissonian

m the A-restricted Hilbert space, provided trie diagonal elements _are of trie _same order of magnitude _as the off-diagonal matrix elements of trie

Hamiltonian.

1. Introduction

The ideas of quantum chaos bave

recently

been

applied

to trie field of

strongly

correlated electron

theory (1-3].

These methods allow for _a

non-perturbative description

of trie statistical

properties

of _a Hamiltonian of

strongly

correlated

electrons,

and _may be _a tool to extract _some information from finite _size systems. ^Trie atm of this article _is to understand whether the methods of quantum chaos _can shed _{some new}

light

_on trie

problems

of

strongly

correlated

bidimensional Fermi systems. Trie

question

whether trie bidimensional Hubbard model is _a Fermi

liquid

_or net is controversial. From _a theoretical point ^of view, Anderson _(4] suggests that the

ground

state of the bidimensional Hubbard model is similar to that of the one-dimensional

Luttinger liquid,

with Z

= o in the

thermodynamic limit,

whereas

Engelbrecht

and llmnderia _[Si argue that the

Fermi-liquid theory

is not

violated,

_so that this

question

is controversial [6].

Numerical

computations

also lead to controversial _answers. For instance,

Dagotto

et ai.

ii,

_8]

observe

quasiparticles

in trie bidimensional t J

model,

whereas Sorella _[9]

emphasizes

trie

Luttinger liquid

behaviour. From trie point of view of quantum chaos, _one bas _{to answer} trie question: does level

spacing

statistics contain information about trie Fermi

liquid

behaviour _or

© Les Editions de Physique 1995

IOURNALDEPHYSIOUEL-T.s.NO7.JIJLY iras

non Fermi

liquid

behaviour of trie _t J model? Before

analysing

trie level

spacing

statistics of trie t J model

(loi,

_we

analyze

in finis paper models with _a well-established

physical

content,

that

is,

trie Fermi

liquid.

As _we shall _{see, we can answer}

only partially

trie

previous

question:

one can detect

integrable

modes at low _energy, but quantum chaos

by

itself does _not

give

information _as to whether these modes _are

quasiparticle

modes _or

Luttinger liquid

modes.

However,

there is _{one case} in which _{one can} condude from quantum chaos: in trie absence of

integrable degrees

^{of freedom} at low energy, one can conclude _to trie absence of _a Fermi

liquid

at low _energy. Notice here trie dilference between trie

approach

of reference

iii

and _Dur

point

of view. In reference

[Ii,

ail trie _energy levels of trie t J model _are

analyzed

_{on an}

equal footing,

whereas _we shall focus

essentially

on trie low _energy

degrees

of freedom of trie Fermi

liquid.

In trie first article of this series

[iii,

_we bave established that trie level

spacing

statistics of trie bidimensional Landau Hamiltonian is Poissonian. In finis

model,

trie

quasipartides

_are in

a one-to-one

correspondence

with trie

non-interacting

_gas of

spinless

electron

excitations,

and thus occupy orbitals labelled

by

trie _same quantum numbers k _as in trie _case of trie _gas. Trie

quasiparticles

interact among themselves in _a

diagonal

_manner

~~~~~~

_~Î =

~

_{£kôYlk +}

~

k

LD

Îkk'ôYlkôn~,

~~'~

l

A numerical

computation proved

that trie level

spacing

statistics of trie Hamiltonian

(1)

is

Poissonian in two

dimensions,

and close to Poisson in _one

dimension,

with _our truncation of trie Hilbert _space. Trie notion of level _spacing statistics _seems to be relevant for quantum

fluids,

and _one bas _to

distinguish

between _one and two dimensions. Trie link between trie

breakdown of trie Fermi

liquid picture

in _one dimension and trie level

spacing

statistics bas

already

been studied (13]. ^{As for} as two dimensions _are

concerned,

_we showed in paper I

il Ii,

that trie Landau

liquid

_was characterized

by

its

generical integrability, namely by

Poisson level

spacing

statistics in two dimensions.

Again,

trie level

spacing

statistics _{are a}

good

trot to _see whether trie Fermi system ^{is at trie} Landau fixed point or net. For trie

liquid

to be

a Fermi

liquid,

_one should be able to generate

quasiparticles

of trie

interacting

system

by

a

switching

_on

procedure.

In trie framework of trie Landau

theory,

trie _success of trie

switching-

on

procedure (14, lsj

suggests ^the conservation of the number of conserved

quantities

at Iow energy

during

the

switching-on procedure, namely

that the level spacing statistics of the _gas and of the

interacting liquid belong

to the _same

universality

class.

Typically, adding

_{a non-}

diagonal perturbation

to equation

(1)

and

obtaining

a Gaussian

Orthogonal

Ensemble

(G.O.E.)

level

spacing

statistics would _{mean a}

departure

from the Landau fixed point. To be at the Landau fixed

point,

_one must exclude strong correlations between levels. The

corresponding

generic level

spacing

statistics is Poissonian.

However,

the _case of the one-dimensional Landau Hamiltonian shows that the statistics may ^not be

exactly Poissonian,

but close _{to a} Poisson law.

Poisson _or close to Poisson level

spacing

statistics _are not a sullicient condition for the system

to exhibit

quasipartides

_{since one} could

imagine

_a situation in which trie _same mechanism

as in _one dimension for trie breakdown of trie Fermi

liquid froids,

_so that trie Ievel

spacing

statistics would remain Poissonian _even in trie non-Fermi

liquid

_case.

However,

trie adiabatic

procedure,

_as

emphasized by

Anderson [14], is

performed

within _a finite characteristic time

Ile,

and trie

thermodynamic

limit is taken for _a finite _e which is then much

larger

than trie

typical

level

spacing.

Thus trie adiabatic continuation does not generate ^true

eigenstates (for

this to be trie _{case, one} should take trie limit _{e -} o

first,

before trie

thermodynamic limit).

This is

why

trie notion of adiabatic continuation advocated to introduce Fermi

liquid

is weaker than _a similar requirement for ail trie

eigenstates

taken

separately. Therefore,

this is

why

_we

should ask whether _a Fermi

liquid

follows Poisson or G-O-E- statistics.

The atm of the present _paper is to

study

the level

spacing

^statistics of the Hamiltonian of

spinless

electrons

H =

H°+H~ (2)

H°

=

£e(k)c(ck ₍₃₎

k

~fl

~

j~ _f~~,

~+ ~+ _~~,~~

j~)

~v ^{>q k+q k'-q}

k,k',q

where V is the volume of the system. ^The Harniltonian

(1)

describes _a Fermi

liquid

for times smaller than the

decay

time of the

quasiparticles,

and leads to

good thermodynamical predictions

_[15].

However,

trie

phenomenon

of

decay

of trie

quasipartides

is not described

by

the Landau form

Il).

Since

quasipartides

_are true

eigenmodes

of

Il), they

have _an infinite tife-time.

By

contrast,

(2)

^takes into account the

decay

of trie

quasiparticles

and it is indeed

possible

to calculate trie

decay

rate of trie

quasipartides

from trie Hamiltonian

(2) _(16].

Even

though

trie Hamiltonian

(2)

is not

diagonal,

it _can be

brought

to a

diagonalform by using

_some

assumptions

and

by bosonizing

trie Fermi surface [18]. ^{We shall review trie main}

assumptions

and trie bosonization of

(2)

in trie

thermodynamic

limit and in trie Iimit of _{a zero curvature} of trie Fermi surface. One of trie

ingredients

of trie solution via bosonization is trie existence of

a momentum A which determmes trie

single partide

states which

participate

in trie formation

of bosons in trie

vicinity

^{of trie} Fermi surface. A is _a necessary

ingredient,

because of trie curvature of trie Fermi surface, but is also _{a source} of

difliculty

since trie

abjects

with _a true bosonic character in trie limit A _{- +co are no}

longer exactly

bosonic in trie limit of _a finite

momentum cut-off A.

Of _course, numerical

computations

cannot be

performed

in trie

thermodynamic

limit since

one can

only diagonalize

matrices of size 2000

by

_a Jacobi method. This technical limitation imposes to work with small systems and to impose a drastic cut-off

A,

_so that trie Hamiltonian

(2)

is _no

longer integrable

in trie framework of trie Hilbert _space of trie numerical

diagonal-

isations. In trie _case of trie _presence of _a

cut-off,

and for finite size systems, _{one may} thus expect ^{trie universal Poisson Ievel}

spacing

statistics to be

replaced by

trie universal G-O-E- level spacing statistics of trie form

P(s)

₌

~s

exp

~~

(5)

2

~

It is necessary ^to determine trie importance of trie cut-off A _on trie level

spacing

statistics and whether trie statistics evolves towards G-O-E- level

spacing

statistics _as A is reduced. This bas net

only

_a

numerical,

but aise _a

physical

interest. For

instance,

lattice models bave _a natural cut-off A

+~

1la,

where _a is trie lattice

spacing.

^To answer this

question,

_{we came} back _to one-dimensional

spinless

fermionic systems, but with _a cut-off A and

study

trie level

spacing

statistics for _gz and _g4 interactions. As _we shall _see, trie level

spacing

statistics is

drastically

affected

by

trie _presence of trie cut-off since trie bosonic

superpositions

of

partide-noie pairs

which guarantee ^trie

integrability,

do not survive in trie presence of trie cut-off. If trie cut-off A is fixed and trie size L of trie system

increases,

_{one expects} for _{a crossover} from _a G-O-E- Ievel

spacing

statistics _m trie

high

_energy part of trie spectrum to _a Poisson level

spacing

statistics at low energy, which shows that finite size effects _are drastic in trie _presence of trie cut-off. In

order to determine trie _crossover scale between trie Poisson

regime

and trie G-O-E-

regime,

_we

introduce

temperature-dependent

level

spacing

statistics. The statistics at finite temperatures is found to pass from _a

non-integrable

statistics at

high

temperatures to Poisson statistics at low temperatures.

As far _as two-dimensional systems _are

concemed,

_{we carry} out numerical

diagonalisations

of _a small system of

electrons,

with _a small number of orbitals. Under these

conditions,

the Hamiltonian

(2)

is _non

integrable.

We found Poisson level

spacing

statistics _even for _a small system. ^{We attribute this} property to the fact that the interactions generate extra

diagonal

matrix elements _m trie Hamiltonian which _are in

competition

with

diagonal

interaction matrix

elements,

but trie interaction connects

only

_a small number of states,

leaving

_a lot of _zero matrix elements in trie Hamiltonian. Provided trie

off-diagonal

matrix elements _are of trie

same order _as trie

diagonal

_ones, trie statistics is dominated

by

trie

diagonal

matrix elements.

However, by keeping only off-diagonal

interaction matrix

elements,

_{we were} able to exhibit G-O-E- level

spacing

statistics.

This _{paper is}

organized

_as ^follows. We first _treat the one-dimensional _case, in the Hilbert space restricted

by

the momentum cut-off A. The results established in _one dimension _are

helpful

in two dimensions since the presence of the momentum cut-off is aise _{a source} of _non

integrability.

In _a second step, we came to the bidimensional Fermi

hquids,

with the

study

of the level spacing statistics of the Hamiltonian

(2).

2. Level

Spacing

Statistics of lD

Spinless

Fermion

Systems

in trie A-Restricted

Hilbert

Space

2.1. BOSONIZATION _AND LEVEL SPACING STATISTICS IN _THE UNRESTRICTED HILBERT

SPACE

(LUTTINGER LiouiD). Using

the bosonization

procedure

of Haldane [22], we cari

salve the one-dimensional

spinless Luttinger liquid

with _a linear

dispersion

relation and _g2 and g4 interactions. The Hamiltonian

reads,

in term of fermions

H

= VF

~lak _kF)

:

cÎ,ack,« +) ~ ~

qlg4qôoe«> +

g~qôa,-«, )pq,ap-q,a,,

₁₆₎

k,a a,a,

where the label _a indexes the branch

a=R(ight),L(eft)=

+1, -1. The interaction _g4 describes the diffusion of two fermions _on the _same

right

or left

branch,

whereas _g2 descnbes the diffusion of two fermions

belonging

to the

right

and left branch. The

density

_{operators are} defined _as

PQ,a "

~ _cÎ+q

_rock,a

.,

Ii)

k

and

obey

bosonic commutation relations

which allows the definition of boson operators

~~ ~/Îl

^~~~~

Î

_{~~~~~~~~~~ ~~~}

and make it

possible

to

diagonalize

the Hamiltoman

(6)

via a

Bogoliubov

transformation:

where

b)

₌ ^cosh

çJqa)

^sinh çJqa-q

(11)

g2q

~~~~

~~~~Î) = ÎÎÎj-19~~~)~[(l~' ~~~

~~ ~ ~j~~ + g~o)

-

g~o)

vN "

vs exp

The interaction functions

g2(q)

and

g4(q)

_are

supposed

to tend to _a constant in the limit _{q -} 0, and to zero in the limit _{q -} +co. Their decrease _is controlled

by

^the

impulsion

scale

27r/R,

where R is _a

given length

scale. In the

thermodynamic

limit

(L

»

R),

_{one can} define _an effective low energy Hilbert _space. To do _{so, we assume} that _{wq m wo} for ail the _wave vectors

q. Of _course, this approximation is

only

valid if the temperature is low

enough.

In this

limit,

the Hamiltonian

(10) depends only

_on two velocities _vN and _vj:

H =

Eo

+

£lvNvJ)~/~lqlblbq

+

)lvNN~

⁺

^{VJJ~) [ii)}

q#o

In _one dimension, the hnk between the breakdown of the Fermi

liquid

picture and the level spacing statistics is well understood. The absence of _a Fermi surface in _one

dimension,

in the presence of

long-range

_g2 and _g4 interactions _was known to

Dzyaloshinskii

and Larkin in the 70's

[17j.

^{The breakdown of the Fermi}

liquid picture

is

already

present at the level of the static

correlation

functions,

and in the Hilbert _space of low energy. The usual infrared

divergencies

are

govemed by

the _{q -} 0 limit of

g2(q)

and

g4(q),

and _are related to the

orthogonality catastrophe

and the absence of _a Fermi surface. The infrared spectrum is described

by

the

Hamiltonian

(Ii).

If the total

charge

and _current quantum ^numbers are

given,

the _q

#

0 excitations _are bosons with _a linear

dispersion relation,

but with _a renormalized

velocity.

The

departure

from the gas behaviour _is measured

by

the anomalous exponents which _{appear in} the static Green functions. The interaction energy scale associated with this static breakdown of the Fermi

liquid

is

given by

_[13]

L -1/2

g]~~~ +~ vF In

(18)

27rR

However,

_as

emphasized

in reference [13], ^{the static} part ^{of the Green function does} not contain ail the

physics

of the breakdown of the Fermi

hquid theory.

From _a calculation of the _two- point ^Green function, and from _a calculation of the

switching-on procedure,

_we found in (13]

a

dynamicai

breakdown of the Fermi

liquid

picture, controlled

by

the interaction scale

~~~~ ~

(kL~ÎÎΰ)1/2'

^~~~~

where _{a is} defined

by

g2q " g2011

lqR)~). (20)

The static process of breakdown of the Fermi

liquid

_picture does _not

modify

the level

spacing

statistics, ^which remam

singular,

_{as m} the _case of the gas, smce the

dispersion

relation of the bosons _remams linear.

By

contrast, ^the

dynamical

breakdown of the Fermi

liquid

_appears to be related to _{a cross-over} in the energy level

spacing

statistics,

namely

tram

singular

level

spacing

statistics to

generical

Poisson level spacing statistics. As far _as symmetries are concemed, the gas ^case and the

Luttinger liquid

_possess the conformai symmetry. The static breakdown of the

Fermi

liquid

_preserves the conformai invariance. The

theory

is the

antipenodic-antiperiodic

sector of the

compactified boson,

with _an

interaction-dependent

radius of

compactification.

The

dynamical

breakdown of the

Luttinger liquid corresponds

to _a massless loss of conformai invariance (12].

2.2. FOURIER TRANSFORM _OF THE CAS SPECTRUM _AND TEMPERATURE-DEPENDENT LEVEL

SPACING STATisTics. We first wish to characterize the spectrum of the gas m the absence of interactions, _a momentum cut-off A and _a finite size L. The idea is to find _a criterion to detect when _a boson of

given

_wave vector q is present _or net in the system. ^Since _g2 ^and _g4 ^interac- tions _are

diagonal

_on the basis of bosonic excitations, this tool is _a

good

_way to characterize the

degree

of

integrability

of trie mortel with _a momentum cut-off. We consider _a one-branch

mortel,

and trie number of quantum states is

simply

trie

integer

part ^of

2AL/27r.

Trie spec-

trum is made _up of levels with _an

equidistant separation

27rvF

IL.

In order _to characterize trie spectrum, we use its Fourier transform

+m

fL(T)

=

/

_e~~~

_{£ à(w E,)}

=

£ e'~~~ (21)

~" lE,j jE~)

In trie

thermodynamic limit,

1

(23)

l~~~~~É ~°

(fm(T)Î

_"

~

2(smvfTq/21

q>o

This function presents

potes

which _are characteristic of trie bosonic modes for Tn,q = 27rn

Iv

_fq,

with _{n an} integer. ^If we rescale _T

by

_a factor 27r and choose vi = 1, _we obtain _a

pote

for _Tn,q

=

n/q,

that is for _every rational number. We

plotted foc(T)(

^{for 5 bosonic modes in}

Figure

1. One _can

easily recognize

^{trie dilferent} bosons in trie _sequence of

potes.

In trie _presence of

interactions,

trie

potes

_are

deplaced.

^{We also studied trie} Fourier transform of trie spectrum in trie _case of _a finite size system. ^{Trie result is}

depicted

in

Figure

2. We _{can see} that trie

potes

are not so well defined. However, we can

recognize

^{trie formation of}

peaks

which

replace

trie

potes,

and attribute well defined boson _wave vectors to _some

peaks.

Trie

corresponding

truncated

density

operators _are ^defined

by

pA(q)

₌

~

_ÙA(k)ÙA

_(k

₊

_{q)c(~~ck, (24)}

k

where ÙA(k) = Ù(A (k

kf().

We _can ask under which condition _a

peak corresponding

to

a truncated boson

pA(q)

_appears on trie modulus of trie Fourier transform of the spectrum

fL (T)

_(. ^To answer this

question,

_{we assume} that trie number nq of bosonic excitations of _wave

vector q

appearing

in the _presence of _a cut-off is such that qnq +~

2A,

_so that

fL(Ti,q)

z~J

2A/q

for trie

peak

at T = ri,q. A boson of _{wave vector q is} well defined

provided fL (ri,q)

_» 1, that is if A _»

q/2.

For _a

given cut-off,

trie statistics is

expected

to be Poissonian at low

energies,

and G.O.E. at

higher

_energies. This _is due to trie fact that truncated bosons _are created

essentially

at low energy. In order to test this

idea,

_we introduce

temperature-dependent

level _spacing

statistics,

which is defined _as follows. If

(E,)

is trie fuit spectrum and

(e,

_are trie _energy levels

io

9

8

7

6

s

4

3

2

1

0

0 1 2 3 4 5 6 7

Fig. 1. Modulus of the Fourier transform of the spectrum m the thermodynamic hmit

(fco(r)(

for 5 basons. Each pale corresponds to _{one or} several definite boson excitations. We have chosen _vi

= 1

and L

= 2x. A given boson _q > 0 generates _{a sequence} of pales at r»,q =

27rn/q.

The pales _are

denoted by their corresponding rational number

n/q.

20

15

io

5

0

0 1 2 3 4 5 6

Fig. 2. Finite size effects of the modulus of the Fourier transform of the spectrum

(fL(r)(.

We

have chosen vi = 1 and L

= 2x, A

= 11. As shown _on the Figure, _{we con} attribute boson quantum numbers to _some peaks. ^We _recognize trie fractions 1/5, 1/4,

1/3,2/5,1/2,3/5,2/3,3/4, 4/5.

after the

smoothing procedure

_[23], the

density

of level _spacing

P(s)

is

-i

Ppis)

=

(~

_exp

^{-,o~~+~~+ ~) L}

_exp

^-à~~+~~+

^~~

ois ie,+i e,)) 125)

~ ,

Moreover, ^for the statistics to be

comparable,

_one needs to scale

Pp (s)

and _s such that

(Pp(s

₌

1, and

(sPp(s))

₌ 1. The _reason

why

_we have to rescale these

quantities

is that

they

_were

equal

to

unity

after the

smoothing procedure,

which _was carried ont in the _zero temperature limit.

This property ^is no

longer

valid with the statistical

weights

of equation

(25).

For _an infinite temperature

(fl

₌

0),

_{we recover} the usual level spacing statistics. As the inverse temperature

fl

increases, the low energy levels _{carry more} and _more statistical

weight.

As _{we saw} from the

description

in terms of the truncated bosons

(24),

the spectrum is

expected

to be

integrable

at low

energies,

so that the level

spacing

statistics should evolve from G-O-E-

statistics,

_or at least intermediate statistics

(that

is, with 0 <

P(0)

<

1),

to Poisson level

spacing

statistics _as

fl

increases.

2.3. LEVEL SPACING STATISTICS _{ÀÀÎITH g2} INTERACTIONS _{IN THE} À-RESTRICTED _HILBERT

SPACE. The numerical method to compute ^{the level}

spacing

statistics of twc-branch models with _a momentum cut-off A and _g2 interactions consists in

tabulating

the states of the Hilbert space

by generating

ail the dilferent

fillings

of the two branches, the number of

partiales

_on each branch

being kept

constant. The second step ^{consists in}

computing

ail the matrix elements of the Hamiltonian and

diagonalising

the Harniltoman

by

the _use of the Jacobi method. The

interactions _are of the form

~~~ ~~° ~~~

lÎÎ~

_~~~~

The scale of the interactions R is choseu

equal

to

A,

_so that ail the matrix elements of the interaction _are important. The evolution of the spectrum _as a function of _{g20 is}

plotted

in

Figure

3. This

figure

is to be

compared

with

Figure

2 of reference

[13j,

^where we have

plotted

the evolution of the energy levels _{as a} function of _g20 but in the absence of cut-off. The obvions dilference between the two

plots

is the _presence of level repulsion with _a cut-off

A,

and the

existence of level

crossings

m the absence of cut-off. The level spacing statistics

corresponding

to a spectrum

analog

to that of

Figure

3 _are

plotted

in

Figure

4.

They

_are in

good

agreement with G-O-E- level

spacing

statistics. In _our

computation, AL/27r

₌ 3, _so that the criterion À _»

27r/L

is not verified. ~Ve could _{not go to}

higher

values of

AL/27r

because

increasing

the ratio

AL/27r

increases the size of the Hilbert _space and numerical

diagonalizations

_are

no

longer possible.

We conclude that the statistics

depends drastically

_on the

length

of the system, ^{if the} momentum cut-off is fixed. If _we

apply

the ideas of

temperature-dependent

statistics of level

spacings

to the system, we find that the statistics of level

spacings

is driven from _a G-O-E- law _at fl

= 0 to _a Poisson law _as

fl

increases, _as depicted in

Figure

5. This evolution shows the fact that

"nearly integrable" degrees

of freedom exist at low _energy. These bosonic

degrees

of freedom _are

exactly integrable

_m the absence of cut-off. In term of classical

trajectones (provided

_{one is} able to find _a classical

phase

_space for the Fermi

liquid !),

this situation

corresponds

to the existence of conserved tori at low _energy, which transform into chaotic

trajectones

_as the _{energy increases.} The _cross-over temperature scale will be derived later.

25

io

s

~ _jj__j~~'~~-'---_

--.---______

~~ ~h'

àÙÎ~))Tlfifi==.

_tm

_jmmm-=- --.."~=

_ _{__} ---i =fi=+1

--= ==_=-

o ---. ..- -~= zm=- ~~~

~~~

~-

~

~

-5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Fig. 3. Evolution of the energy levels in trie _presence of _g2 interactions with _a momentum cut-off.

Trie parameters ^{of trie model} are ~i = 1 and L

= 2x. Trie cut-off is chosen equal to 3, _so that _we bave 3 fermions _on each branch, for 6 available quantum ^{states. Trie} interactions _are of trie form g2q =

g20exp(-q/R).

with R

= 6. Trie momentum sector is 1, leading to 45 states _m trie Hilbert

space. Because of the partiale-hale symmetry, ^trie spectrum is symmetric, but with _no level crossings.

2.4. LEVEL SPACING STATISTICS WITH g4 INTERACTIONS _{IN THE} À-RESTRICTED HILBERT

SPACE. We _now consider the _case of _g4 interactions. The interaction Hamiltonian is

H~

=

£ £ g4qc)~~c$_~ck,ck (27)

q#o k,k,#k+q

The interest of the g4 ^term is that _{we can use}

only

_a one-branch model, and _{we can} reach

higher

values of the ratio

AL/27r

without

increasing

the _size of the Hilbert space. We could

reach

AL/27r

= 12 in _a sector of total momentum P ₌

24.27r/L.

P is the

impulsion

with

respect to the fundamental. As

long

_as P <

AL/7r,

_one has

generated

the

complete

Hilbert space in the absence of interactions. This fact motivates the choice of P

=

AL/7r.

The condition

AL/7r

= 24 _» 1 is

respected. However,

_we did not find Poisson statistics, but

ours is intermediate between _a Poisson law and G-O-E- statistics.

Namely,

the value of the

density

of normahzed _zero

crossings

_is not 1 but o-à- The statistics is

plotted

in

Figure

6. This

computation gives

_an idea of the extension of the _{crossover as a} function of AL

/27r,

^since we

obtain intermediate statistics for

ÀL/27r

₌ 24. We also

computed

the

temperature-dependent

level spacing statistics. We found _{a cross-over to} Poisson statistics _as the inverse temperature

mcreases, as

plotted

in

Figure

7. This _{cross-over is} much _more

rapid

than in the _case of

the g2 interaction.

However,

the values of the cut-off in the g2 case and the g4 case are not

comparable.

2.5. CRoss-OVER SCALES. In order to get an idea of the temperature cross-over between the G.O.E. and Poisson regime, we look for the _energy scale

kBT*

below which the _energy levels

1

Poisson G-O-E-

0. 8

0 6 ",

',, ^'

d

',,

',

~

',,

~.

0 4 ~~~~~~~~ ^"

0 2

"""""

0

0 0.5 1 1,5 2 2.5 3 3.5 4

normalized level spacing _s

Fig. 4. Level spacing statistics of ID spinless fermions with g2 interactions and _a cut-off. The parameters _are ~i = 1, L ₌ 17r. In order to eliminate the particle-hale symmetry present in Figure 3,

we chose _a non-symmetric cut-off for particles and hales: the number of fermions _on the right branch

is 4, and the number of quantum states is 9. On the left branch, the number of fermions _is 5 for 9 quantum states. The Hilbert _space contains 1052 states in the sector of momentum 1. The level

spacing statistics is found to be well fitted by G-O-E- statistics.

are uncorrelated. To do _so, we make the

approximation

^{that trie}

only

effect of interactions is to compress trie spectrum

by

_an amount

((1+ g4/ui)~ (g2/vf)~))~/~,

where _g2 and g4 are

typical

energy interactions. This

compression

of trie spectrum means that trie effective _mass increases if _g2 is

large compared

to g4. This approximation

corresponds

to

taking

local interactions _in

trie real _space. Then _{we see} that trie

required integrable

modes _are such that

vi (q( ^<

1

+ ~~ ^~ ~~

)

^~~~

kBT.

(28)

Vi Vi

These modes _are

integrable provided

that

they

lead to well-defined bosons. We say that _a boson is well-defined

provided

that _a sufficient number of

partiele-noie

excitations enter trie summation

(24),

that is, ^{if 2À} _(q( >

27ra/L,

where _a is _a dimensionless

coefficient,

which

counts trie number of _non~zero terms in

(24),

and is thus _{a measure} of

integrability.

We get

kBT

t 12À

27ra/L) (il

₊

g4/vf)~ ig2/vf)~)~~~ 129)

The energy scale kBT* thus increases _as trie cut-off A increases, and decreases if trie interaction

strength

_g2 ^increases.

1

Poisson

0.8 ^~'~'~'

Î ,'

Î

--~

Î

j ~

, ,

0.6 '

: ,

',

: .,.

"

w '-.

£ ^". ,

", ,_,

., ,

0,4 ',

', N

j .,

, ',

' ...

1

, _.,

1 '

0

level s

Fig. 5. -

emperature. Trie are vi = 1, L = 27r. In

order

to _eliminate

trie symmetryresent in Figure ^3, we _{chose a non} ut-off for particles and noies: trie number of fermions on ^trie right _{ranch is} 4,

and the number of uantum states is 9. On the _left branch,

trie number

is 5 for 9 _quantumstates. Trie ilbert space contains

1052

tates in the sector of omentum 1. _Trie

level

tatistics is plotted for inverse temperature equal to fl = 0, 0.1, 0.5 A cross-over is found

from _G-O-E- statistics _for fl = ₀ to _Poisson tatisticsas fl

3. Level

Spacing

Statistics of _a 2D

Spinless

Fermion

System

in trie A-Restricted Hilbert

Space

3.1. BOSONIZATION OF THE BIDIMENSIONAL FERMI SURFACE. TÎ1e bosonization of tÎ1e

Fermi

hquid (19-2 Ii

^involves a

covering

^of trie Fermi surface

by spheres

of radius A. Each small

sphere

is labelled

by

_an

integer

_a and _{one assumes} that trie Fermi surface _is flat at trie scale A.

Assuming

that A _» 27r

IL,

trie commutation relations of trie operators

pq,a =

£ _9«lk

₊

_{q/2)9alk q/2)nqlk), 130)}

Wiill

'lq(~) C~-q/2~k+q/2>

^(~~~

must indude _a

Schwinger

term,

leading

to trie commutation relation

lpq,~, p(,~>l

=

ôa,~,ôq,q, £ _{c~jk, q)jnl+}

j

nl- j), 132)

k

where trie constraint

C~

_is written _as

CoElk>~)

_" ÙoElk +

))ÙoElk ))ÙoElk'+ ( _)ÙOElk'~ ( _).

₁₃₃₎

1

Poisson

", G.O.E.

0 8 1,

~, "..

', ." ".

',

," ",

~,, ^{." "}

0 6 ,~'

"_

Îi _".

iÎ _'.

0 4 ,/

~~

~.

0 2

"'

0

0 0.5 1 1-S 2 2.5 3 3.5 ₄

normalized _{spacing s}

Fig. 6. Level _spacing statistics of ID spinless fermions with g4 interactions and with _a momentum cut-off. The parameters are chosen such that _~F

= and L

= 2x. The value of the momentum cut-off A is 12 and the total momentum is 24. The Hilbert _space contains 1185 states. The statistics

is found to be intermediate between

a G-O-E- statistics and

a Poisson statistics.

To

simplify

_expression

(32),

_one makes the

assumption

of _a flat Fermi surface _in each

sphere

of radius A. In order to

specify

this

condition,

_one

imposes

that _no bosonic excitations with

an

angle

>

7r/2

exist, where denotes the

angle

between kf and q. The maximal

angle

is such that

tan

Ù 1)

§

), ₍₃₄₎

which leads to

A 1

l~~~

^{)~/~ 135)}

Equation (35)

_means that the curvature _is

negligible

in _a

sphere

of radius

A,

and is

compatible

with the condition

A » 27r

IL (36)

provided

^the hnear

length

is

large enough:

L »

7r/kf.

If A _< 27r

IL

and (q( «

A,

^the

leading

term _m

(32)

is of the form

[pq,~,p(,

~,] =

ô~,~,ôq,q,Va(q.n«).

We evaluate the corrections under the conditions A »

27r/L anl

(q( < A. Under these assumptions, the

leading

term

represents ^the number of states m the

parallelogram

^of

Figure

8. To obtain the corrections,

one has to subtract the number of states contained in the small shaded

triangle.

The number

of _states to be removed is

approximately equal

to

ÎÎ~

_{~~~~ ~°~} ~ _~~~ ~

~~~'~~~'~~~kÎ~

~~~~

1. i

1

Poisson

0.9

0.8

o.7

0.6 f

~ 0.5

0 4 "._

0 3 ,,___,~~_{".._}

0 2

"",,

o.1

0

0 0.5 1 1.5 2 2.5 3 3.5 4

normalized spacing _s

Fig. 7. Level spacing statistics of ID spinless fermions with _g4 interactions and with _a momentum cut-off. The parameters _are chosen such that _~F

= 1 and L

= 2x. The value of the _momentum cut-off A is 12 and the total momentum _is 24. The Hilbert _space contains 1185 states. The statistics is plotted

for the _inverse temperature equal to fl

= 0 and fl

= 0.3. The statistics for fl

= 0.3 is close to the Poisson statistics.

The commutation relations _are thus of trie form

jpq,~, pj,,~,j

=

ô~,a>ôq,q,Vajq.na)jl

+

Oj'~ ~~')j. _j38)

Followmg

reference

(21j,

we define

aqlkf)

"

~ _4Allk kil)Inqlk)Ùlq.Vk)

₊

n-qlk)Ùl-q.Vk)1>

₁₃₉₎

and

bq(kf)

=

(NAV(q.vkl)~~~~aq(kf), (40)

where _{ç§A is a}

smearing

^function such that #A _- ôk,k~ ^{if A} - 0, and NA is trie local

density

of states:

NA _"

) _{L i4Aiik kfi)~ôilL £k).}

141)

Provided trie curvature of trie Fermi surface is

negligible,

that is

provided

condition

(35)

is

satisfied,

_{one can} bosonize trie Harniltonian

(2)

to obtain

H =

~j £ (q.vk16( (kf)bq(kf)

⁺

£ _fk

k,

qn-q(k)nq(k'). (42)

~ ~

2V

~ ~,

' '

f q,q.vk>

, ,q

sphere

_n

sphere

n+I

S

to be removed Fernù

surface

Fig. 8. Representation of _a sphere ai trie Fermi surface. Trie commutation relation

(32)

is pro-

portional

to the number of states contained in trie intersection of the

parallelogram

and trie sphere _n.

Trie leading order term takes into account ail the states in the parallelogram.

Since trie interactions _are

quadratic

in trie boson operators, one con

diagonalize

them via a

generalized Bogoliubov

transformation. However, ⁱⁿ numerical

computations,

one con

only

treat _a Hilbert _space of size 2000. This _means that trie cut-off A _must be reduced

drastically,

_as

well _as trie Fermi _wave vector

kf.

In

particular,

conditions

(35)

and

(36)

_{are no}

longer

valid and

one cannot

diagonalize

trie Hamiltonian _as described above. Two scenarios _are candidates for trie appearance of chaos m trie spectrum ^{of trie} bidimensional Fermi

liquid.

Trie first _scenario is trie effect of curvature. In trie limit L _- +co, and for _a cut-off

independent

of

L,

condition

(35)

is _no

longer valid,

and trie model _cannot be solved

by

bosonization. We shall not

study

this effect in trie present _paper. Trie second scenario comes from trie fact

that,

_even with _a flat Fermi

surface,

condition

(36)

_may be

violated,

_so that trie system _{is no}

longer integrable by

bosonization. We shall

study

trie latter type ^{of elfect} in trie rest of trie _paper.

3.2. LEVEL SPACING STATISTICS OF A 2D SPINLESS FERMION SYSTEM _{IN THE} À-RESTRIC-

TED HILBERT SPACE. We _{now tum} to trie bidimensional _case in trie presence of _a momentum cut-off. Trie Fermi _sea in its fundamental state _is

pictured

in trie inset of

Figure

9. We treated

a Fermi _sea of 5 electrons for _a total of 29 available quantum states. Trie Hamiltonian is

given by expression (2).

Trie interaction _{term can} be

split

into two terms as follows

H~

=

HI

₊

H) ₍₄₃₎

HI

=

) _~j fk,k+q,qnk+qnk

₁₄₄₎

k,q

HI

=

~~j _~j

fk,k,,qc(~~c(,_~ck,ck ⁽⁴⁵⁾

~

q#o k,k,#k+q

The term

HI

is of trie _sonne nature _as trie

diagonal

Landau interaction between

quasiparticles.

Trie

only

^dilference is that

ônk

_represents

occupation

numbers of renormalized

quasipartides

in trie Landau

theory,

whereas _nk is trie number operator of bare fermions. Trie Hamiltonian made up of trie kinetic term H°

(3) plus

^trie term

H( (44)

bas

already

been studied in reference

iii]