Glauber Dynamics in a Zero Magnetic Field and Eigenvalue Spacing Statistics

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Glauber Dynamics in a Zero Magnetic Field and Eigenvalue Spacing Statistics

R. Mélin

To cite this version:

R. Mélin. Glauber Dynamics in a Zero Magnetic Field and Eigenvalue Spacing Statistics. Journal de

Physique I, EDP Sciences, 1996, 6 (4), pp.469-492. �10.1051/jp1:1996225�. �jpa-00247198�

Glauber Dynamics in

_a

Zero Magnetic Field and Eigenvalue Spacing Statistics

R. Mélin

(*)

CRTBT-CNRS,

BP 166X, 38042 Grenoble

Cedex,

France

(Received

21 J1Jne 1995, revised 13

November1995, accepted

4

January1996)

PACS.75.10.Hk Classical

Spin

Models

PACS.75.10.Nr

Spin-glass

and other random models

Abstract. We disc1Jss the

eigenvalue spacing

statistics of the Glauber matrix for _various models of statistical mechamcs

la

_one dimensional

Ising

model, _a two dimensional

Ising

model,

a one dimensional model with _a disordered

ground

state, and _a SK model with and without _a

ferromagnetic bios).

The

dynamics

of the _one dimensional

Ising

model

are

integrable

and the

eigenvalue spacing

statistics _are non-umversal. In the other _cases, the

eigenvalue

statistics in the

high

temperature

regime

_are intermediate between Poisson and G-O-E

(with P(0)

of the order of

o-à)-

In the intermediate temperature _regime, ^the statistics _are G-O-E-- In the Iow temperature

regime,

the statistics have _a

peak

at _{s =} 0. In the Iow temperature regime, and for disordered systems, ^the

eigenvalues

condense around

integers,

due _to the fact that the local field _{on any spm nover} vanishes. This property _is still valid for the

Ising

model _on the

Cayley

troc, _even ^{if it is} not disordered. WC also study the spacing between the two

largest eigenvalues

as a function of temperature. This quantity _seems to be sensitive to the existence of _a broken symmetry

phase.

Résumé. Nous discutons les

statistiques

d'écart entre valeurs propres de la matrice de Glau- ber _pour différents modèles de

mécanique statistique (modèle _d'lsing unidimensionnel,

modèle

d'lsing

bidimensionnel, modèle umdimensionnel _{avec un} état fondamental désordonné, modèle

SK _{avec ou sans} biais

ferromagnétique).

La

dynamique

du modèle

d'lsing

umdimensionnel est

intégrable,

et la statistique d'écart entre valeurs propres ^est non universelle. Dans les _autres cas, la statistique de valeurs propres dans le

régime

haute temperature est intermédiaire entre la loi de Poisson et la loi G-O-E-

(avec P(0)

de l'ordre de

o-à)-

Dans le

régime

intermédiaire, les

statistiques

sont G-O-E-- Dans le

régime

de basses

températures,

les

statistiques présentent

un pic pour s = 0. A basse

température

et pour des

systèmes

désordonnés, les valeurs propres condensent autour des entiers, à

cause du fait _que le

champ

local

sur aucun spin _ne s'annule

jamais.

Cette

propriété

est _{encore vraie} pour le modèle

d'lsing

_sur l'arbre de

Cayley

bien

qu'il

_ne

soit _pas désordonné. Nous étudions

également

l'écart entre les deux

plus grandes

valeurs propres

en fonction de la

température.

Cette

quantité

semble être sensible à l'existence d'une

phase

_avec

brisure de

symétrie.

(*)e-mail: mehn@crtbt.polycnrs-gre.fr

©

Les

Éditions

_{de Physique 1996}

1. Introduction

The ideas of level spacing statistics

emerged

^for the first time in the context of nuclear

physics [1-3],

where

Wigner proposed computing

statistical

quantities

from consideration of de- tern1inistic

spectra. Later,

these ideas _were

applied

to

quantunl systems

whose classical

analogs

are chaotic

[4, Si.

^{The idea of level} spacing statistics is to calculate the difference between two consecutive

levels,

and to

study

the

probability

of _occurrence

P(s)

of _a level

spacing

_s. The diiferent

generic

behaviors of

P(s)

_are dassified

according

to random matrix

theory [4,

_fil. A

generic

case is the

integrable

spectrum. Each level is labeled

by

_a set of quantum

numbers,

the _energy levels _are decorrelated and the statistics _are Poissonian:

P(s)

= exp

(-s).

If the number of conserved

quantities

is too

small,

it is not

possible

to find _{a set} of

quantum

numbers for each

level,

and the levels _are

correlated,

that

is,

there _exîsts level

repulsion.

The

repulsion

is

linear and the level

spacing

statistics have the Gaussian

Orthogonal

Ensemble

(G.O.E.) shape:

P(s)

=

jse~l~~ ^(l)

If time reversal invariance is

broken,

and if the

systenl

is

chaotic,

the level

spacing

statistics have _a Gaussian

Unitary

Ensemble

(G.Il.E.) shape:

Pis)

=

~(s~e~l~~, (2)

7r

where the

repulsion

is

quadratic.

The ideas of

quantum

^{chaos have been}

applied

to various fields of condensed marrer

physics,

such _as disordered

systems iii.

Another field of

application

is

strongly

correlated electron

systems [8, 9],

^{where the}

hope

is to extract more information from finite _size

systenls.

In the

present

paper we wish to

analyze

the

dynamics

of classical

spin

systems

using eigenvalue spacing

statistics. We _use here the term

"eigenvalue"

rather than

"level" since there _{are no} energy

levels,

_as in quantum mechanics. We consider the

2~

_x

2~

Glauber

matrix,

with N the nunlber of

Ising

_spms, and

diagonahze

it for snlall dusters. The

only symmetries

_are the lattice

symmetries,

that _we treat

using

_group

theory,

and the

global 22 symmetry.

^We can thus

only study

the

dynamics

for _a small number of

sites, typically

_on

the order of10 sites. The

eigenstates

_are not

physical,

except for the Boltzmann distribution which

corresponds

to the _upper

eigenvalue

₌ 0. The other

eigenvectors

_are not

probability

distributions,

_since the _sum of their components ^is zero, so that their

interpretation

is not obvious.

We have studied two diiferent

quantities.

The first

quantity

is the distance

AN (fl)

between the _two

largest eigenvalues (one

^of them

being

_zero for all

fil

for N sites. In the infinite

temperature limit, AN loi

= 1

(see

below for _a

proof

of this

fact).

Notice that in what follows

one unit of time

corresponds

to _a

single

_spin

flip step;whereas

_a Monte Carlo

Step

would

correspond

to N

single spin flip

_steps. As _a consequence, with the normalized lime units the

spectrum

lies in the interval

[-1,1],

and the normalized distance between the two

largest eigenvalues

is

ÀN(fl)

"

AN(fl)/N.

In

particular, ÂN(0)

=

1IN

in the infinite

temperature

hmit. One would

expect

^{the function}

AN (fl)

to decrease _as the inverse

temperature increases,

since the relaxation times _are

expected

to be

larger

^for smaller

temperatures.

^{In the} presence of

a broken symmetry

(for

instance in the _case of the _two dimensional

Ising model),

_one

_expects that,

_in the

thermodynamic limit, Acc(fl)

= 0 if

fl

_>

pc

_since the broken

symmetry

^states

becomes

degenerate

with the Boltzmann distribution _m the range of

temperatures p

>

pc. By

contrast, ^{in the} absence of _a broken

symmetry (for

instance for _{a one} dimensional

Ising

model

or for _a model with _a disordered

ground state),

the

quantity Acc (pi

should be

finite, _except

in the hmit

p

= +cc.

Ho~ever,

_our finite size

study

_is far from the

thermodynamic

hmit _{smce we}

could

only diagonalize

the Glauber matrix for about ten sites. We _were not able to

distinguish

between the diiferent

conjectured

behaviors in the

thermodynamic

limit.

Nonetheless,

_we

study

finite _size eifects and show that

AN (fl)

decreases _as the number of sites N increases. For _some models

(one

dimensional

Ising

model and the frustrated _one dimensional

Ising model),

_we find that

AN (fl)

is close to an

exponential.

In the _case of the two dimensional

Ising model, AN (fl)

is

dearly

non

exponential.

Another

quantity

of interest is the

eigenvalue spacing

statistics of the full

spectrum.

^{We first} consider the _one dimensional

Ising

model. In this _case, the

dynamics

_are shown to be inte-

grable.

The

corresponding eigenvalue spacing

statistics _are found _to be

non-universal,

with _a

peak

at s = 0 which increases _as the

temperature

^{decreases. In the} case of the two dimensional

Ising

model with nearest

neighbor coupling,

the statistics _are intermediate between Poisson and G-O-E- for _very small

fl,

with

P(0)

_ce o-à- As

fl increases,

the statistics evolve towards a

G-O-E-

shape là

r-

1).

At low temperatures no

eigenvalue repulsion

exists, and the statistics exhibit _a

peak

at _s

= 0. The

weight

of the

peak

increases with

fl. Next,

_we consider _a frus- trated _one dimensional model with _an extensive

entropy

at low temperatures. ^The evolution of the

eigenvalue spacing

statistics _is similar to the _case of the two dimensional

Ising

model.

In the _case of the

Sherrington-Kirkpatrick (SKI model,

_we also have the

following

evolution of the

eigenvalues

_spacing statistics: _no

repulsion

at very low

temperatures, repulsion

for inter-

mediate

temperature là

r-

ii,

and _no

repulsion

at low temperatures. ^An

important property

of disordered models is that their

eigenvalues

condense around

integers

at low

temperatures.

This is due to the fact

that, _except

for disorder realizations with _zero

probability

_measure,

the local field _{on any} site is _{never zero.}

However,

such _a behavior also exists for _non random

systems,

^for instance for the _nearest

neighbor Ising

model _on the

Cayley

tree.

2. The Glauber Matrix

Glauber

dynamics _[11]

_{are a}

single

spin

flip dynamics

with _a continuous time. If

p((a), t)

is the

probability

to find the spin

system

in the

configuration (a)

at time t, ^the master

equation

for the

single spin flip dynamics

is

~

p((OE),t)

_"

If _~((OE))~ p((OE),t)+( _w~((OEl,

_~OE~,

,

CEN))P((OEl,

, ~OE~,..

,

UNI, t).

~~

~=l ~=l

(3)

The

single

_spin

flip

transition

probabilities

_are defined _as the

probabilities

that the

spin

_a~

flips

from _a~ to -a~ while the other

spins

remain fixed. Since the Boltzmann distribution _{is a} fixed

point

of the

dynamics (3),

the transition

probabilities

have the form

w~jjajj

=

jjl

a~ tanh

jpJ ~ _{ajjj, j4j}

JEV(~)

where

Vii)

is the set of

neighbors

of the site 1.

Denoting

the

2~

_{vector of the}

p((a), t)

_as

p(t), equation (3)

can be written _as

Î~~~~ ^~~~~~'

^~~~

where the matrix G is the Glauber matrix. Since the Boltzmann distribution is _a

steady

state of the

dynamics,

its

corresponding eigenvalue

is _zero

regarless

of temperature. ^{The matrix G}

is not

symmetric.

It _can however be related to _a

symmetric

matrix M. To do _{so, we} notice that the Glauber matrix satisfies the detailed

balance,

that is

Ga,ppf~

=

Gp,apl~,

where

p(°)

is the Boltzmann distribution. As _a consequence,

(0) _~ (0) (0)

~ (0)

jfi)

~~°

~~~

"~

~~fl

~~~

~~fl

~~~

~"

~~°

~~~

We call M the matrix defined

by

Map

₌

(p(°))

^~~~

^G~p (p(°~)

~~~

ii)

Then,

M is

symmetric.

^If _p ^is a

right eigenvector

of the Glauber

matrix,

then

£ _Gappp

=

Àpp (8)

fl

is

equivalent

to

[ _{Maô lP1°~)}

^~~~_Pff =

lPi°1)~~~

Pa, 191

so that

(pΰ1)

~~~

p~ is _an

eigenvector

of M. We condude that G is

diagonalizable,

and that all its

eigenvalues

_are real.

The

spectrum

^{in the infinite}

temperature

limit _can be understood _as follows. If _we call

llfil =

fl _fllalllail

_{@ @}

_{ION), li°1}

iai

then the

dynamics

read

(ltfil

=

)ltfil

⁺

flalltfil, Îi Iii)

so that the

eigenvalues

of the Glauber matrix _at infinite

temperature

_are of the form

=

(

+

~i £

_/t~,

₍₁₂₎

where _/t~

= +l. The

spectrum

in the infinite temperature ^{limit is thus made of}

eigenvalues

at

integer

values between -N and

0,

with _a

degeneracy

_given

by

the binomial coefficients.

Another

property

of G is that for

bipartite lattices,

such _as the _square lattice _or the

Cayley tree,

the

spectrum

of G is

symmetric:

if

belongs

to the spectrum then -N is _an

eigenvalue

also. The

proof

is _as follows. Let

X(a)

be _an

eigenvector

^of

M,

with _an

eigenvalue

À:

~~~~~ i _~~

~~~~~~~~~~~~~~~~~~ ~

2coslÎflJh~~~°~"

^'~°~"

^{>°N), (13)}

where h~ is defined

by

h~ =

£

aj.

(14)

jEV(4 Let

Y(a)

be defined _as

Y(a)

=

(-1)~1°lX(à), ₍₁₅₎

where

u(a)

is the number of _up

spins

in the

configuration (a). (à)

is deduced from

(a) by flipping

the

spins

^of one of the two sublattices.

Then,

N

(MY)(a)

=

-£-(1-a~tanh(flHh~))(-1)~1°lX(à) ₍₁₆₎

~=i~

~Î~

N ~~~~~~~ ^{~ ~~'}

~~2CoshÎpJh~j'~l~i~..~~~~,

,àml

=

(-1)~~°~ IN ~ _(l

_{+ â~ tanin}

_{(flJh~ II X(à) (1il}

~=l

~ coshÎflJh~)~~~~'"

'

~~' ~~~j

=

-iN

+

Àii-ii~l°lXlàl

=

-iN

+

ÀiYlal. i181

Given _an

eigenvector

X for the

eigenvalue

_{À, we} have built _an

eigenvector

Y for the

eigenvalue

-N À.

The diiference between

(Si

and the

Schrôdinger equation

is that quantum ^mechanics pre-

serves the scalar

product, leading

to Hermitian Hamiltonians. More _over, the _space of

physical

states is _a Hilbert _space, and each _state of the Hilbert state is

physical.

In the _case of the Glauber

matrix,

no vector space is present in the _sense that the _sum of two

probability

dis- tributions is not a

probability

distribution.

However,

_some

quantities

_are conserved

by

the

dynamics.

It is easy ^to show that the

eigenvectors

of G for the _non-zero

eigenvalues

have the

property

^that

~ _P(°)

" °.

(19)

loEl

This is _a

simple

consequence of the fact that the Glauber matrix preserves the

quantity

flPlal. _12°1

iai

3. One Dimensional

Ising

Model

3.1. INTEGRABILITY _OF THE DYNAMICS. In trie _case of trie _one dimensional

mortel,

trie Glauber

dynamics

_are

integrable.

To show

this,

_we follow Glauber and write evolution _equa- tions for the correlation functions. We call

RÎ)~_

~

(t)

trie

n-point

correlation

function,

with

ia ~

_{ip if a}

~ fl,

that is ^{' ' "}

j~~~~ (~) = ~

(~)) (~~)

~i ~~~

~

~~

~

Then

following Glauber,

_we write trie evolution

equation

of

R))~

~

(t)

under trie form

jRli,1

,~~lu =

-21a~~ iti..a~n

_{lu iW~~}

ilali

_{+ + W~n}

ilaliii, (221

where the transition

probabilities

_are

given by (4).

We notice that in trie _one dimensional _case, each

spin

has two

neighbors,

_so that

w~((a))

_can be written _as

W~llall

₌

(i _)a~la~+i

+ a~-i

Il

,

1231

with _{~ =} tanh

2flJ. Here,

we take

periodic boundary conditions,

but the _case of _open

boundary

conditions is similar.

Inserting (23)

into

(22)

_we

get

~~ÎÎÎ

.,~n ~~~

~~ÎÎ~

>~n~~~ ~

~ Î~~'"~~ ~ _'P~'

_~~~~

e=+1n=1 p#n

The terms with

e = 1 collect the

right neighbors,

and _e

= -1

corresponds

to the left

neighbors.

The correlation function in

(24)

leads to _a

in 2)-point

correlator if

àfl,i~+~

=

ip

or to _a

n-point

^{correlator if} not. The

expression (24)

can be

brought

under the form

d (~) ^~

l~

~~~~~

_'~" ^~~~

~~~~~"'~"

~~~ ~

(~ ^{~ ~ Ùp,~~+eRÎ)}

^~Î~_~,~~~~

~ ~

j~~

Em OE=1 p=i

' P-1, p+1>. .,~n

n

~ i à

fl(n)

j~~

j~~~

~

M>~OE+E _ii,- toe-i>tm+E>~n+i>. _->in

'

~~~

If _none of the sites

ii,

...,

in

_are

neighbors,

the term

containing R("~~l

vanishes.

However,

^if at least two sites in the set ii,

..., in ^are

neighbors,

we have to take into account a term

containing R("~~l

in the evolution of

R(").

_{It is clear that}

(25)

is

nothing

but _a

rewritting

of

(3)

in the

case where all the sites have

only

two

neighbors.

The number of distinct correlation functions

is

~j

~ ^~

=2~, _(2G)

~=o

~"

which is

equal

to the number of _spin

configurations.

The

system (25)

is

integrable.

Glauber

gives

^the

explicit

solution for

RÎ~I(t).

The

equation giving dR(~)(t)/dt

_contains

_only

linear combinations of

R(~).

_{In order to solve for the three}

_points

correlation

functions,

_we

inject

Glauber's solution into the evolution

equation

^for

R(~), diagonahze

the associated matrix and

get

_a ^first order diiferential

equation,

which is

explicitly integrable

and

yields

the second order correlation functions. The entire

hierarchy

_can be solved

by

this method _since

dR(~) /dt

does not contain

R(P)

_{with p > k.}

3.2. EIGENVALUE SPACING STATISTICS. In order to calculate the

eigenvalue spacing

statis-

tics,

we need to take all the

symmetries

^{of the} lattice into account.

Here,

the

symmetries

are so

obvious that _we do not require a group

theory

treatment. We work with _an open

Ising

chain.

This

graph

is invariant under the reflection and the

identity

_operators. We denote the basis of _our "Hilbert" space as

(a)).

We _use quotes since there _{is no} vector space ^structure on the

probability

distributions.

However,

to

diagonalize

the Glauber

matrix,

_{we can use an}

analogy

to quantum ^{mechanics. If R is the} reflection operator, we form the combinations

iialie

=

) _niait

+

éRiialii

₁₂₇₁

This

operation

leads to states with _a well defined behavior under the reflection. The

resulting

state is either

symmetric le

₌

1)

or

antisymmetric le

₌

-1).

The

antisymmetric

state may be

zero if

((a))

is invariant under the reflection. The dimension of the

antisymmetric

sector is

(2~ 21~))

,

(28)

0 2 0

-l 3 -1

flfi

-2

3 5 3

-4 '_ -6

-5 _-7 -5

-6 -8 -6

' fi

-7 '

8

.ol ù-1 1

inverse _inverse emperature

(a) (b) (c)

Fig. i. Evolution of the

eigenvalues

of the Glauber matrix _{as a} function of the _inverse temperature for:

a)

_an 8 site

Ising

chain _m the reflection

antiperiodic

sector and the 22

antiperiodic

sector. No

avoided crossings _are present, _as

expected

for _an

integrable

system;

b)

the 3 _x 4

Ising

mortel _{on a}

square

Iattice,

in the representation number 3 and the

antisymmetric

22 sector. The evolution of 32

eigenvalues

_is

plotted

in this

figure; c)

trie frustrated

one dimensional mortel with _nearest

neighbor

interactions, for 8 sites in the antisymmetric reflection sector, ^{and the}

antisymmetric

22 sector.

and the dimension of the

symmetric

sector is

~

ij

-(2

_{+2 2}

), ₍₂₉₎

where

[...]

^{denotes the}

integer part. Finally,

we take into account the

global 22 symmetry

of the Glauber matrix.

(The

matrix elements of the Glauber matrix _are invariant under the

transformation

(a~)

_-

(-a~)). Taking

into account all the

symmetries,

_we

diagonalize

the Glauber matrix in the reflection

symmetric

and the

antisymmetric

sectors, and in the

22 symmetric

and

antisymmetric

sectors. The evolution of the

eigenvalues

of _one sector _{as a} function of the inverse temperature _is

plotted

in

Figure

la. No avoided _{crossings are} present, which is what is

expected

for _an

integrable

_system. The diiference between the two

largest eigenvalues

is

plotted

in

Figure

2 _{as a} function of the _inverse temperature.

Eventhough

_some

deviations _are

visible,

the behavior of

AN là)

is close to _an

exponential decay.

Finite _size eifects

are visible:

AN (fl)

^decreases _as ^{the number of} sites N increases.

N=6 - N=7 -~-

N=8 a

N=9 x

N=10 -~-

o-i

O.oi

', u

'#

0 001

0 0 5 1.5 2 2.5

Inverse temperalure

Fig.

2. Evolution of the parameter AN

(fl)

_{as a} function of the _inverse temperature fl, for different

sizes

(open Ising chain).

1.6

Poisson

14

12

f

08 ^'

06

0.4

02

0

0 0.5 5 2 25 3 3 5 4

normalized separalion

Fig.

3.

Eigenvalue

_spacing statistics of the _one dimensional

Ising

mortel with nearest

neighbors couphngs.

The _inverse temperature _is

fl

= i. The statistics _are non umversal.

The

eigenvalue spacing

statistics _are found to be _non universal. For

instance, Pis)

is

plotted

in

Figure

3 for

fl

= 1. The

height

of the

peak

at _{s =} 0 decreases _as the

temperature

decreases.

Eventhough

the

dynamics

_are

integrable,

the

eigenvalue spacing

statistics _are not of trie Poisson form

il).

Such _a behavior for the

spectrum

of

integrable systems

^has

already

been found in

the context of

integrable quantum

^fluids

(see [9]).

4. Two dimensional

Ising

Model

4.1. DYNAMICS. In this _case, the

dynamics

_{are no}

longer integrable.

The evolution of the correlation function is _{not a} linear

equation,

as it _was in the _case of the

Ising

chain. This

is

essentially

due to the fact

that,

with four

neighbors,

_one bas to introduce _a cubic term in

w~((a)) given by equation (23)

tanin

(flJ(ai

+ a2 + a3 +

a4))

"

a(ai

+ a2 + a3 +

a4)

+

a'(ai

_{+ a2 + a3 +}

a4)~, (30)

with

o =

j _(8tanh2flJ

_tanh

_{4flJ) (31)}

a'

=

(tanh 4flJ

2 tanh

2flJ). (32)

In the _one dimensional _{case, we} could

integrate

the

dynamics

because

dR(~)/dt

_was

_only

a

function of

R(~)

_{with k <}

n. In the two dimensional _case

dR(~)/dt

is also _a function of

R(~)

with k _{> n, so} that the

hierarchy

is _no

longer integrable by

this method. It is _not because

one does not know how _to solve trie

dynamics

that these

dynamics

_are not

integrable.

The

analysis

of the

spectral

statistics of the Glauber matrix may be useful to determine whether

or not there exists _some conserved

quantities

in the Glauber

dynamics

of the

Ising

model.

4.2. USE _OF GROUP THEORY. We _use group

theory

to find the

symmetries

of the clusters for which _we shall

diagonalize

the Glauber matrix. Notice that _{we are} restricted to small sizes since the _size of the "Hilbert" space is

equal

to

2~.

_In

practice,

and to have reasonable

execution

times,

_{we are} restricted to N < 13. The first step ^is to determine the

symmetry

group of the

lattice,

that is to enumerate all the

permutations

that leave the lattice invariant.

To do _{so, we} do _not test all the NI

possible permutations

_since trie

computation

time may be

huge. Instead,

_{we use} trie

following procedure.

We first label trie lattice sites and

give

the list of bonds. We determine all the

possible images a(1)

of trie site

1,

that is all the N -1 sites.

Then,

for each of the

possible images

of trie site

1,

_we determine trie

images a(2)

of site 2 which

respect

the lattice

symmetry:

if there is _a bond between 1 and

2,

^there must be _a bond between

a(1)

and

a(2).

If there is _no bond between 1 and

2,

^there must be _no bond between

a(1)

and

a(2).

ht this

point,

we bave _a list of

potential permutations beginning

with

ail)

and

a(2).

Then,

_we determine all the

possible images

of site 3 which leave the lattice invariant. We thus

get

a tree structure,

but, during

the

construction,

_some branches shall

stop.

^{At the end of the} process, that is when

a(N)

has been

determined,

we

get

^{all trie}

permutations

which leave trie

graph

invariant. The second

step

is to determine the classes and the table of characters of the group. We _use the program in

[12],

^which

automatically

determines the classes and the table of characters. In _a third

step,

_we have to determine the size of the blocs

corresponding

to the irreducible

representations,

and how _many times _a

given

irreducible

representation

_appears.

The dimension of the blocs

corresponding

to the irreducible

representation (j)

is

equal

to

Dim(J~

=

£ xlàlx~~~làl. ₁₃₃₁

geG

j

^{is the}

representation

of trie _group element g in the "Hilbert" _space, h is the cardinal of the group

G, x(§)

is the trace of

§

and

x(Jl(g)

is read from the table of characters at the

intersection of the line

corresponding

to the

representation (j)

and the column of the Mass of

j.

The fourth step ^is to

implement

_a Gram-Schmidt

procedure

to determine the basis of _one bloc

corresponding

to the

representation (j).

We first _use the

projector

p(J)

=

fl ~(J)jjjj_ j~~j

gEG

A basis element of the Hilbert _space is coded _{as a}

binary

number of size N. Zero

corresponds

to _a down

spin,

and 1 codes an up

spin.

In order to label the basis vectors, _we use the decimal

representation

of the

binary

number of size N. We denote the

corresponding

vector

(~lk).

The

procedure

consists of

scanning

all the _states (~lk) and _to determine

ko

such _as

P(J)(~lk)

= 0

if k <

ko

and

P(J)(~lko) #

0. The _state

P(J)(~lko)

is the first vector of the basis that _{we are}

looking

for. Once _we have found the first _vector of the

basis,

we continue to scan all the states

(~lk),

^but we

project

them with

PÎ~~

"

Llilkolàlilko)à.

1351

gEG

If

P)~~

(~lk) =

0,

_we

forget

about (~lk) and

project (~lk+i).

If

P)~l

(~lk)

~ 0,

we

try

to

incorporate P)~~(~lk)

into the basis

using

_a ^modified Gram-Schmidt

procedure _[13].

If

P)~~(~lk)

^is a linear combination of the basis vectors, then _we discard it and

project (~lk+i).

If it is

not,

we

incorporate ^{it into the}

basis,

after

having orthogonalized it,

and _we make the

projection

test for

(~lk+i).

At the end of the

procedure,

the dimension of the basis _must be

equal

to

(33).

We note that it is not

possible

to store all the components of the orthonormal basis because of limited

storage capacity.

In order to _save memory, we stored

only

the _{non zero} components.

The fifth

step

^is to take into account the

global 22 symmetry.

The sixth and last

step

is to

diagonalize

the Glauber matrix

using

the basis that has been determined at the fourth step.

The size of the matrices to be

diagonalized

_are small

enough,

_so that _{we can use} the Jacobi method.

4.3. RESULTS. We work with _a 3 _x4 lattice with

periodic boundary

conditions. The number of

representations

is

equal

to

15,

and the maximal bloc dimension is 335. The spectrum in a

given ^sector of

symmetry

of the 3 x 4 square lattice is

pictured

in

Figure

16 _{as a} function of the inverse

temperature.

^In the limit

fl

_-

0,

_{we recover}

degeneracies

for

integer eigenvalues (see

relation

(12)).

As the _inverse

temperature increases,

the

degeneracies present

for

fl

= 0

are

lifted,

but the

eigenvalues

^from two diiferent

degeneracies

are not free to cross, due to

eigenvalue repulsion.

We studied the evolution of

AN(fl)

_{as a} function of

fl

for the 3 x 4 lattice and for the 3 _x 3 lattice. The result is

plotted

in

Figure

4. Finite size eifects _are

visible; AN (fl)

^decreases if N

increases, and the evolution of

AN là)

is

clearly

_non

exponential.

In the

thermodynamic limit,

Acc là)

= 0 if

fl

>

Pc

and

Acc là)

> 0 if

fl

<

pc,

which _means that In

Acc là)

_{- -cc} if

fl

_-

flj.

Dur finite _size

study

is consistent with such _a behavior.

However,

it would be useful _to

analyze larger samples,

^which will be done in the _near future.

We _now discuss the

shape

of the

eigenvalue

_spacing statistics

Pis).

If the inverse temperature is very

small,

the

degeneracies

of the

fl

₌ 0 _{case are} lifted and trie _once

degenerate eigenvalues

spread

out

hnearly.

Trie

corresponding

statistics _are

plotted

in

Figure

5 for

fl

= 0.01. In this case, we find

P(0)

_ci o-à and trie statistics _are close to trie Poisson law for

large

_s. If

fl

increases, one reaches the

eigenvalue repulsion regime (see Fig. lb).

Linear

eigenvalue repulsion

3x3 - 3x4 -~-

o-i

o.oi

o coi

0 0001

~

',,,

'~+,~

l e45

",,

1e.06

0 O.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

inverse temperature

Fig.

4. Evolution of AN

(P)

_{as a} function of the

inverse temperature fl for the 3 _{x 4} Iattice

(N

₌

12)

and the 3 _x 3 Iattice

IN

₌

9).

Finite _size effects

are

visible,

and the variations of

àN(fl)

_are not

exponential.

12

beta =001 beta=1 Po>sson GOE

08

j~ ~~

~

04

,~-f~~

0 2

0

0 0 5 5 2 2.5 3 3 5 4 4.5

normalized spacin9

Fig.

5.

Eigenvalue spacing

statistics of the 3 _x 4 square Iattice for fl

= 0.01 and

fl

= 1. The

Poisson and G-O-E- distribution _are

plotted

in dashed Iines. In the regime where

imtially degenerate eigenvalues spread hnearly (fl

=

0.01), P(0)

_flf ù-à and the statistics _are close to the Poisson Iaw for

large

_s. In the

eigenvalue

repulsion _regime

(fl

₌

1),

the statistics exhibit

eigenvalue

repulsion

(P(o)

=

0),

but the

shape

of

Pis)

is

distinctly

different from the G-O-E-

shape.

2

beta=10 Poisson 18

16

1.4

~ 0.6

0A

02

0

0

Fig.

6.

Eigenvalue

_spacing statistics of the 3 x 4 square Iattice for

p

₌ 10. The Poisson distribution

is

plotted

in dashed fines.

15 visible _m

Figure

5 _smce, for

fl

=

1, Pis)

r- s for small _s.

However,

for _s of order

unity, large

deviations to the G-O-E- law _occur. For

large fl,

the statistics _are not universal

(see Fig.

6 for

fl

=

10),

with _a

peak

at _s

= 0. The

weight

of this

peak

increases _as

fl

increases.

5. Frustrated One Dimensional Model

5.1. THE MODEL. We consider the _one dimensional

antiferromagnetic Ising

model with

antiferromagnetic next-nearest-neighbor

interactions. This model _can be _{seen as a succession} of

triangles,

_as

pictured

in

Figure

7 and _can be solved ~ia _a transfer matrix

formalism,

with

trie sites

gathered

_as shown in

Figure

7. Trie transfer matrix bas trie form

T=( ~), ₍₃₆₎

~~~~

4pJ 2pJ 2pJ

À

"

~2pJ

^~ _{i ,} ^{~ ~}

~ÎJ (3i)

In

(36),

trie _{states are} ordered in trie form

Î, Î), Î, i), i, i), i, ii.

Because of trie form

(36)

of trie transfer

matrix,

if

(~l,

çJ) ^is an

eigenvector

of

T,

then (~l + çJ, ~l ⁺ çJ) and (~l çJ, ~l çJ)

are

eigenvectors

^{of T for trie} same

eigenvalue,

_so that trie

eigenvalues

of T _are trie

eigenvalues

of A + B and A

B,

and trie initial 4 _x 4

problem

is reduced _to two 2 _x 2

problems,

^due to trie time reversal invariance. Trie

partition

function _is

simply ZN

"

TrT~/~

_for

a N site chain

IN

is assumed to be

even).

The _zero

temperature entropy

_is ^found to be

extensive,

^{of the form}

S(0) IN

₌ In

2/2,

whereas in trie

corresponding ferromagnetic problem,

the

entropy

^{is finite} at

zero

temperatures.

^This one dimensional

antiferromagnetic

model bas thus trie _same

properties

as the

triangular antiferromagnet [14], namely

the number of _zero temperature

ground

states is

proportional

to exp

aN,

with _{o a} constant.

Fig.

7. The _one dimen5ionaI

Ising

model with

next-nearest-neighbor coupling,

and its representa~

lion _{as a} succession of

triangles.

The dashed Iines represent how the sites _are

gathered

in the transfer matrix formalism.

5.2. RESULTS. We work with _an open chain version, so that the

only symmetries

are

trie inversion and trie

global 22 syInmetry.

We bave

already explained

how to treat these

symmetries (Sec. 3.2).

The evolution of

AN (fl)

_as a function of the _inverse

temperature fl

is

plotted

in

Figure

8 for diiferent values of N. We observe finite _size eifects:

AN(fl)

decreases with N. Trie variations

N=6 _-

N=7 _-~~

N=8 _u

N=9 x

N=10 -~-

o-i

o.oi

0001

0 0.5 5 2

inverselemperature

Fig.

8. Evolution of AN

(PI

_{as a} function of

p

for N

= 6,7, 8, 9,10 in the _case of the disordered _one dimensional model of

Figure

7. If N >

N', AN(P)

<

AN'(fl)1

AN

(P)

decreases

exponentially

with the

inverse temperature

p.

beta=001 beta=1 Poisson G-O-E 08

06

04

02

0

0 05 1.5 2 2.5 3 3 5 4 45

normaiized spacin9

Fig.

9.

Eigenvalue

_spacing statistics of the frustrated _one dimensional model with next nearest

neighbor interactions,

for _{an inverse} temperature

fl

= 0.01 and

fl

₌ 1.

of

AN (fl)

_{as a} function of

fl

_are linear in _a

semi-log plot,

so that

AN (fl)

decreases

exponentially

with

fl.

The evolution of the

eigenvalues

^of the Glauber matrix as a function of the inverse

temper-

ature is

plotted

in

Figure

lc. Level

repulsion15

visible for 0.2 _<

fl

< 1.

Trie

eigenvalue spacing

^statistics in the

high temperature regime

are

plotted

in

Figure

9

là

₌

0.01).

In this _range of

temperatures,

^trie statistics _are intermediate between Poisson and G-O-E-

Pis)

bas _a

maximum,

but is _occurs for smaller

separations

than in the G-O-E-

case. If the inverse

temperature

_increases, ^the maximum occurs for

larger separations,

and

the

eigenvalue

_spacing statistics _are close _to the G-O-E- law for

fl

= 1

(see Fig. 9).

For

larger temperature là

= 3 in

Fig. loi, P(0)

is not _zero

(for

_instance,

P(0)

_t 0A for

fl

₌

3)

and

Pis)

is close to the Poisson law for

large spacings.

6. SK Model

6.1. THE MODEL. This model was

proposed

in 1975 as an

"exactly

solvable"

spin glass

model

[15].

^For

general

reviews _on the

problems

of

spin glasses,

we refer the reader to references

[16-18].

The SK model _is defined

by

the

disordered,

infinite _range interaction Hamiltonian

H

=

£ _{J~a~aj, (38)}

(~Jl

with

quenched

random interactions

J~j

^with a Gaussian distribution

P(J~j)

=

~~)

^exp

_(-~~~~

~

~~), (39)

27rJ

~

2J

beta=3 Poisson G OE

0.8

0.6

j~

~ ,,~

0.4

'

0 2 "'~,,

0

0 0.5 1.5 2 2.5 3 3.5 4

normaiized separation

Fig.

10.

Eigenvalue

_spacing statistics of the frustrated _one dimensional model with next nearest

neighbor

interactions, for

an inverse temperature fl

= 3.

In this

section,

_we

study

the

symmetric

_case à

= 0. The _case à _> is the

object

of Section 7.

We refer the reader to the reviews

previously quoted

for the solution of this model. For T <

J,

the model is _a spin

glass,

and _a

paramagnet

^for T _> J. In the

glass phase,

the model has _a

large

number of

thermodynamic phases,

with _no

symmetry connecting ground

states. The diiferent

ground

states _are

separated by large barriers, proportional

to the system size. Another feature of

glassiness

is the _presence of

ageing,

associated with slow relaxation _processes.

6.2. EIGENVALUE SPACING STATISTICS. In the _case of the SK

model,

_we do not need _to

look for the lattice

symmetries since, except

^for some

special

_cases ^of _zero

probability,

the random infinite range interactions break all the lattice

symmetries.

^The

only symmetry

to be taken into account is the

global 22 symmetry.

The evolution of the

eigenvalues

as a function of the _inverse temperature ^is

plotted

in

Fig-

ure 1la in _a

given symmetry

sector. We observe

that,

in the limit

fl

_{- +cc,} the

eigenvalues

condense around

integers.

This

property

is due to the fact

that,

in this

hmit,

the

symmetric representation M,

of

G,

_is

diagonal

since the local field _never vanishes

(except

for _some disor- der

configurations

of _zero

probability measure),

and that the

diagonal

coefficients _are

integers.

This cluster

property

m the _zero

temperature

limit is _an eifect of

disorder,

and is not related to the existence of

glassiness. However,

_{as we} shall _{see in} Section

8,

_we find _a similar behavior for _a

glassy ferromagnetic system

^with no disorderl

The evolution of

AN là)

_{as a} function of

fl

is

plotted

in

Figure

12. The variations of

AN là)

_as

a function of

fl

_are

approximately

linear in _a

semi-log plot,

_so that

AN là) decays exponentially

as a function of

fl.

The

eigenvalue

_spacing statistics

Pis)

in the

high

temperature

regime

are intermediate be- tween the Poisson law and the G-O-E- law

(see Fig.

13 for

fl

=

0.01).

In this regime,

P(0)

_t ù-à and

Pis)

is close to the Poisson law for

large separations.

If the _inverse

temperature fl

in-

o o

-i -1

2

=. -2

-3 -3

-4 -4

-5 -5

-6 6

-7 7

0.01 Ô.l 10 100 0.01 Ô.l 10 100

inverse temperature inverse temperature

(a)

(b)

Fig.

ii. Evolution of the

eigenvalues

_{as a} function of the _inverse temperature

fl, a)

in the 22

antisymmetric

sector of _a 7 site SK

model,

and for _a given disorder

configuration; b)

_m the 22 antisymmetric sector of the 7 site SK model with _a

ferromagnetic

bios, ^and for _a

given

disorder

configuration.

The

ferromagnetic

bios is à

= I.à.

creases, the _maximum of

Pis)

_occurs for

larger separations (see Fig.

13 for

fl

=

0.1)

and

finally

@

_{reaches the G-O-E- law}

_{(see Fig.}

_{14 for}

_fl

=

1).

In the _very low

temperature limit,

the statistics _are close to the Poisson

law,

with _a

peak

for _s

= 0

(see Fig.

14 for

fl

=

10).

The

height

of this

peak

increases _as

fl

increases.

6.3. ZERO TEMPERATURE DENSITY _OF ZERO EIGENVALUES. At _zero temperature, ^the

eigenvalues

of the Glauber matrix of the SK model _are

integers,

and the Glauber matrix _is

diagonal

_in the natural

basis, excepted

for _some disorder

configurations

of _zero

probability

measure. As _a matter of

fact,

_some

eigenvalues

are zero. The states

corresponding

to the _zero

temperature

zero

eigenvalue

_are the metastable _states of the SK

model,

and their

degeneracy

is

expected

to increase

exponentially

with the

system

_size, as

expected

from the number of solutions to the TAP

equations [16].

^{It is thus}

interesting

to calculate the

density

of _zero

eigenvalues

_in the _zero temperature ^limit as a function of the system

size,

even if _our

approach

is restricted

only

to small

system

sizes.

First,

we

analytically

treat the _case of the 3 site SK

model,

_{as a warm up exercise.}

',,

delta 0 -

&,

delta ₌ -~-

delta l.5 u

'~

_x ^{delta=2 x}

, ..

delta 3 _-~

X u

',

~, X a

O,I ', ^~

' X

~,

~

~

', ^~ ",~

o ",,

à X

~,,

"~

'

~ "',~

', ^{~ 13} '~"~

n ù ",,

', a

~",

'+

0 01

'n ',

'&

', ',

'~

O.COI

0 0 2 0.4 0.6 0.8 1.2 4 6 1.8 2

inverse temperature

Fig.

12. Evolution of

àN(P)

_{a5 a} function of p for N

= 9 in the _case of the SK

model,

for different values of the

ferromagnetic

bios à.

àNfias

been

averaged

over 200 realizations of the disorder. We

plotted àN(P)

for à

= 0,1,1.à, 2,3.

àN(P)

decreases if the

ferromagnetic

bios à increases, ^which is

consistent with the fact that the transition temperature _{mcreases as a} function of à.

beta O.Ol beta=0.1

Poisson G O.E

08

0.6 [[

;lÎ

_ÎÎ

q~ Ii

ÉÎ 1,11^Î(j

0 4

Ill

0.2

0

0 0.5 5 2 2 5 3 3 5 4 4 5

normalized spacin9

Fig.

13.

Eigenvalue

_spacing statistics of the SK model for _{an inverse} temperature

p

= 0.01 and

p

₌ o-1- The G-O-E-

shape

and the Poisson

shape

_are

plotted

in dotted fines.