Energy-Energy Correlations in Square and Cubic Ising Models

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Energy-Energy Correlations in Square and Cubic Ising Models

Roland Netz, Ingo Gersonde

To cite this version:

Roland Netz, Ingo Gersonde. Energy-Energy Correlations in Square and Cubic Ising Models. Journal

de Physique I, EDP Sciences, 1997, 7 (7), pp.827-832. �10.1051/jp1:1997204�. �jpa-00247366�

Short Communication

Energy-Energy Correlations in Square and Cubic Ising Models

Roland R. Netz

(~,~'*) ^and Ingo

^H.

Gersonde (~)

(~)

Department

of

Physics,

Box 351560,

University

of

Washington, Seattle,

WA 98195-1560, USA

(~) ^{Service de}

Physique Th60rique, CEA-Saclay,

91191 Gif _sur Yvette, France

(Received

9

September

1997, ret,ised 4

February1997,

accepted 6

May 1997)

PACS.05.70.Jk Critical

point phenomena

PACS.64.60.Fr

Equilibrium properties

_near critical points, critical exponents

PACS.02.70.Lq

Monte Carlo and statistical methods

Abstract. Cluster-Monte-Carlo simulations _are used to calculate the structure factors _as- sociated with

order-parameter

and energy correlations for _square and cubic

Ising

models, from which the

corresponding

correlation lengths ( and (E,

respectively,

_are determined. The _am-

plitude

ratio ~KE +

((/(~

is

asymptotically

universal but

strongly

influenced

by

the _crossover from mean-field behavior. The

numerically

determined XE therefore does not reach its universal

asymptotic

value but _agrees well with

e-expansion predictions including

corrections to

scaling.

The consequences for experiments _on near-critical fluids and colloidal systems are

briefly

dis- cussed.

On

approaching

_a continuous

symmetry-breaking phase transition,

the order parameter field

#(r) characterizing

the ordered state exhibits

diverging

correlated fluctuations. The structure factor

G(q, t),

which is the Fourier transform ofthe bilinear correlation function

Giq, tl

_C~

/ d~r e~~'~1410)41r)1, il)

has been of central interest in the

study

of critical

phenomena [il.

In _an

experiment

^where

a

scattering probe (such

_as

X-rays

_or

neutrons) couples

to the order

parameter,

the result-

ing scattering intensity

is

proportional

to

G(q,t),

which allows accurate tests of theoretical

predictions _[2].

The local energy

density

is

proportional

to the square of the order

parameter, #~(r),

and therefore shows

strong

fluctuations

sufficiently

close to _a critical

transition,

too. In this article

we discuss the structure factor

GE(q,t)

of connected energy-energy

correlations,

GE(~,~)

_C~

/ _~~r

_e~~

~(~~(0)~~(r))C. (~)

(* Author for

correspondence (e-mail: netztlmpikg-teltow.mpg.de)

Present address: Max-Planck-Institut fir Kolloid und

Grenzflhchenforschung,

Kantstr. 55, 14513 Teltow,

Germany

©

Les

#ditions

_de

_Physique

₁₉₉₇

828 JOURNAL DE

PHYSIQUE

I N°7

To _measure

GE (q, t) experimentally

_one needs _a

scattering probe

which does not

couple

to

the order

parameter

^{field itself but}

only

to the _square of it. One such

example

is sound

adsorption

in near-critical fluids _or solids

[3,4].

^{A different} mechanism is

provided by systems

with

distinct, coupled

order parameters;

scattering

from _a

secondary

order

parameter

which shows critical behavior

only

due to _a

coupling

to the square of the

primary (critical)

order

parameter

^should then be

proportional

to

GE (q, t). Experimental

realizations include colloidal

particles

in near-critical

liquid binary

mixtures

[5,

6] ^and the so-called sponge

phase

found in microemulsions

[7].

One of the

early

theoretical _successes consisted of

postulating

_[8]

land

later

deriving _[9])

for the structure factor

ii

the

scaling

form

Gjq,tj

=

xlt)glq~f~), _13j

valid in the

asymptotic regime, I.e.,

^{for small}

temperatures

t and for _wave vectors much smaller than the inverse lattice constant a,

I-e-,

_{q «}

a~~.

Unless at _a critical

point,

^defined

by

t ₌

0,

the

scaling

function

g(x)

_can

always

be

expanded

_{as a}

jower

_{series of its}

_argument.

_{For small}

arguments,

^{the inverse}

scaling

function is

commonly approximated by

g~~lq~f~)

= i +

q~f~

+

Olq~f~),

₁₄₎

which defines the correlation

length (

_as the second

spatial

moment of the correlation func- tion. Close

enough

to

criticality,

the

susceptibility

and correlation

length

_can be written _as

tit)

=

xot~~

and

fit)

₌

(ot~~, respectively.

The

amplitudes

_xo and

to

_are non-universal

parameters depending

_on details of the

interaction,

etc. The exponents _+t ^and u are universal and determined

by

the dimensionalities of space and order

parameter

^field.

Generalizing

the

scaling hypothesis

to the

biquadratic

correlation function

GE(q,t),

_one

writes the

biquadratic

structure factor

(2)

ⁱⁿ

analogy

to

(3)

and

(4)

as

GE iq, t)

₌

Cit)gE iq~fi)

_~f

~ ~~

ji)j

j~~ j~

^,

is)

which defines the energy correlation function

(E

_as the second moment of the energy-energy correlation function. In the

asymptotic limit,

the

specific

heat and the _energy correlation

length obey-

the

asymptotic

^laws

C(t)

=

Cot~°

and

(E It)

=

(E,ot~~E, respectively. Here,

_o is the universal

specific

heat exponent and _uE is the

exponent

^{of the} _energy correlation

length

which could in

general

be different from _u.

Within the

e-expansion technique (where

_e

= 4 d _measures the deviation from the _upper

critical dimension

four)

the energy-energy correlation function

GE(q, t)

can be calculated for the standard

#~ theory using

_a double

expansion

in _e and the

quartic coupling strength [ii.

The actual calculation is rather

long

and

technical,

but among the results _one finds that the two correlation

lengths

^scale with the _Same

exponents,

I.e.

u/uE

= 1+

fJ(e~ [10].

^{A different} conclusion was

initially

drawn from

X-ray scattering

results for similar

biquadratic

correlations at _a smectic-nematic transition in _a

liquid crystal sj,stem [11],

^but later reconciled with the

above theoretical result

[12].

Furthermore, the ratio of the correlation

lengths asymptotically approaches

_a universal

amplitude

ratio

[10]

j2

~

~~ (~

6~

^~

~~~~~'

_~~~

Although

the

equality

of the two correlation

lengths,

I.e. _u

= uE, has

only

been

proved

_up to the first order in _{e, we} note that _a naive

scaling theory

would

suggest

^that this

equality

is exact.

Also,

it could be

speculated

that the relation

(6),

which has been calculated _up to

e~, might

hold _to all orders in _e, and thus be

applicable

at all dimensionalities. This could of _course

only

be substantiated

by higher-loop

calculations. As is well

known,

the

specific

heat is

strongly

influenced

by

the _crossover from mean-field _to

asymptotic

critical behavior

[13].

^{It therefore}

might

be hard to observe the

asymptotic

correlation

length

ratio

(6)

in _an

experiment

_or a

simulation.

Defining

the effective

temperature-dependent exponents +te~(t)

e d In

x(t) Id

In t and _ae~

it)

_e d In

C(t) Id

Int _one derives to

Die)

the relation

[10]

x~jt)

-

(

=

je~((~ ^j7)

which holds

beyond

the

asymptotic regime

since it includes corrections to

scaling.

It is this

expression

which would be measurable

experimentally

and which we test in this article.

In our simulations _{we use} the standard

Ising model,

defined

by

the Hamiltonian

~

IT

₌

-J£j~~~

s~sj ^with si = ~1 and the _sum

running

_over

nearest-neighbor pairs

_on square or cubic lattices. We

empliy

_a Cluster-Monte-Carlo

algorithm [14],

^{which has}

recently

been

successfully

used to calculate bilinear correlation functions

[15].

Figure

I shows results for square lattices of linear size _up to L

=

200; subaveraging

_over

progressively larger

time intervals has been used to calculate statistical _errors

(which

_are

always

smaller than the

symbol size).

The

susceptibility

_x

(solid circles, Fig. la)

_compares well with exact

results,

+t ⁼

7/4

and _to

= 0.962582

[16] (solid line).

The

specific

heat C

(open circles), predicted

to exhibit

logarithmic

behavior

(a

=

0)

indicated

by

the broken line with

zero

slope,

has _not

quite

reached the

asymptotic scaling regime,

in accord with theoretical

expectations [13].

^{For the} calculation of correlation functions

along

a certain direction

(here

chosen to coincide with _a lattice

principal direction)

_we

_separate

the

spatial

coordinates into the

parallel, _rii

_{e r Q, and}

perpendicular part,

ri % r

ripe,

^{and rewrite}

^equation ii)

_as

G(q, t)

«

V~~

drjie~o~"

dr(j ~ d~~~r[#(r(j,r ^[) ~ d~~~r14(r(j

⁺

rip,ri) ), (8)

thereby minimizing

the numerical effort.

Figure

16 shows the inverse

scaling

functions

g~~

and

50gj~

as a function of

q2

for t

= 0.11438.

According

to

(4)

and

is),

the

corresponding squared

correlation

lengths

_are ^determined

by

the

slopes

for small momentum

(solid lines).

As _seen in the

figure,

the energy-energy

scaling

function

(open circles) extrapolates

not as

nicely

to _a

straight

line at the

origin

as the bilinear correlation function

(closed circles) does,

and the extraction of the energy correlation

length

^is harder. This is in full accord with the field-theoretic

calculation,

which

predicts higher-order

terms in the

expansion

of the

scaling

function in

(5)

to be

larger

than for the bilinear

scaling

function

[10].

^To access even smaller q values _one would need to take

larger system sizes,

which increases the

computational

effort.

The

resulting

bilinear correlation

length ( (circles, Fig. 1c)

is much smaller than the system ^size

(so

that finite-size effects _can be

neglected)

and _agrees well with series

expansion estimates,

u = 0.632 and

to

₌ 0.495

ii?] (solid line).

The correlation

length

ratio

XE (for

which the _error bars would be hard to

estimate)

is

plotted

in

Figure

id. With _o

= 0 the

predicted asymptotic

value is

XE

₌

0, according

to

(6)

^and

using

the

working hypothesis

that the _e

expansion

results

are also valid in dimension two, I.e. for _{e =} 2. The actual values for

,KE, though being

_very

small,

_are far from _zero and in contrast to the

asymptotic prediction

_agree rather well with the effective

exponent

^{ratio determined}

by (7) (solid line).

The effective exponents +te~ and _{ae~ are} determined from _our data

by fitting

the

double-logarithmic

data in

Figure

la to _a

polynomial

of fifth order.

830 JOURNAL DE

PHYSIQUE

I N°7

x

a)

c

^<2°

js0gi

^°

^'

~ ~

_

,~ ~~

0

' 0 2 4 6 8 lo

0.03 0.< 0.3 2

t

~l

20

C) ~

,~ ^E

I.

.

~~~

a)

^~~ 60

~0g/

~

~f

-c--_

~

o I

° ~

~~~

~

^~ ~ ⁴⁰ o

.

<00

o

o

^g-1

~

b)

20 ~

0

<0 0.06

C) XE

j

.

~ _.

d)

2

~~~

832 JOURNAL DE

PHYSIQUE

I N°7

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