Field Theory of Finite-Size Effects in O(n) symmetric systems

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Field Theory of Finite-Size Effects in O(n) symmetric systems

X. Chen, V. Dohm, A. Esser

To cite this version:

X. Chen, V. Dohm, A. Esser. Field Theory of Finite-Size Effects in O(n) symmetric systems. Journal

de Physique I, EDP Sciences, 1995, 5 (2), pp.205-216. �10.1051/jp1:1995121�. �jpa-00247050�

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Classification Physics Abstracts

05.70 64.60A 75.40C

Field Theory of Finite,Size Elllects in O(n) syuunetric systeuls

X.S.

Chen,

V. Dohm and À.

Esser(*)

Institut für Theoretische Physik, Technische Hochschule Aachen, D-52056 Aachen, Gerrnany

(Received

18 October 1994, received in final form 21 October 1994, accepted 25 October

1994)

Abstract. We _propose _a _new perturbation approach ta finite-size elfects _near the critical point within trie _q~~ field theory for _an n-cornponent order pararneter with periodic boundary conditions. Our approach is applicable bath above and below Tc. Calculations of finite-size

scaling functions of trie specific heat, susceptibility, order pararneter, and _a cumulant ratio _are

compared with Monte Carlo data for trie three-dimensional XY model. Gard overall agreement with trie Monte Carlo data is found.

l. Introduction

The çJ~ field

theory

of finite-size effects _near _the critical point _as

proposed by

Brézin and Zinn-Justin

(BZ) Ill

and

by Rudnick,

Guo and Jasnow

(RGJ)

_[2] has

recently

been further

developed by Esser, Dohm,

Hermes and

Wang

_[3] and

by

Esser and Dohm [4]. ^While ^the çJ~

theory

of

BZ,

in its present

formulation,

is restricted to the

region

T > T~ the

approach

of RGJ is in

principle applicable

both above and below T~ but ^ils results _are non

quantitatively

reliable for T < T~ [4]. ^For ^the case of _a one-component ^order parameter

(Ising-like systems)

with

penodic boundary conditions,

the _new

perturbation approach

of Esser et ai. [3, 4] leads to _an effective Hamiltonian which _can be considered _as _an appropnate

generalization

of the effective Hamiltonian of the _çJ~

theory

of BZ a~ld _can be used both above and below T~. The results obtained

by

this

approach

_agree well with Monte Carlo data for the three-dimensional

Ising

mortel [3-5].

In this paper we consider the _more

general

_case of

O(n) symmetric

systems near T~ with

an n-component

(n

>

1)

order parameter. ^It ^is well known that for _n _> 1 the

long-distance properties

^{of the} ^bulk system below T~ _are

governed by

massless spm-wave

(Goldstone)

modes which get a small finite _mass in _a finite volume

Ill.

This _case _was studied

previously

_neon T~

by

BZ

Ill by

_means of _a

low-temperature expansion

within the nonlinear a-model. Further studies have been

performed

in [6]. Here _we shall treat these modes in _a different _way within the

O(n) symmetric

_çJ~ model which is well suited for

quantitative

calculations of finite-size

scaling

functions in three dimensions. It _tutus out that there exists _no

straightforward

extension of the

(* ^Present ^address: Fachbereich Physik, Philipps-Universitàt Marburg, Renthof 6, D-35032 Marburg, Germany.

Q Les Editions de Physique 1995

(3)

effective Hamiltonian of the previous

(n

₌ ₁₎ _case _[3,_4] because of

spurious

infrared

singularities

in the bulk limit

arising

from the massless

(transverse) order-parameter

fluctuations below T~

[7]. ^Instead we propose an

appropriate perturbative

treatment of these transverse modes different from that of the

longitudinal order-parameter

fluctuations. As _an

application

_we

calculate various

thermodynarnic quantities

for the _case _n

= 2 in

one-loop

order of renormalized

perturbation theory

^and _compare our field-theoretic

predictions

^with Monte Carlo data of the three-dimensional XY mortel

[8-12].

2. Problem of Perturbation

Theory

for _n _>

We _use the standard

Landau-Ginzburg-Wilson

Harniltonian

~

Î _~~~

ÎÎ~°~~

^~

^~~~~~ ~'~°~~Î

~~~

where

çJ(x)

is _an n-component field in _a finite cube of volume V

=

L~.

_For

periodic boundary

conditions the field

çJ(x)

_con be

expanded

as

§~(X) "

L~~ L

_§~k~~~~

₍₂₎

k

where the summation _runs _over discrete k vectors with components k~ =

27rm~/L,

_m~ ₌

o,+1,+2,..,

i

=

1,2,..,d,

in the range (k~( ^< A.

Eventually

_we shall let A _- _cc in the renormalized

theory.

Followmg

BZ and RGJ _we

decompose çJ(x)

_as

~J(x)

₌ 4l +

a(x) (3)

where

4l =

L~~çJo

₌

L~~ / _{d~x çJ(x)} ₍₄₎

is the _most

important

mode and

a(x)

₌

^L~~ £

_çJke~~'~

₍₅₎

k#0

contains ail modes

higher

than the zerc-mode 4l. We

decompose a(x)

further into

longitudinal

and transverse parts

a(x)

=

aL(x)

₊

aT(x) (6)

where

aL(x)

and

aT(x)

_are

parallel

and

perpendicular

_to the _vector 4l,

respectively.

Corre-

spondingly

the Hamiltonian

(1)

is

decomposed

_as

H = Ho(4~~) +

H', (7)

l~'

"

I~Î(>aL)+HÎ(>aT)+H2(~,aL>aT), ₍₈₎

where

Ho(4l~)

₌

L~(~ro4l~

₊ _~lo4l~),

(9)

2

Hi(4~~>aL)

=

/ d~x1(ro

⁺

12~lo~~)ai

+

(T7aL)~l, (10)

H/(4l~,aT)

"

/ d~x((ro

⁺

^4~1o4l~)a(

⁺

(VaT)~j, ⁽¹¹⁾

2

H2(4l, aL,aT)

=

/ _d~x 4~1o4laLa~

+

~loa~j (12)

(4)

with a~

=

a(

₊

a(.

_The calculation of the

partition

function Z =

/

_DçJ

_exp(-H) ₍₁₃₎

requires

functional

integrations

with respect to aL

(x)

and _aT

(x)

and _an

integration

with respect to 4l.

Performing

the _aL and _aT

integrations jirst

^would

yield

the

partition

function in the form

Z ₌

/

_d"4l

exp

(-Ho(4l~ r(4l~)j ⁽¹⁴⁾

where

r(4l~)

= In

DaL DaT exp(-H') (15)

contains the contributions of the

hîgher

modes. For _ro _> o _a

perturbative

treatment of

r(4l~)

based _on the

decomposition (8)-(12)

is

straightforward.

For _ro <

o, however,

the propagators

corresponding

to

Hi

and

Hf

_may _become

_negative

which is the _same

problem

_as described in references [3, 4] for a one-component ^order parameter. ^Therefore one may attempt to avoid

this

problem by extending

the

general

_n, i e., one may introduce the

positive

parameters

roL " ro +

12~1oM/ (16)

and

roT = ro +

4~1oM/ (17)

with

Ml

= < 4l~ >o =

/ d"4l4l~ exp(-Ho(4l~)j, ⁽¹⁸⁾

Zo

Zo _"

/d"4lexp(-Ho(4l~)j, ⁽¹⁹⁾

and _use

Hi

=

/ _d~x

(rota)

⁺

^(VaL)~

⁺

^roTa(

⁺

(i7aT)~j ⁽²⁰⁾

2

as the

unperturbated

part ^of ^the Hamiltonian H'. The

resulting perturbative expression

for

r(4l~)

would read _to

leading

order

L-dr(~2)

=

)L-d L in(ro~

₊

k2)

+ ^~

j L-d L _in(ro~

₊

_k2)

k#0 k#0

+ 2~1o(4l~

Ml) _(3Si(rot)

₊

(n 1)Si(roT)j

4~Î(~~ ~Î)~ ^~S2(TOL)

₊

(n _1)52(TOT)j (~l)

with

S~n(r)

=

L~~ £(r

₊

_k~)~'* (22)

k#0

(5)

But _now a serious

problem

arises from the last _term in equation

(21) containing

the _sum

52(roT).

This _sum

diverges

for _ro < o in the limit L _- _cc because the transverse modes become

massless,

1-e-, roT - o for _any _ro < o in the bulk limit.

(Note

that

M]

tends to the

mean field value

-ro/4~1o

_m the bulk limit for _ro <

o).

Similar infrared

divergences arising

from the contributions of the transverse part _aT will appear m

higher

order. Since the

partition

function _is _an

O(n)

invanant

quantity

which should not have Goldstone

singularities

_[13] _we

consider these

divergences

not _as true

physical

Goldstone

singulanties

but

only

as a spunous effect due to _an

inappropriate

formulation of the

theory. (These divergences

would not appear

m a

theory

at finite externat field h if the bulk limit L _- _cc is

performed

before

letting

h _-

o.)

This _means that

(our

above

generalization of)

the effective Hamiltonian of the çJ~

theory

of

BZ does not

meaningfully

exist below T~ for _n _> 1 due to the contributions of the transverse

part _aT.

3. New Perturbation

Approach

In this paper we offer _a solution to this

problem.

The main ideas _are to introduce _an effective Hamiltonian

arising only

from the contributions of the

longitudinal

part _aL and to

change

the order of

integration:

we shall

perform

the

problematic

_aT

integration ajier

the _aL and 4l

integrations

have been carried ouf. Then _no _spurious Goldstone

singulanties

will arise in the limit L _- _cc.

The first step ^is to

perform

the

integration

over aL m a

perturbative

_way. As

previously

[3, 4] we take

H) (Ml, aL)

_as the

unperturbed

_part and

AHI(4~~,aL)

=

Hl(4~~,aL)-Hl(MÎ,aL)

=

/ d~x(6~1o(4l~ M/)a(j ⁽²³⁾

as the

perturbative

part, in addition _to H2.

Integration

over aL then

yields

the

partition

function to

leading

order

Z = DaT d"4l

exp(-Ho(4l~) H)(4l~,aT) _rL(4l~) R(4l~,aT)j ⁽²⁴⁾

where

L-~r~(~2)

=

(L-~ L _in(ro~

₊

_k2)

₊

_6~to(~2 _Mi)si(ro~)

k#0

36u((4l~ M])~S2(roL)

₊

O(~lo,~1(4l~,~1((4l~ M()~) (25)

is of the _same form _as

previously

[3, 4] and contains the

purely longitudinal

contributions of the

higher

modes

arising

from

AH)

₊

H2.

The _term

R(4l~, aT)

in

equation (24) corresponds

ta other contributions of

higher

order.

Following

Esser et ai. [3, 4] we define _an effective

longitudinal

zero-mode Hamiltonian

H(~(4l~) according

to

Ho(4l~)

₊

FL(4l~)

"

Hi~(4l~)

₊

FL(°)

,

(26)

H(~(4l~)

₌

L~( r(~4l~

+ ~1[~4l~)

(27)

2 with effective parameters

r(~

= ro +

l2~1oSi(roL) +144~1(M(52(roL), (28)

~1(~ = ~lo

36~1(52(roL). (29)

(6)

Then _we have

Z ₌

e~~L(°) / _DaT /

_d"4l

exp

(-H(~(4l~ H/(4l~, aT) R(4l~, aT)j. ⁽³⁰⁾

Next _we shall

perform

the 4l

integration

while

treating Hf

₊ R in _a perturbative _way. For this _purpose _we define trie effective _average

Mj~

₌ < 4l~ _>~a

=

£ /d"4l4l~ exp(-H(~(4l~)j, ⁽³¹⁾

eR

Z~a =

/d"4lexp(-H(~(4l~)j. ⁽³²⁾

~~~ ~~~~~~~~

jfT( jf2~ ~~) /

_d~X

(T~~a~

⁺

(À7aT)~j

^~~~~

l _e ' ~ ^°

with

r((

= ro + 4~1oMj~

(34)

as the

unperturbed

part of the contribution

Hf

₊ R. Thus _we rewrite in equation

(30) I~Î(~~, aT)

"

I~Î(ÀÎÎOE, aT)

₊

~I~Î(~~> aT) (35)

with

~I~Î(~~,°T)

"

I~Î(~~>aT) I~Î(ÀÎÎOE,aT)

=

d~x

^2~1o(4l~

Mj~)a(j ⁽³⁶⁾

/

and

expand

part of the exponent ^of equation

(30)

to

leading

order _as

exp

(-AH)(4l~, aT)j

^" ^1-

AH/(4l~, _aT)

₊

(AH/(4l~, aT)j

~

(37)

Now the 4l

integration yields

Z = Z~i e~~L(°~

/

_DaT

_il

+ 2~1( ^< (4l~

Mj~)~

>~i

/

d~xa()~j

x exp

(- ^HI ^(Mj~, _aT)j ⁽³⁸⁾

Note that _we have avoided _any

expansion

^{of the} exponent ^of

equation (30)

^with respect ^to ^the

longit~ldinal

contribution

H(~(4l~)

for _reasons

given previously

[3, 4].

The final step is to

perform

the Gaussian _aT

integrations

m

equation (38). Using

the relation

~ (4~~ MÎOE)~~~ >eOE "

(~2L)~/)~É~~'

^16'~ ^~

^il' ⁽³⁹⁾

we obtain the result for the

partition

function Z of the finite system _up to

one-loop

order

In Z ₌ In Z~i

rL(o)

^~

£ ln(r((

₊

_k~)

~ k#0

4(n 1)~Î (~~2(TÎÎ)

⁺

^(il l)~~~l(TÎÎ)~j

^~

~/~É~~ ^(~°)

0

(7)

Using

the _same

perturbation approach

_as for the partition function _we find the _averages

(up

ta

one-loop order)

< (4l(~' > =

j /DçJ(4l(~'exp(-H) ⁽⁴¹⁾

= < (4l(~' >~a +

4(n -1)~1oSi(r(()~

^~

(~~

^~~~

r~

+

~(~ l)~Î~

^~

_(~~2(TÎÎ)

₊

_(~ l)~~~l(TÎÎ)~j

~~

()/2~ ^(~~)

0

Equations (40)

and

(42)

_are the _main results of _our

(bare) perturbation approach

whose extension to

higher

order is

straightforward.

The basic difference with the

approach

of RGJ [2] is the

following:

_we evaluate the _averages

m

equation (31)

and

equations (38)-(42)

with the effective statistical

weight

_exp

(-H(~(4l~)j,

^without ^further

^expanding

^this

^weight,

^which ^ensures ^that

as much

(perturbative)

information _as

possible

is

exploited.

Results based _on the

approach

of RGJ will be

presented

and discussed elsewhere

il?].

In order to describe the critical behavior

appropriately

it is necessary ^to ^tutu ^to the _renor- malized

theory.

We have

employed

^the minimal renormalization scheme at fixed d _< 4

[14-16]

without _using the _e

= 4 d expansion. ^Some ^of our results for the _case _n ₌ 2 and d

= 3 _are

presented

in the _next Section. The details of the renormalized

theory

_are

parallel

to those in reference _[4] and will be _given elsewhere

Il?].

4.

Applications

From In Z, equation

(40),

_we _can calculate the

specific

heat

using

the usual definition [16]

C ₌ GB +

(a(C( ⁽⁴³⁾

with

~

~~

"

~Î

_~~~'

~~~~

From the _averages < (4l(~' >,

equation (42),

_we obtain the

susceptibility

X~

"

L~~ / _d~xi / _d~x2

<

çJ(xi

_)~9(x2 _>

(45)

=

L~

_< _{4l~ >,}

(46)

the _square of the

magnetization

M~

=

j

<

/ _d~xçJ(x)(~

>

(47)

= < 4l~ >,

(48)

and Binder's [18] ^cumulant ^ratio

U=1-)~(~ ^~. ₍₄₉₎

(8)

The definitions _we have taken here _are

equivalent

to those used in Monte Carlo simulations

[8-12].

On the basis of the renormalized

theory

_we can show

that,

for

suiliciently large

L and small (t( =

(T

_T~(

/T~,

the

quantities

mentioned above _can be written in the finite-size

scahng

form

C =

L"/~P~(tL~/~)+ÔB, ₍₅₀₎

x+

=

L~/~P/(tL~/~), (51)

M~

=

L~~~/~Pj~~(tL~/~), (52)

U

=

U(tL~/~). (53)

We have calculated the

asymptotic scaling

functions

P~, PI, Pf~

and U in

one-loop approxi-

mation _m three dimensions. The

analytic

form of these functions will be

given

ⁱⁿ ^reference

il?].

There _are _no

adjustable

_parameters other thon those of the bulk

theory (1.e.,

the

amplitudes

of the

asymptotic

bulk

expressions

of the

susceptibility

and of the

specific

heat _as well _as

ÔB).

For

n = 2 and d

= 3 _we have chosen these parameters ^such ^that ⁱⁿ ^the limit L

- cc our

one-loop

results for the

specific

heat and

susceptibility

above T~ _agree with the bulk

amplitudes given by

Ferer et ai. [19]

(for simple

cubic

lattices).

The bulk

amplitude

of the

magnetization

can

then be calculated from universal combinations of

amplitudes

_[20].

In

Figures

1-8 _our results for _n

= 2 _are

plotted (in

units of the lattice

constant)

and _are

compared

with Monte Carlo data of the XY mortel with cubic

(L

_x L _x

L)

_geometry and with

periodic boundary

conditions [8-12] ^for varions values of L.

(More precisely,

the

following

dimensionless

quantifies

_are

plotted

in

Figures

1-G:

Cla, Pcla, x+la~, P~la~, M~a, _Pj~~a,

where _a is the lattice

spacing

with

(o

_" o.486a

[19].)

4

MCDa1a L=48 o

16

35 x

4 A

w

3

2.5

~ x

D

~J 2

u

i à

o

o o

D.à

o

o. i o oà o O.Oà o i

t

Fig, l. Theoretical results of the specific heat C from equation (50) and Monte Carlo data from reference [8] (X and

stars),

reference [9]

(n

and Zh), reference

[loi (o),

and references [11]

(+)

and [12]

(+)

us. reduced ternperature t for L ₌ 4,16, 48. Dashed fine: bulk result frorn reference [19].

(9)

18

MCDa1a. L=48 _o

16

185 x

4 à

w

.19

o x

o

Pc

,igà

-20

o

21

22

.8 6 4 -2 0 2 4 6 8

~Î/V

Fig. 2. Scabng function

Pc(tL~/~)

ofthe specific heat C defined in equation

(50)

corresponding to

Figure 1.

iooo0

,, MC Data L

=64 _o

', 48 +

', u

',, 32 x

', 16 à

8 w

X~

~°°°

à ~

ioo

o o0ooi o oooi o.coi o oi o i

Fig. 3. Theoretical results of the susceptibility X~ frorn equation (51) and Monte Carlo data from reference

[loi (D),

references [11, 12] _us. ^reduced temperature t for L

= 8,16, 32, 48, 64. The data _on the left-hand side correspond to T ₌ Tc. Dashed fine: bulk result frorn reference [19].

(10)

16

MC Data L=64 o

48 ₊

14 32

16 A

8 w

12

P(

02

0

0 2 3 4 5 6 7

t~l/V

Fig. 4. Scaling function

Pf(tL~/~)

of the susceptibility X~ defined in equation

(51)

corresponding

to Figure 3.

o-à

MC Data L₌ 16 o 0 45

04 ₊

+ 0 35

fif2

₀₃ + ^~

+ +

°

+ +

025

02

à

o15 ^~

~ à

~

o i

._

o 05

1,

o

o i -o oà o o oà o i

t

Fig. 5. Theoretical results of the rnagnetization _square ^M~ frorn equation

(52)

and Monte Carlo

data from reference [8] us, reduced ternperature t for L ₌ 4,8,16. ^Dashed ^fine: bulk result frorn references [19, 20].

JO URNAL DEPHYSIQUE Il T5, N° 2, PEBRUARY 1995

(11)

7

McDala L=16 _o

8 A

4 +

à

o à

p(2)

~

o +

3

a +

2 ^~

i

0

6 4 2 0 2 4 6

t~l/V

Fig. 6. Scaling function

Pf~(tL~/~)

of the rnagnetization _square M~ defined in equation

(52)

corresponding to Figure 5.

07

MC Data L=64 o

32 _D

0 65 ^~

~

à ~

o-à

o àà

~

U O.à

à

045 ^°

à

04

035

03

004 003 002 001 0 001 002 0.03 004

t

Fig. 7. Theoretical results of the cumulant ratio U frorn equation

(53)

and Monte Carlo data frorn reference

[loi

(star) and reference [12]

(o,

n, and Zh) us. reduced ternperature t for L

= 16, 32, 64.

Asyrnptotic bulk values _are

2/3

for t < o and

1/3'for

t > o.

(12)

07

MCDa1a L=64 o

32 D

065 16 à

06

*

o.5à

~

n

U ₀₅

à

0 45 ^°

a

04

0 35

03

8 6 4 -2 0 2 4 6 8

t~l/V

Fig. 8. Scahng plot of the cumulant ratio

us, tL~/~ corresponding to Figure 7.

We _see that there _is

good

overall agreement ^between our field-theoretic

predictions

and the Monte Carlo data. The

existing

differences _may

partly

be due to _our

one-loop approximation;

furthermore,

_as

previously

_[3,_4], there _are deviations due _to

nonasymptotic effects,

in

particular

those of the

magnetization

outside the

asymptotic region

well below T~

(Figs.

5 and

6),

which

are not yet ^induded in the

(asymptotic)

evaluation of _our

theory.

Acknowledgments

We thank SFB 341 for support.

Note added in

proof:

In

Figures

1 and 2 the

original

Monte Carlo data

given

in references [8-12] ^have ^been

multiplied by

_a factor (T~

/T)~

_so _as _to be in accord with _our definition of the

specific

heat in

equations (43)

and

(44).

References

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