Critical exponents from a regularized field theoretical model

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Critical exponents from a regularized field theoretical model

C. Bervillier, C. Godrèche

To cite this version:

C. Bervillier, C. Godrèche. Critical exponents from a regularized field theoretical model. Journal de Physique, 1982, 43 (2), pp.243-250. �10.1051/jphys:01982004302024300�. �jpa-00209391�

Critical exponents ^from

^a

regularized field theoretical model

C. Bervillier

Service de Physique Théorique, ^Centre d’Etudes Nucléaires de Saclay, 91191 Gif sur Yvette Cedex, ^France

and C. Godrèche

Centre d’Etudes de Limeil, 94190 Villeneuve Saint Georges, ^France

(Reçu le 14 août 1981, accepte le 28 octobre 1981)

Résumé. ^- Nous utilisons une théorie des champs

(~2)2

régularisée pour déterminer les exposants critiques y ^{et v}

en dimension trois. La régularisation ^est réalisée par le caractère gaussien ^du propagateur. ^Cela permet de calculer aisément à tout ordre _{en g} toute fonction de corrélation, ^{mais introduit un deuxième} paramètre ^de développement.

Malgré les difficultés inhérentes à l’étude des séries doubles, ^{nous obtenons} des valeurs pour les exposants critiques

en bon accord avec celles obtenues à l’aide de la théorie renormalisée. Cela, ^{avec la} discussion _que nous exposons du lien entre la théorie renormalisée et notre modèle, illustre le concept d’universalité des phénomènes critiques.

Abstract ^- We use a regularized

(~2)2

^field theory ^{for the} determination of the critical exponents 03B3 and 03BD in three dimensions. The regularization ^{is done} by ^{a Gaussian} propagator. This enables any order in g for any ^corre- lation function to be easily calculated but introduces a second expansion parameter. ^In spite ^{of the} difficulties encountered when dealing ^{with double} series, ^{we obtain} values for the critical exponents ⁱⁿ good agreement ^with those found with the renormalized theory. This and the discussion we give of the link between the renormalized

theory ^{and our model} illustrate the concept ^of universality ^{in critical} phenomena.

Classification Physics ^Abstracts

02.60 ^- 64.70 ^- 11.10

Introduction and summary. ^- Since the

proposal

of Parisi

[1] in

1973, it has been

possible

^{to determine}

from the three-dimensional euclidean

cp4

field theo- ry

(~p3),

^{the critical} exponents [2-4] ^{and some uni-}

versal

quantities

associated to

leading

^and

subleading

critical

amplitudes [5, 6].

The methods used

rely

upon the calculation of the renormalization functions at a

given

^order

[7]

ⁱⁿ powers of the renormalized

coupling

constant u associated to the

~p4 term

of the Landau-

Ginsburg-Wilson (L.

^G. W.) Hamiltonian. The _quan- tities of interest for the critical behaviour ^are derived from these functions calculated at the fixed

point

^u*.

The use of the renormalized

~p4

^field

theory

^{for the}

study

^{of critical} behaviour is based upon ^a

long

^and

complicated

series of arguments

[8].

^{Let us sketch}

them as follows :

i)

^In

principle

^{the true} Hamiltonian has the most

general

^{form one can write}

(i.e. :

^not restricted to the L. G. W. Hamiltonian).

ii)

All the relevant and dimensioned parameters ⁱⁿ the critical domain are measured in terms of

only

one

length

^{~l -1 which} is smaller than _any other

relevant

length (~1-1

is of order of the intermolecular

distance).

iii) According

^{to the}

hypothesis

^of

universality

some characteristics of the critical behaviour do not

depend

^{on the}

length

^{~l -1.} This idea combined with

point ii)

^{leads to the}

study

of the limit A - oo

and to the use of the renormalized

theory.

^{In the}

renormalization _process, the contributions of _ope- rators of

higher

^dimension

(higher

^{than the dimension}

of

~p4)

^are not, a priori, ^{zero. It can be} shown, ^{at least}

in the framework of the

E-expansion (E = 4 - d),

that the _presence of such operators ^{does not desta-} bilize the _pure

(p 4fixed point.

^{However we do not}

know whether this is true at d ⁼ 3 (c

= 1).

We _use, in this _paper, a Hamiltonian

density

^which

contains such operators ^of

higher

dimensions,

namely

H(x) ^{reads :}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01982004302024300

244

d is the dimension of the space and x refers ^{to a}

point

of this _space.

If, ⁱⁿ

H(x) (Eq. (1)),

^one

^expands

^the

exponential,

one finds

formally

^{the L. G. W.}

density

Hamiltonian

plus

^{an infinite sum of terms of}

higher

dimensions.

This very

particular

choice for

H(x)

^{leads to a}

perturbative expansion

in powers

of g easily

^cal-

culable at any order or dimension,

provided

^{that one}

expands

^{also the}

theory

ⁱⁿ powers of t. Furthermore the

large

order behaviour of the series in _powers

of g

is known

[9]

^and

provides

^{us an}

opportunity

^of

using

the

sophisticated

methods of resummation

already

used in the determination of critical indices

[4].

Unfortunately, dealing

^with double series causes some trouble, ^{and the} way ^we have

incorporated

this information does not

improve

the result obtained

by

another method

previously

considered

[10].

^We

think that the

knowledge

^{of the}

large

orders beha- viour would be ^more

efficiently incorporated by using,

^for

example, partial

differential

approximants

[11].

In

spite

of the difficulties inherent in the resumma-

tion of double series, ^we find evidence for a critical behaviour similar to that

given by

the renormalized

~p3 theory.

^{We thus}

give

estimates for critical expo- nents which are in _very

good

agreement ^{with those} obtained from this latter

theory.

^The progress ^we have made since our

previous

^work

[10]

^concerns

several aspects : ^{first we extend our}

study

^{to the}

case of an

0(n) theory

^{for n = 0 to 3 at d = 3; next}

we have checked that the value of the

coupling

constant which we had selected is in agreement with the fixed

point

^value of the renormalized

theory;

finally

^{we discuss at more}

length

the calculational aspect ^{of our work and we}

publish

^our series for n ⁼ 1.

The article is divided in two main parts. ^{Part A is} concerned with a discussion of ^our model

given by equation (1) :

^{its link to} the renormalized

cp4 theory,

the

computation

^{of the}

perturbative

^{series. Part B} presents ^{the methods used to} calculate the critical exponents and the results.

PART A

1. The bare and the renormalized theory. ^-

Equa-

tion

(1)

represents ^a Hamiltonian

density

^{with an}

explicit

^and

particular dependence

^{on the}

length A-I.

In the

language

^{of field}

theory

^{it can} be considered

as a bare Hamiltonian,

leading

^{to a}

regularized ~4 theory.

^In the usual scheme

[8],

^{the limit ~1-~ oo}

plays

a central role. Let us consider

formally

this limit on

equation (1).

^{We find :}

According

^{to the}

general

^ideas

concerning

^{the use}

of field

theory

^{for the}

study

of critical

phenomena,

the parameter t appears ^{as a} linear measure of the temperature ^T and has the

value tc

^{at T =} Tc

( T~

^is

the critical

temperature).

Following

^{the standard}

procedure

^{of renorma-}

lization in field

theory [8],

^we define the renormalized

quantities : §5~

(we ^{take n = 1 for}

simplicity), m

^and

u

through

the relations :

In

equation (5)

^we have introduced a dimension- less bare

coupling

^constant uo, the dimension

of g being expressed by

the sole dimensioned parameter A which exists.

The functions

Zi(u), Z3(u)

^and

6Au)

^{which appear}

in

equations (3-5)

^{have to} be further detailed. In order to facilitate further numerical

comparisons

with other works

[2-4]

^we choose the

following

conditions on the renormalized vertex function

rRN)(pl~ ..., pN-1 ~ ~ u) ~

(i)

^We define the vertex functions

T ~N~({ p 1, nl, u, A) ({ p }

stands for _{p 11}

... I PN- 1)

^{as :}

with

where

T (N)(~ p ~ ; t,

^uo,

A)

^{are the bare vertex func-}

tions.

(ii)

^The renormalized vertex functions of the _pure

cp4 theory

^with subtraction conditions

given by equations (7-9)

^{are thus}

given by :

F(’)(I p 1; m, u)

^{= lim}

p m, u, A) . (10)

vi-~oo (m,u fixed)

As will be

explained

below, ^{the bare vertex functions}

related to our model are

given

^{as double series in}

powers of the dimensionless parameters t ^and uo.

Let us define the dimensionless functions

Fi(t, uo)

(i

⁼ 1, 2,

3)

^{as follows :}

From these definitions and

equations (6-9),

^{it is}

easy ^to deduce that :

Since we want to compare values of the

coupling

constant u of the renormalized

cp4 theory

with values of _uo, we take the limit ~1-~ oo at m and u fixed.

Equation (14) gives t,

^{in this} limit, ^{as a} function of _uo.

Thus

equation (15)

^{combined with}

equation

(14)

lead to the relation between u and _uo. This relation will be used to compare the value u* of the

cp4 theory,

estimated

by

Le Guillou and Zinn-Justin

[4]

^{to our}

estimate of

uó.

2. Calculation of the

perturbative

^{series. - We}

shall take, ^for

simplicity, n

^{= 1.}

The Feynman propagator associated to

equation (1),

is, ⁱⁿ p-space :

The numerical calculation of

perturbative series

with

equation (16)

^is difficult, ^{so we}

expand AF(p)

in _powers of t :

in which

Equation (17)

^{can be}

expressed diagrammatically

as follows :

With the propagator

equation (18),

^each

Feynman diagram

represents ^{a Gaussian}

integral easily

compu- table

using

^the

following identity :

In order to have a

systematics

^for calculations, ^we

use an

analogy

^{with the}

Schwinger parametric

repre- sentation [12] ^{of usual}

Feynman integrals.

^{Let us}

recall what this

representation

^{is :}

The usual Feynman propagator

(p2 + m2) -1

^is

written in the

integral representation :

For a

given

diagram ^{D with I} internal lines and L

independent loops,

^the

corresponding Feynman

^inte-

gral reads :

Q(a, { p })

^{reduces to zero} when all external momenta

{ p }

^{are taken}

equal

^{to zero.} Since that will be essen-

tially

^{the case} in this work we do not consider this

quantity

^further.

P(a)

^{is a}

polynomial

^of

degree

^L in the a’s defined

as :

{ T }

^{is the set of tree}

diagrams

contained in D.

For instance, ^the

diagram D, :

1

contains three trees :

mid

equation (23)

^{leads to :}

We

easily

^{see that the} propagator

(18)

^can be consider- ed as

being

^the

Schwinger parametric

form of the usual propagator ^{with no}

integration

^{on the a’s}

(a

⁼

1 ~~12)

and m ⁼ 0. As a consequence it is

straightforward,

in our model, ^to write the contribution

ID({

⁰

})

of _any

diagram :

Pp(a

⁼ 1) ^{reduces to} the number of trees contained in the

diagram

^D.

Using

^the

topological

^relations

(with

^the

~p4 inter- action)

^{for a}

diagram

^{D of order K with N external}

lines :

Equation (25)

^{reads now as :}

When

~’p(a)

is known it is _easy to determine the contributions which come from the

expansion

equa-

246

Table I. ^- Double power series in g and t

of

^the

function F 1 (g,

t) ^as

defined

ⁱⁿ equation (11)

for

^{d = 3 and n = 1.}

The rows

correspond

^{to the}

different

powers in g, the columns to the _{powers in} t. The numbers are written in the standard IBM notation,

for example :

0.1 D-0 I means 0.1 x 10 - 1.

Table II. ^- Double power series in g and t

of the function F2 (91

t) ^as

defined

^{in equation}

(12) for

^{d = 3 and n = 1} (same presentation ^{as in table} n.

Table III. ^- Double power series in g and t

of the function F3(g, t)

^as

defined

ⁱⁿ equation

(13) for

^{d = 3 and n = 1} (same presentation ^{as in table} I).

tions

(17,

19). ^{The set} of values for the a’s will differ from

unity according

^to the number of crosses on

the

corresponding

internal line.

In

pratice,

^{we have} calculated for d ⁼ 3 the _quan- tities

Fk(t, uo) (k

^{= 1, 2, 3 ; see} definitions in _equa- tions

(11-13))

^{up to :}

- five orders in _powers

of g

^and twenty ^{three in} powers of t for

F1

1 and F2

(Cf-

^{tables I and II for the}

results in the case n ⁼ 1).

- four orders in _powers

of g

^and twenty ⁱⁿ power of t for

F3

(cf ^{table III} for the results in the case

n=1).

The

coupling

constant 9 ^is dimensionless and related to g and uo ^{as :}

PART B

l.

Computation

of critical

exponents

y and ^{v as} power series in _g. ^- We shall restrict the discussion to the inverse of the two

point-correlation

^function

r ~~~(0, 1, g).

If the action of the operator ^of

higher

^dimensions

than

~4

^{can be}

neglected

^{in the}

vicinity

of the criti- cal

point,

^{as it is}

supposed

when renormalized

9’3 theory

^is used, ^we expect ^that

The task is to

determine t~, g*

^{and y from the truncat-}

ed double series :

and to compare the values obtained for

g*

^and y to the

corresponding quantities

^{u* and y of the renor-}

malized

theory [2-4].

As in our

previous publication [10],

^{we choose the}

following

^{form for}

T~2~(0, t, g) :

in order to determine

t,,(g)

^and

y(g) by

^the

Dlog

^Pade

approximants

method. Of ^course

equation (32)

^is

expected

^to coincide with

equation (30) only

^for

g ⁼

g*.

^{As a} consequence, ^we

hope

^{that the functions}

t~(g)

^and

y(g)

obtained from

equation (32)

^{will have}

a

particular

^behaviour for g ^near

g*.

Since we know the

large

^K behaviour of _axL

[9],

the natural _way would be to resum first

in g

^{the series}

equation

(31). ^{This would}

give

^a

simple

^{series in}

powers of t :

from which we could extract

t,,(g)

^and y(g)

by

^the

Dlog

Pade method mentioned above.

Unfortunately,

^{a lack of terms} in the series in _powers

of g

^{does not}

permit

^a

good enough

accuracy ^on the coefficients

aL( g)

^{and Pade}

approximants

^introduce

instabilities which

spoil

the result. So we postpone the

g-resummation

^{and we} handle series in _powers

of 9

instead of numbers. More

precisely,

^{the mean-}

ing

^of

aL(g)

^{will be}

We have thus considered the Pade

approximants [L - j, j] ( j

⁼ 1, 2, 3...) ^on the series in _powers of t obtained from the

logarithmic

derivative with respect

to t of

equation (31).

We obtained

t,,(g)

^and

y(g)

as series in _powers

of g depending

^{on two} parameters Lmax

and j.

^We shall write the series for

y( g)

^{as follows :}

A

g-resummation

^on these series now has to be

performed.

Our first idea was to use the resummation method

developed by

^{Le Guillou and} Zinn-Justin

[4]

which uses information ^on the

high

orders behaviour of the series. But as

already

^{mentioned we have to}

use it on the series

equation

(34). ^A priori, the coeffi- cient

TK(Liiiax5 j)

have, ^{at fixed} Lmax’ ^a behaviour for

large

^{K similar to} that of the

aKL’s.

^In

particular

^the

nearest

singularity

of the Borel transformed series would be located at the same

place.

^{Thus the resum-}

mation method of Le Guillou and Zinn-Justin can

be used at least in its

simplest

form. The

problem

comes from the fact that we have no theoretical infor- mation about the behaviour of the resummed _quan-

tity

^when Lmax increases. In the

following

^{section we}

shall see that this parameter

plays

^a central role.

Nevertheless we have

applied

this method on the series

equation

(34) and found results [13]

roughly

in agreement with those

presented

^below.

2. Resummation of _power series for critical exponents.

- The method used to resum series of the type ^of

equation

(34) ^to obtain estimates of the critical _expo- nents is based on the work of

Fogli,

Pellicoro and Villani

[14].

^{In a}

previous

paper

[10]

^we gave the results for the case n ⁼ 1 obtained

by

^{this method.}

Let us recall the basis of it.

If a function

F(g)

^{is known to be lim}

F(g, c),

^and F(g, 8)

given by

^a

Stieljes

^{series : £-0}

in which 8 is a parameter ^{such that}

248

then the [N, ^{N -} 1 ] (g, s) ^Pade

approximant

^has

F(g,

8) ^{as a} limit when N goes ^to

infinity :

Fogli et

^al.

[12]

^have

proved

^{that :}

where

EN(g)

is the maximum

(at

^{fixed N and}

g)

^of

[N,

^{N -} 1]

(g,

E) (see

Fig.

1).

Similarly,

^the

diagonal

^Pade

approximants

[N,

N] ] (g, c)

^{lead to :}

where, this time,

EN(g)

is the inflexion

point

^of

[N, N]

(g,

8)

(see Fig. 1).

Fig. ^{1. -} Qualitative ^{behaviour of} diagonal ^{and sub-} diagonal ^Pad6 approximants ^{as e varies.}

This

technique

^is

particularly

^well

adapted

^{to our}

case since we know that

t~(g)

^and

y(g)

^{have no}

Taylor expansion

around g ^{= 0}

[15].

This means that the coefficients

TK(L,,,.,, j)

^{of the}

expansion equation

(34) go ^to

infinity

^as Lmax grows

for j

^{and K} fixed. This

1/L,~

^is

equivalent

^{to e.}

Considering

^{the number of terms that we have}

calculated for the

perturbation

^{series in} g, ^{we can use} the

following approximants :

^[1,1],

[2,1], [2,2]

^and

[3,2]

^{for the series}

equation (34).

^{We thus} expect ^that

they

^are

organized

^{as shown in}

figure

1 with the

replace-

ment : 8 ^-~

11 Lmax; F(g) ~ Y(g); F(g, s)

^~

y~j(~).

Hence the parameter Lmax ^{will be fixed}

by taking

^the

value which

corresponds

^to the maximum

(or

^the

inflexion

point)

^{of the}

appropriate

^Pade

approximant.

We shall discuss the results and the choice of the

parameter j

^{in the next section.}

First we shall consider the determination of the fixed

point g*.

3. The fixed

point

value. - In reference

[10]

^we

gave ^a criterion for the choice of

g*, namely :

^the

value

of g

for which the inflexion

points

^{are the flat-}

test. We found two

possible

^values

g*Y

^{= 0.343 or}

g*v

^{= 0.372}

respectively

^{for the} exponents y and v.

With

equations

(14, 15) ^{we can} compare these values with those of the renormalized fixed

point

of Le Guil- lou and Zinn-Justin

[4].

^These authors have consider- ed the

following coupling

^{constant :}

and found

(d

⁼ 3, n ⁼ 1) :

Equations (14,

^{15) lead to a relation}

VLmax(g)

^between

g(=- uo/(4 n)d/2)

^{and v which}

depends

^on Lma}( ^{As for}

the series

Y Lmax ,j( g), V Lmax ( g)

^{goes to + oo as Lmax}

goes ^to

infinity.

The result obtained

by

resummation of these series shows the best agreement ^with equa- tion

(41)

for g around 0.37.

Figure

^{2 shows the resum-}

Fig. ^{2. -} Values of the renormalized coupling ^constant

as Lmax ^varies for g ⁼ 0.37. The symbols ^{*, o, A} represent respectively the orders 2, 3, ^{4. The} horizontal hatched

area corresponds ^to the estimate of v* of Le Guillou and Zinn-Justin (Ref. [4]).

mation of the three orders

(in

powers

of g)

^of

vLmax(g)

as Lmax ^varies

(for g

⁼ 0.37). This value indicates that

our

previous

determination of

g*

^was reasonable and that the value of

g*Y

^{seems to be too small.}

With some more order we would _very

likely

^obtain

a value between these two.

4. Results for the exponents ^and conclusion. ^- This section is devoted to a

presentation

^{and a discussion}

of three

figures

^{for the critical} exponent y ^{at n =} 1.

Those

figures complete

^those

published

^{in refe-}

rence

[10].

^The

striking point

is that the situation is identical for the other values of n.

Fig. ^{3. -} Behaviour of diagonal ^and subdiagonal ^Pade approximants ^as 1/ Lmax varies for the series

7L-..,j(g) at

g ⁼ 0.37. a) j ^{= 2;} b) j = 3. ^The symbols ^{*, 0, A} represent

respectively the orders 2, 3, ⁴ in powers of g.

For n ⁼ 1, ^{we have}

fixedg

^{to the} value 0.37.

Figures

³

show the behaviour of the Pade

approximants

^for

7~,j

for j

^{= 2}

(Fig.

3a)

and j

^{= 3}

(Fig.

3b). ^{The last}

order does not have a very flat inflexion

point

^since

g is not at the value 0.343. As indicated in reference

[10],

v has a flat inflexion

point

^{for g = 0.37.}

In

figures

^{3 we see that} as j grows, the number of

points

^which

belong

^{to the inflexion}

region

^decreases.

But this

region

stays ^{around the same value of y.}

We have chosen for _y, the lower value of the inflexion

region. If g

grows the curve has a maximum followed

by

^{a minimum more and more} distinctive. We report, in

figure

4, ^the variation of the last order as g is varied.

This

figure

is identical to

figure

³ of reference

[10]

but its scale is

larger.

Fig. ^{4. - Behaviour} of the last order of

YLmax~~~g~

^as

Lmax ^varies for j ⁼ 2, at g ⁼ 0.24, 0.343, ^0.44.

""’

Considering

the small number of orders in _powers

of g

^{that we deal} with, ^{we cannot} say

that g

^{= 0.37}

is out of the critical

region

^{for our model. In refe-}

rence

[10]

^{we used this}

uncertainty

^to

give

^{an effec-}

tive scale for the.error of the estimation of the critical exponent y and v

(for n

⁼ 1). ^{In this} paper ^{we want} to illustrate that our model does not lead to a contra- diction with the renormalized field

theory

^frame- work, ^{so we do not indicate error} bars for the values of the critical exponents. ^{We have}

simply looked,

for each value of _{n, at} the inflexion

points,

compared

the values of the

corresponding g*

^{to the} renormalized values

given by

^{Le Guillou} and Zinn-Justin and indicated

(Table

IV) ^the

corresponding

critical expo- nents values. The situation is identical whatever the value of n. This leads to the values

given

^{in table} IV,

the agreement

being

very

good

with the results of Le Guillou and Zinn-Justin.

Let us say, ^{as a} conclusion, ^{that we have not found}

evidence for a destabilization of the

cp4

^fixed

point

anyway. The two

possible

^fixed

points

(0.34 ^and

0.37)

indicate a lack of terms in our work rather than a

contradiction with the standard field

theory approach.

Furthermore, ^we think that our values indicated in table IV are upper bounds in our model, ^{since the}

true value for

g*

^{seems to} be, very

likely,

^{smaller than}

0.37 for n ⁼ 1. We know that the estimate for y(n = 1)

250

Table IV. ^-

Comparison

between the values obtained in this

work for

^the exponents y and ^v with these given

by

^the

renormalized

~p4 field

theory (Renorm.

Group).

^The parameter ^{n is} the number

of

components

of the field.

obtained

by

the standard

high

temperature

(HT)

series

analysis,

^was around 1.250. We do not think that this value can be reached

by

^{our model. More}

recent

analyses [16],

^{based on}

longer

^HT series,

lead to an estimate for

y(n

⁼

1)

around 1.2440. The present work does not

give

^any argument ^{for the}

breakdown of the field theoretical

approach

^to

critical

phenomena.

Acknowledgments.

^{- We want to} thank J. Zinn- Justin for his continuous interest in our work.

References

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39

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