On the Large Order Behaviour of the Potts Model

(1)

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On the Large Order Behaviour of the Potts Model

Domenico Maria Carlucci

To cite this version:

Domenico Maria Carlucci. On the Large Order Behaviour of the Potts Model. Journal de Physique

I, EDP Sciences, 1996, 6 (8), pp.983-992. �10.1051/jp1:1996111�. �jpa-00247235�

(2)

J.

Phys.

I £Fonce 6

(1996)

983-992 AUGUST1996, PAGE 983

On the Large Order Behaviour of the Potts Model

Domenico Maria Carlucci

(*)

Scuola Normale

Superiore,

Piazza dei Cavalieri 7, 56100

Pisa, Italy

(Received

8 March 1996, revised 22

April

1996,

accepted

2

May1996)

PACS.05.20.-y

Statistical mechanics PACS.11.10.z Field

theory

PACS.11.10.Jj Asymptotic problems

and

properties

Abstract.

Following

^thiwork

by Houghton,

Reeve and Wallace about _an alternative for-

mulation of the _n _- 0 limit of the

In +1)

state Potts model in field

theory

for the

large

order behaviour of the

perturbative

expansion, _we

generalise

their

technique

to all

n

by establishing

_an

equivalence

in

perturbation theory

order

by

order with another bosonic field

theory. Restricting

ourselves to _a cubic

interaction,

_we obtain _an

explicit

expression

(in

terms of

n)

for the

large

partition

function.

R4sumk. Suivant le travail de

Houghton,

Reeve et Wallace concemant une formulation

alternative de la limite _n

- 0 du modAle de Potts _en thdorie des

champs

_pour le comportement

aux

grands ordres,

_nous

g6n6raIisons

leur

technique

h _n

quelconque

_en 6tablissant _une

6quivaIence

ordre par ordre _en thdorie de

perturbation

_avec _une autre thdorie de

champs

de bosons. En _nous

restreignant

h _une interaction

cubique,

_nous obtenons _une

expression explicite (en

fonction de

n)

_pour le comportement _aux

grands

ordres de la fonction de

partition.

Introduction

The Potts model _{[1] is} _one of the most

fascinating topics

in statistical mechanics. It _can be

regarded

_as the

generalisation

^of the

Ising

model [2] ^to ^a

generic

^number _q ^of

components.

Originally

this

problem

_was

proposed

to Potts

by

his PhD Professor Domb

[3],

^who

suggested

him to

investigate

the critical

properties

^of a

system

^of

interacting spins

confined in _a

plane, pointing

in _q

equally spaced

directions

Hi

₌ ^~" i

=

o, I,..

,q I

ii

q

with _an Hamiltonian

depending only

_on the relative orientation of _two

neighbouring spins

of the form

7i =

~j _J(81, _8j) _J(e~, _8j)

c~

cos(81 8j) (2)

lsPinsl

(* ^e-mail: carluccitlux2sns.sns.it

Q

Les

(ditions

_de

Physique1996

(3)

Actually,

the

planar

Potts model.

Nevertheless,

the most famous model is the _one where two

spins

interact

only

when

they

_are

present

ⁱⁿ ^the same

state,

that is

J(81,8j)

_cc

b([l~,. ⁽³⁾

The interaction

(3)

_can be re-written in such _a way as to reflect _a

full-symmetry

ⁱⁿ _a

(q11)-

dimensional space

by making

use of _q vectors

(q-1)-dimensional

e°

pointing

in the _q

symmetric

directions of _a

hypertetraedron

ⁱⁿ

(q -1)

dimensions.

Indeed,

the interaction

(3)

can be _re- written _as

b(81,8j)

=

[1

+

(q -1)e~~ _e~3). ₍₄₎

q

The

increasing

attention to the Potts model _was due to the fact that it has _proven to be related to a

large

class of

outstanding problems, especially

when

particular

values of _q, _even _non

integer,

are taken. For

example,

the

percolation problem

_can be formulated in terms if _q

= 1 Potts model

[4-6].

^Also Fortuin and

Kasteleyn

_[7] showed that the Kirchhoff's solution [8] for _an ensemble of resistors is

strictly

related to q = o limit of the Potts model.

Moreover,

the critical

behaviour of the dilute

spin glasses

is well understood in terms of the _q ₌

)

limit

[9,10]. Finally, Lubensky

and Isaacon

[11]

^showed ^that

interesting

_processes in

gelation

and vulcanization fall in the _same class of

universality

of o < q ^< 1 Potts model

(For

_a

complete

review of the Potts

model _see Wu [12]

).

A convenient formulation of Potts model _as _a field

theory

_was first

performed by

Zia and Wallace

[13],

^who

investigated

the critical

properties

in the framework of the renormalization group.

Actually

in the field

theory

literature it is

customary

to refer to

in +1)

Potts model

in

+1 ₌ _q in the

previous notation).

The

symmetry

^of the model allows _a trilinear interaction which is the

leading

term in the renormalization _group framework. Because of _a _pure cubic

interaction,

the

high

perturbative expansion

is

clearly non-oscillating

and therefore _non Borel-sommable.

Nevertheless,

when

particular

limits _are taken

(e.g.

_n _- o

percolation problem)

the

large

order behaviour is found to be

oscillating

and Borel-sommable [14] as obtained

by Houghton,

Reeve and Wallace

(from

_now _on referred _as

HRW).

In Section

1,

we recall the Hamiltonian of the

in +1)

Potts model and the alternative

approach

to n = o limit used

by Houghton,

Reeve and Wallace. In Section 2 its

generalisation

to each value of _n is showed. Section 3 will be devoted _to the evaluation of the

large

order behaviour for such _an

alternative field

theory

and the results will be shown to agree with the

In +1)-State

Potts Model

We consider the

in +1)

state Potts model in d

dimension,

described

by

_an Hamiltonian

[13]

of the form

7i

[#]

=

/ _d~x

(T7#~ (T7#~ +

ii11

₊ ^, _{p~jk 4~}

4j 4k (5)

2

2 3.

where _sum _over

repeated

indices is

implied

and _runs from to _n. The

system

_possesses an

hypertetrahedric symmetry, provided

that p~k is of the form

n+i

p~k "

~ _e)e)e( ₍₆₎

o=1

where the e's

satisfy

n+I n+I n+I

~j

_ES

= o

~j _e$ej

=

In

₊

1)b~ ~j _e$ef

=

In

₊

1)b°~

1

ii)

a=I a=1 ~=l

(4)

N°8 ON THE LARGE ORDER BEHAVIOUR OF THE POTTS MODEL 985

Physical quantities

can be extracted from the

partition

^function

(the

discussion about the

right

contour in the

path integral

for _a _pure cubic

theory

_can be found in

[15] expanded

in powers of _g:

~n(9)

"

/ _l~~e

~

"

i~ _~K(n)9~~ ₍₈₎

~,_~

We _are interested in the

asymptotic

behaviour of

ZA-(n)

as K goes ^to

infinity.

We _can extract

ZK In)

from

(8) by

the usual

dispersion

relation

[16,17]

ZK(n)

=

/ _Vi ) ^~ _dg~ )e~~l"1 ₍₉₎

m g

where the closed

integral

is

performed

in the cut

complex plane g~.

In the

large

K

limit,

the

previous integral

_can be evaluated

by

the

steepest

^descent ^method ⁱⁿ ^the space

Iii, g)

and the

saddle

point equations

_are

T7~#1 ₌

e~k4j4k (10)

2

Equations (lo)

and

(11)

_can be

decoupled easily by setting

with the

~ti's satisfying

lli " fijklljllk.

(13)

Equation (13)

has _n solutions ~t(~~, with _r

=

1, 2,..

_,n,

r

(r) jr) _a

j~~)

~ " a §

~

i=1

where

a(r)

=

j15)

(n

+

I)(n

₊ 1

2r) Working

in d

= 6

dimension,

^the ^saddle

point equations (lo, it)

then read

V~4clx)

=

411x) l16)

~

~j~ti~ti / _d~x#)(x)

=

-Kg~ Iii)

3

whose

general

solution is

~~~~~

[12(x

~~~~

+ 1]~

~~~~

where I and _To _are constant

parameters.

(5)

From

equations (17, 18),

we obtain

~~

~~~ /K _I$~ ~~l

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

~ ~, 5

~

~ ~ '~

j20)

3 28~3

minjrj (£~

ll)~~ll)~~

For the

in

+

I)

state Potts model

In

_>

integer)

the minimum in

(20)

is for _r

=

I, giving,

as

expected

for _a _pure cubic

theory,

_a

malign

behaviour for the coefficients

ZK

which do not oscillate

[15, 16]

~~~~~

'~ ~~

.n

~~~~

For _n

= o

(corresponding

to

percolation problem)

the saddle

point

cannot be obtained

by

_a

naive

analytic

continuation of

(20)

because

evidently

_r

= I is not the

dominating

saddle

point.

HRW [14] found _an alternative formulation for the

percolation problem

and

they pointed

out that in the _n _- o limit the

Feynman

rules _are the _same _as those of the Hamiltonian with two scalar fields

~i14,#,9)

_"

/ _d~x

I)jvi)~ ^jjv~l)~

⁺

jji~ ~l~)

+

j

g

ii

+

l)~j j22)

provided

that ~ one

keeps only j-connected diagrams

as those shown in

Figure

1.

By

_means of

path integrals techniques, they

found the coefficients of the

expansion

of the

two-point

correlation function

Gix

Y) -

(~ ^GK ^9~~ ^GK

^+~ ^K<

Al

^~

123)

which

fortunately

^show _an

oscillating

behaviour.

2. HRW Method for

Arbitrary

_n

As mentioned in the

introduction,

it would be useful to

generalise

the

equivalent

Hamiltonian in

analogy

with

(22).

In order to

gain

_some

insight,

let _us compute ^the

two-point

correlation function _up to fourth order

(~extPijk~i~j~kPlmn~l~m~nPabc~a~b~cPdef~d~e~f~ext) (24) by performing

all the

possible

Wick contractions. Because of the

properties (7),

_we _can

get

terms as

L

n+I

_~l _lli~

₊

i)b~~ ii _lli~

+

I)b~~ ii

_lli~ ₊

I)b~~ ii

a,#n,6=1

x

in

+

I)b~~ lliin

₊

I)b~~ _II f[ ₁₂₅₎

Inspired by

the

previous discussion,

_we

try

to

interpret (n +1)b°~

and

(-1)

_as

propagators

of two different

(possibly multi-component)

scalar

fields, #

and

# respectively.

The several

(6)

,- ,,

, ,

#' "

# I

la) 16)

,'h

,' ',

~'

ⁱ

I '',

',

ic)

Fig.

1. Some

diagrams

built

by

_means _{of ifi}

(solid line)

and ifi

(dotted line)

propagators.

Only

the

diagrams la)

and 16) are suitable because # connected.

terms _we obtain

by expanding

the

products

ⁱⁿ

(25) correspond

in the second

theory

to all the different

diagrams

with the _same

topology

_as the

original

_ones. As in the _n _- o _case

only diagrams

with external

# legs

connected

by

_a

j's path

do contribute.

In

Figure

1 we have drawn three of the

possible graph

built

by #

and ifi

propagators. By requiring

that the

diagrams

in the _two theories have the _same numerical

coefficients,

_we obtain the

Feynman

rules.

Specifically,

from the _term

involving only #

^fields

(Fig. 1a)

_we

get

n+1 n+1

£ _{El in}

₊

1)5b°~b°~bbb~~ e)

=

in

₊

₁₎₅ £ _e)e) ₍₂₆₎

a,#,~,6"1 a=1

In these _case, the _presence of b's eliminate the _sum _over the four indices. If _we associate _a factor

in +1)

to each

# propagator,

_we obtain the _same results without

introducing

_any

weight

for the vertices.

Let _us _now consider the second

graph (see Fig. lb)

n+I n+I

L _~l

_li~ ₊

i)~bbbbl~i) _~)

"

l~i)li~

+

1)~ L _~l~) ₁₂₇₎

a,#,~,6 ^a=1

as we

said,

we associated _a factor

(-1)

for the ifi

propagator

^and we do not need _any factor for the vertex

#2#.

The situation is

quite

different when _we deal with

graphs

where _we have _a

#3

_vertex _as in

Figure

1c which derives from

n+I n+I

n+I

l~i)l~i)l~i) ~ _~l

_li~ ₊

i)~bb~)

"

l~i)

_li~

+1)~ L L l~)~

128)

a,#,~,6=1 ~=l a=1

We note that _we _are left with _a _sum _over _~ which

gives

_a factor

in +1).

So

a1fi3

vertex

requires

_a factor of

in

+

1).

At

higher

order _we also have _non-zero

diagrams

with also

#

connected

sub-graphs

_as shown in

Figure

2.

(7)

,,--,,~

,' '~ l' ',

' , ' ,

# ' I

Fig.

2.

High

order

diagrams

with

a # connected

subgraph.

The filled

region

_means _any _non-zero

diagram.

It _can be rather

simply

_proven that each

j-connected subgraph gives

a factor of

in +1).

Summarising

the

Feynman

rules for

in +1) _component

Potts model _are

1) Only j-connected graphs give

_a non zero contribution.

2)

Each

# propagator gives

a factor of

in

+

1).

3)

Each ifi

propagator gives

a factor of

(-1).

4)

Each

ifi~vertex gives

_a factor of

in

₊

1).

5)

^Each

j-connected subgraph gives

_a factor of

in

₊

1).

These

Feynman

rules

correspond

to the

following

Hamiltonian

i~ m

/ _d6z lit jjiv<~)2

₊

iii)

~i

~)lV~)~

⁺

^-~fi~)

⁺

) ^~~ ~ _(la

₊

~fi)~l129)

By following exactly

the _same

procedure

_as HRW'S in order to extract the

j-connected diagrams,

we

get

^for the

two-point

correlation function

/ _Vi _#(x)#(y) exp[-7i*(#)] ^e~~"

~~~

~~~~~~

/ _ ^~~

^~~~

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

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

~

) _~ ~j

~~~~

where

7i*(#)

is the reduced Hamiltonian

7~*ii,

_~fi)

=

/dx6 )j _{(iv«2 +12)} jj _(iv<)2

+ ~fi2) ⁺

j _ii

₊

~fi)~

131)

and

W*(ifi)

is defined

by

exp

iw*tiff)i

₌

/ _vi

exp -7i*

ii,

_ifi)

₁₃₂₎

(8)

3.

High

Order Behaviour

~s in the _n _- o case, it _can be shown that

only

the second term _on the left hand side of

(30)

contributes to the

high

perturbative expansion

^because ^{of the}

particular

form of the cubic interaction and _we have to evaluate

/ _p~pj(i _j~ _j(2) _j(i) _j~) _~(2)

~)

^l

)

_~

~2

1

21ri

g2K

for large K.

Our

goal

~V~#1

"

)9

(ifi +

Ii

_)~ i

"

1,

2

(34)

~

T7~4i + g(ifi +

Ii

_)~ ₊ _g_(ifi ₊

d2)~

₊

Ii n)

^~~~ ₌ o

(35)

il + 1 2 2

3~l

j / _d~x _j~

+

~i

_)~ +

lift

+

42)~

+

II

_~l)

~)

₊ ^~~

" °

136)

It is

simple

to evaluate the

partial

derivative of W* with

respect

to g and

# by steepest

descent method _too and _we obtain

and

~~~

=

/d~x(ifi

⁺

^o)~) ^Ii

⁺

^O(g~))

fig

3'

provided

that

lo obeys

to the

following equation

fi~~'°

=

j9140

⁺

^~)2

After the

re-scaling

#~ - 4l~ = #~ i

=

o, 1,

² 9

1 ifi ~

~Y"~ifi

the saddle

point equation

_are

finally

Ah = (Ah +

ill)~

i

=

0,1,

2

(37)

n + 1 2

T7~fll + _(Ah +

ill)~

+

(4l2

+

ill)~

^~~ ^~~

(4lo

+

ill)~

= o

(38)

(9)

2Kg2

=

j _/d6x

j14~i

⁺

^4/)~ +14~~

+

4/)~ ii -n)14~o

+

4/)~j ¹³⁹⁾

In order to solve the

equations (37, 38, 39),

let _us make the ansatz

4~i =

~liicix)

~P =

uicix) ₁₄₀₎

where

#c(x)

is the solution of

(16). Therefore,

_we have reduced _our differential

equations

to

algebraic

_ones.

(n 1)

^~~ ^~~ ^~ ^~~~~

~ ~ ~ ~

(1 n) In

+

1)

^~

In

+

1)

^~

In

+

1)

^~°

3, ~6y~3

Kg~

=

[(u

+ ~t1)~ +

Iv

+ ~t2)~ +

In I)(u

₊

_~to)~j

5

Solving

the first

equation gives

two solution

~~~~

l~ _in _l)

^~

_in _l)

^~~

^~~~~

^~ ^~~~^~~~ ^~~~~

Only

four choices _are

possible 1)

~~ =

~l+)

~~ = ~~ =

~l-) 2)

_~ti ₌ ^~t~~) _~t2 _" _~to _"

^~t(+)

4)

_~to _"

^~t(~)

~ti = ~t2 "

~t(+)

From the

1)

^and

2)

_we obtain _two

equivalent

solutions

provided

that _n

#

1

~

in

₊

~~n

1)2

^~

~

_~~~~

The choices

3)

^and

4) gives

other _two

equivalent

solutions for _u when _n

#

3

~ "

j~

/(

j~

_~~~ ⁿ

^~

³

^j4~~

The

give

two saddle

points

for

g2

~ 1 3 231r~ _n

~

~ j44)

~(~) _K ₅

in

+

1)2 In _1)~

and

~ ~

9~2) ^"

(

^~ ^~

j~ /)) _j/~

~)2

^~

~

(~~)

A this

point,

it is

interesting

to note that the saddle

points

g(~~ and

g(~~ ^are

respectively

the

r = 1 and the _r

= 2

points

^of

(18). Anyway,

_now, _our final aim is _to evaluate the

large

perturbative expansion,

that is the coefficients

ZK

in the

expression (8).

^So

(10)

a little _care needs _to understand which saddle

point

dominates _at _n

fixed.

We _can obtain the two

following

behaviours

and

~

~~~

'~

3

91r3

~~ ~

~~ ~)

_~~~

~

~ ~~~~

For _n

= o

(percolation)

the second saddle

point

dominate _over the first _one and _we _recover the HRW results.

Actually

there exist other formulations for the

percolation problem disagreeing

with HRW

[18],

^but ^it can be shown

[19]

^that ^the ^different ^result was due _to _an

unappropriate

choice of _an

integration

contour.

Nevertheless,

the _nature of the

percolation

behaviour still

remains controversial because there is another result

[20]

^where ^the r = saddle

point

_seems to dominate. For _n

=

(Ising model)

_we cannot choose the coefficient

Z(~

and

Z(~

is

identically

zero

according

to the fact that the trilinear interaction vanishes _as _n

= 1. For _n

integer

_we have _to take into account the

(46)

and this is in

agreement

^with ^the calculations

performed

in

Section 1.

Conclusions

We

performed

the

computation

of the

large

perturbative expansion

of the

In +1)

state Potts model with _a cubic interaction for _a

generic

value of _n,

by mapping

the

original problem

in _a

simpler

_one. We showed that _our

equivalent

field

theory

has _two saddle

points depending

_on _n and _one has _to choose the smallest _one in order to extract the dominant term at

large

order. A natural

question

can arise

looking

at the different number of the saddle

points

in both the formulations. Indeed in Section 1 it has been shown that the Potts

model,

when studied

by

the usual

techniques,

has _n saddle

points

characterized

by

_an

integer

r, whereas _our alternative formulation

gives only

two

inequivalent

saddle

points. Indeed,

the

large

order behaviour is related _to the cut

discontinuity

of the correlation function for small value of the

coupling

constant, that is the

imaginary part

^of ^the Green functions

generated by

the

Taylor expansions

^around ^the ^saddle

point.

For _n

=

2,3,..

the

imaginary part

^of ^the correlation functions is dominated

by

the _r

= 1 saddle

point.

When _an

analytic

continuation in _n is

performed

and _some

particular

limit _are

taken,

the _r

= 2 saddle

point

becomes the

leading

term. All the other saddle

points

_are

sub-leading

and

they

do not affect the

large

order behaviour. This

generalisation

can

help

to

compute

^critical

exponents

^for

theory

where _n is not

integer

_as those mentioned in the introduction

eventually by adding higher

order interactions.

Acknowledgments

It is _a

great pleasure

to thank

Giorgio

Parisi for

having suggested

and

supervised

this work and G.F. Bonini for scientific discussions.

(11)

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