Crossover and scaling in one dimension

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Crossover and scaling in one dimension

S. Barišić, K. Uzelac

To cite this version:

S. Barišić, K. Uzelac. Crossover and scaling in one dimension. Journal de Physique, 1975, 36 (12),

pp.1267-1271. �10.1051/jphys:0197500360120126700�. �jpa-00208373�

CROSSOVER AND SCALING IN ONE DIMENSION

S.

BARI0160l0106

and K. UZELAC

Institute of

Physics

^{of the}

University Zagreb, ^Croatia, Yugoslavia (Reçu

^{le 12 mai}

1975, accepté

^{le 4 août}

1975)

Résumé. ²⁰¹⁴ Nous étudions un système de chaines faiblement

couplées,

décrites par ^un modèle

Ginsburg-Landau. ^{Nous montrons} que l’indice de crossover ~ ^est égal à l’indice 03C8 qui ^{décrit la} dépendance ^{de la} température critique ^au couplage interchaine. La valeur de ces indices est deux pour

n > 1. Nos résultats sont obtenus en utilisant l’analogie ^du problème Ginsburg-Landau ^au problème

des oscillateurs anharmoniques faiblement couplés.

Abstract. ²⁰¹⁴ The crossover and the scaling ^{laws are} derived for the n ~ ¹ Ginzburg-Landau ^model

of weakly coupled linear chains. The results are obtained through ^the analogy ^{with the} weakly coupled

anharmonic oscillators, ^the spectrum ^{of which} obeys ^a homogeneity relation of the Widom type.

For n > 1, ^{the crossover index} ~ and the transition temperature index 03C8 satisfy ^~ = 03C8 ^{= 2.}

Classification Physics ^Abstracts

1.680

1. Introduction. ^- The

purely

one-dimensional systems ^{with an} n-component ^{vector order}

parameter

and short _range forces do not exhibit a

phase

^transition

at finite

temperature unless [1] n >

^{1. The} introduction of a weak

coupling

among linear chains shifts the critical

temperature Tc

^{from zero to a finite value.}

The aim of the

present

^{work is to examine the corres-}

ponding

^laws

using

^the

Ginzburg-Landau (G-L)

functional

approach.

There are a number of works based on G-L

[2]

or classical

Heisenberg [3]

^and

Ising [4] models,

which have examined this

problem

^{in various}

approxi-

mations. In the two most recent papers, which concern

the G-L n ⁼ 1

[5] and/or [5, 6] n

^{= 2} cases, the inter- chain

coupling

^{was treated}

through

^{a mean-field}

approximation.

^The present ^{method is based on the}

exact

homogeneity properties

^{of the}

thermodynamic

functions and

yields

^{the exact value of the crossover}

index.

Contrary

^{to the}

conjecture

of reference

[5]

this value does not differ from its mean-field deter- mination.

T,

of reference

[5]

^{exhibits the exact}

scaling properties, provided

^{that the}

temperature

^{scale is not} fixed

by

^the

single

^chain mean-field transition

tempe-

rature

To,

^but

by

^the characteristic temperature

Tb

which involves the anharmonic

coupling

^{of one-}

dimensional fluctuations. The

discrepancy

^{can be}

traced back to some minor errors in reference

[5].

After correction the mean-field

expression

^for

Tc

^is

consistent with the exact

homogeneity

^relations.

The

homogeneity

^{of the relevant}

thermodynamic

functions is derived here from the

homogeneity properties

^{of the}

equivalent

anharmonic oscillator Hamiltonian. To our

knowledge,

^{this is the first}

time that such laws are obtained with no reference

to the renormalization _group

approach.

Our results

represent

^{therefore a}

possible

^{check on that more}

general

^method

[7].

2. Général. ^- We consider a set of

weakly coupled

one-dimensional

chains, represented by

^{the free}

energy functional in the G-L form :

The chains are labeled

by

^{the index i. The summation}

over ô is taken over

only

the first

neighbours,

^i.e.

we

keep only

^one interchain interaction constant À.

The other coefficients have their usual

meaning.

(x)

^{is a}

n-component

classical vector, i.e. ^the quantum fluctuations are

neglected.

^We also notice that the functional

(1)

is associated with both the infinite

longitudinal

cut-off and the finite transversal cut-offs. This type of cut-off asymmetry ^{is unim-}

portant [8]

^{for the crossover}

problem

^considered

below.

As

argued previously

^{in the n = 1 case}

[2],

^the

G-L functional is associated

by

^the

analogy x H

^it

with a quantum mechanical

problem

^defined

by

^the

Hamiltonian.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197500360120126700

1268

The

length

xo is exhibited

explicitly

^{in order to stress}

that the Planck constant of the

equivalent

quantum mechanical

problem

^is

One can see

immediately,

that the low

temperature

limit of the

problem (2) corresponds

^{to the WKB}

limit of the system ^of

coupled

n-dimensional anhar- monic oscillators.

Analogous

^{to the n = 1} case, the

ground

^state

Go

of the

operator

^{G =}

Hlxo represents

^{the free energy}

per unit

length

^{of the} system ^for

general n,

^{while the}

energy of excitation to the first excited ^state determines the

longitudinal

correlation

length

The main

point

^{we wish to make here is that the}

free _energy

density

operator ^{G =}

H/xo obeys

^the

homogeneity relation,

where

This is obvious _{from eq.}

(2).

^{Since the two} operators which _appear on both sides of _eq.

(5)

^are propor-

tional, they

^{have the same}

eigenfunctions.

^Therefore

the same

homogeneity

^relation

given by

eq.

(5)

^holds

for all

eigenvalues Gm.

In

particular,

^the

homogeneity

^relation

(5)

^holds

for the free _energy

density Go

^{and for the difference}

G l-G 0

^{involved in the} correlation

length

^{as well as}

for the

quantities

involved in

higher

^order correlation functions.

The

temperature

appears in two

coefficients,

a ⁼

a’(T - To)ITo

^{and K =}

c/(kB T)2.

^{For rea-}

sons which will become clear

below,

eq.

(5)

^{is imme-}

diately

useful for

studying

^the

phase

^transition

only

^when

Tc À5 0,

i.e. when the

temperature depen-

dence

entering through K

is dominant. We shall therefore restrict our discussion to the

range n > 1 only.

^{The case n = 1} will be considered

separately,

since then the usual critical _power laws become

exponential.

3. n > 1. - Let us start with the

perturbations

 and h

equal

^{to zero} and consider _eq.

(5)

^{in the} para- meter space

(a, b, K). Using

^the

language

^{of the}

renormalization _group, the critical surface in this space is the surface K ⁼ oo. The

point (oo,

^oo,

oo)

on this surface can be considered as a once unstable fixed

point

^with

respect

^to which a and b are irrelevant

parameters,

^{while K is the relevant one.}

Following

the usual

procedure

^for

obtaining

^{the critical}

depen-

dence upon the relevant parameter, ^we put s ^{= K} and obtain

for AG, 0

⁼

G1 - Go

Since the

point (oo, oo, 1)

^{does not}

belong

^{to the critical}

surface,

it is reasonable to ^assume that

iBG1o

^{has a}

finite

limiting

^{value in this}

point

^{and we introduce}

the critical

index VG

^{= 2}

(K - 1 - T2)

associated with

iBG10.

From eq.

(4)

it therefore follows that

ç(T) 1’-1 T- 1,

^{i.e. v = 1. This} agrees with the

previous

result

[1].

In the case of the free _energy

density Go

^{we have to}

remove the part

Goo,

which cancels out

automatically

in the difference

(7) :

^{We know that when}

kB Txo -

⁰

the

particle

^exhibits

only

^{a weak zero}

point

^motion

around the bottom of the

potential well,

^{which is} harmonic in the lowest WKB

approximation.

^This

approximation corresponds

^to

[1] ]

Obviously,

^{the term}

AG,

⁼

Go - Goo

^{contains a} temperature ^behaviour

comparable

^{to that found}

for

AG,,g

^of eq.

(7).

^It should be also mentioned here that the

point (oo, oo, 1)

^{is not a}

regular point

^of

Goo(a, b, K).

Using

^{the same}

procedure

^{as in} eq.

(7)

^and

retaining

À and h we find

This is to be

compared

^{with the usual}

scaling

^relation

It therefore follows that _aG ⁼ 0.

Since cv

⁼

T aT2 , a2G

^{ÔT 2}

we have a ^{= -} 1. This _agrees with the

previous

results

[1]. ^Also,

^it shows that the

singular

part ^of c,

near T ⁼ 0 has the same temperature ^{behaviour as} the part associated with

Goo [9].

The critical index associated with the symmetry

breaking

field is A ⁼ 2. The lattice

anisotropy

^cross-

over index _ç is

equal

^to two, i.e. the ^{crossover tem-}

perature

^{T* at} which À becomes

important,

^because

KÂ in _eq.

(9)

^reaches the values of the order of

unity

is

given by

Above T* the system ^behaves

one-dimensionally.

Tc

follows the same law.

Namely,

^{below the cross-}

over temperature ^the system exhibits three-dimen- sional behaviour. In this

region,

^{the free} energy

obeys

three-dimensional

scaling relation, expressed

^{in terms}

of new set of variables and critical indices

Here 1" ⁼ T -

Tc(À.) and at and ah

^are the three- dimensional critical exponents.

According

^{to the}

renormalization _group

approach [10],

^a coefficient of

a

gradient

^{term  is a}

marginal

variable in the three- dimensional

regime.

^{On the other}

hand,

^{near the} fixed

point

^the

scaling

^relation

(5)

^{can be} put ^{in the} form

where io ⁼

k. T/Cl/2.

The simultaneous

validity

^of

eq.

(12a)

^and

(12b)

^{was studied}

previously [11] ]

^and

shown to lead to

and to

In contrast to the usual

results,

^which

give

^the

depen-

dence of T* and

Tc

upon the ratio of the

perpendicular

and the

longitudinal

temperature

independent

^corre-

lation

lengths

which is similar to

Âlc,

eq.

(11)

^and

(13) give

^the

dependence

^{of T* and}

Tc

upon the

quantity

Àc which is similar to the

product

^{of these}

lengths.

In

fact,

^{it can be}

easily

^{seen from the}

scaling properties

of À and K that

Âlc

^{is not a} natural variable for T*

or

Tc

^{in the} present situation. This latter conclusion agrees with the calculation for the

anisotropic

^B-E

free _gas

[8].

^The B-E calculation is in the one-to-one

correspondence [8]

^{with the Hartree} calculation of

T

^{in the G-L}

problem,

^which

applies

^{to the limit}

n = oo of this

problem.

^Both calculations lead to eq.

(13)

and determine the

missing

^{factor of} propor-

tionality.

This factor is

essentialy equal

^to

a/nb.

^{It is}

very

unlikely

^{that the} exponent ^of

bla depends

^{on n.}

It can be therefore

reasonably conjectured

^{that for}

an

arbitrary n

> 1

with

Tb

^defined

by

where

Kc

⁼

K(Tc).

^{A similar}

equation

should hold for T*. The

replacement

^of

Tb by To

ⁱⁿ eq.

(15)

^is

inconsistent with _eq.

(13),

^i.e. eq.

(5).

Tuming

^{now to the}

scaling ^laws,

^we notice that two sets of critical indices have been defined above.

LE JOURNAL DE PHYSIQUE. ^- T. 36, ^NI 12, ^DÉCEMBRE 1975

One set

(VG

aG, ^,

...) corresponds directly

^to the spec- trum

Gm

^{of the} quantum mechanical

problem.

^The

other

(v,

^a,

...)

^{describes the} temperature ^behaviour of the

quantities ^ç,

^{c,, ..., the} definition of which involves an extra

kB

^{T factor. As the}

homogeneity

relation

(5) directly

^{concems the} spectrum

Gm,

^the

scaling

^{laws hold} among the critical indices of the first set.

They

^are

only exceptionally

^{valid in the}

second set.

E.g.

gives

The correlation function

g(O) = kB Tx(0) diverges

with _y ⁼ 1. This _y is not related

by

^a

scaling

^law

(17)

to a ^{= -} 1 of the

specific

^{heat. The crossover index 9}

is related

by

^{the usual}

[12]

^{relation to} y. and ^{not to} y,

This _agrees with our

previous conjecture [8].

^Further-

more the

Josephson

relation is valid

only

in the first set

The Fisher relation

givres 11

⁼

1,

ⁱⁿ agreement ^with

previous

calculations.

This relation is valid for both sets of critical indices.

4. n ⁼ 1. - The case n ⁼ 1 must be considered

separately

because in this case the power laws for the critical behaviour become

exponential [13].

Although

^the

homogeneity

^relation

(5)

proves useful

even in such a

situation,

^we

prefer

^to

give

^{here the full}

WKB solution of the crossover

problem.

The

procedure

consists of two

steps :

^{First one} finds the WKB solution for the two lowest levels of the

purely

one-dimensional

problem [14]. Second,

one uses the

equivalence

^{of the crossover}

problem involving

^two energy levels and the two-dimensional

Ising

^{model in the} transverse

magnetic

^field

[15].

This latter

problem

posesses ^{an accurate} solution

[16].

The first step ^{was carried out in reference}

[1] ]

except for the determination of various factors. The second step ^was

accomplished

^{in reference}

[2] by using only

the numerical solution of the olne-dimensional _pro- blem

[13].

^{Here we determine the}

lacking

^constants

in the one-dimensional part ^{of the}

problem

^and

combine this

analytic

solution with the

analytic

solution of the

Ising

^{model in order to determine the}

crossover behaviour.

# being

^{a scalar order} parameter ^the

potential

energy

V(#)

^of eq.

(1)

^{with = h = 0}

1270

contains two minima at

with

separated by

^the barrier of

height (- Vo).

The pres-

cription

^{of the BKW} method for such a case

[17]

^is

i)

^{to solve the}

ground

^state

problem

for infinite

separation

^of

potential wells ;

^{and :}

ii)

^to

bring

^the

potential

^{wells to a} finite distance and

to allow for

tunneling by

^{the BKW} version of the

tight-binding

^method.

The

tunneling gives

^{rise to the}

splitting G 10

of the _energy level

Go, doubly degenerate

^{at infinite}

separation.

We

proceed according

^{to above}

prescriptions : i) The frequency

^{of the} harmonic oscillator in each

potential

^{well is}

We note that ev does not vary with b even in the limit b -

0,

^which

corresponds

^{to an infinite}

separation

^of

the

potential

^{wells and an} infinite barrier between them. We conclude that

Goo

^{and the}

corresponding

wave-function

give

^{the correct solution to the first} step ^{of our} programme. Since we show below that the

splitting

^of

Go

^is

exponentially

small when T ->

0, Goo

^{is the}

leading

^{term in the}

ground

^state energy of _eq.

(1)

^{with = h =} 0. This is

analogous

^{to our}

separation

^of

Go

^into

Goo

^and

AGO

^{for n} > 1. The BKW

approach explains why

^the

simple

^harmonic

approximation

^{leads to the}

essentially

^{correct T ---+ 0}

behaviour for the free _energy,

specific

^{heat and}

2 > _ /2 for n >,

^1.

ii)

^The correlation

length ç

^is determined

by

^the

splitting AGio

of the level

Go,

^{rather than}

by

^the

distance of the

ground

and the first excited state of the harmonic oscillator.

According

^{to reference}

[17]

The

phase integral

^{in the}

exponent

^is

elementary

since we

replace Go by

^{its classical value}

Vo, which is legitimate

ⁱⁿ the T ---+ 0 limit. The argument ^{of the}

exponential

^becomes

Thus,

ⁱⁿ agreement with reference

[1]

^{we find that}

the inverse temperature ^enters

linearly

^the exponent.

This

disagrees

^{with the}

expression

^for

ç(T)

^{used in}

reference

[5]

^obtained

by

^an

analytical

fit of the numerical n ⁼ 1 results of reference

[13].

^{The two}

expressions

also differ in the

prefactor

^{of the} expo- nential.

Eq. (24)

^{is exact and}

obeys

^the

scaling

^rela-

tion

(5).

Both the mean-field

approximation

^and

the

^accu-

rate treatement via the

Ising

model in the transverse

field,

^{lead to}

essentially

^{the same}

equation

^for

T,.

Using ç(T)

determined

by

eqs.

(24)

^and

(25)

^{in this}

equation,

^{we find}

for

T,,ITB «

^{1. This agrees with the}

scaling

^behaviour

of the involved parameters ^a,

b,

^{K and Â. The most} salient feature of the result

(26)

is that the relevant ratio is

Àla

^{and not the ratio}

À.lc (which

^{here has a}

dimension).

^{We notice that}

Tb

^of eq.

(25)

^and

(26)

appears ^{as a} characteristic temperature ^{for n = 1 too,} while

To

is irrelevant.

5. n 1. - The

homogeneity

^relation

(5)

^{is not}

suitable for an

unambiguous

^{treatment of the}

phase

transition at a finite

temperature.

^{The reason is that}

eq.

(5)

^{concems the} variable a and not aa

= a - a*

where ag

is the value of a at an

appropriate

^stable

fixed

point.

^The

example

^of eqs.

(12a)

^and

(12b)

^has

already

^{shown us that different sets} of scaled variables

are in

general

^{related to different sets} of critical indices.

Still,

^{it should}

perhaps

be mentioned that the substi- tution s-1 = Aa in _eq.

(5)

^leads to correct

[1] :

^critical

indices for the one-dimensional case, if the

multi- plicative

factors of Aa are assumed finite in the limit of small Da.

But ac

^{= 0 is}

certainly

^{not a value of a}

at a stable fixed

point,

^{and this agreement should be}

considered as fortuitous.

6. Conclusion. ^- We have shown that the

simple homogeneity

relation holds for the _energy levels of the system ^of

weakly coupled

anharmonic oscillators.

Such a system

corresponds

^{to the}

Ginzburg-Landau description

^{of the set of}

weakly coupled

linear chains.

The derived

homogeneity

relation bears a close resemblance to the laws

usually

obtained from the renormalization _group

approach.

^The

language

^{of the}

renormalization _group, i.e. the

concept

of the critical surface and a fixed

point proved

^{useful even}

^though

the fixed

point

^{is not a finite}

point

^{in the}

parameter

space, since we could define an

unambiguous limiting

process ^to reach it. A similar

approach

may prove

helpful

ⁱⁿ

problems

^{which are more}

complicated

^than

the

present

^{one. As a result of our} discussion we have obtained the critical indices for the T ⁼ 0

phase

transition of the one-dimensional G-L model. These critical indices _agree with the results of

explicit

^calcu-

lations when

they

^{exist. The} critical indices are shown to

obey

the usual

scaling

^{laws and some}

previous

difficulties encountered for ^the

Tc

^{= 0}

transition,

are resolved

by introducing

^an

appropriate

^{set of}

critical indices. The crossover critical exponent ^is

determined and shown to agree with a

previous conjecture.

Acknowledgment.

^{- A} discussion with G. Toulouse is

gratefully acknowledged.

References

[1] BALIAN, ^{R. and} TOULOUSE, G., ^Ann. Phys. ⁸³ (1974) ^28.

[2] DIETERICH, W., ^Z. Phys. ²⁷⁰ (1974) ^239.

[3] RICHARDS, ^P. M., Phys. ^{Rev. B 10} (1974) ^4687.

[4] FISHER, ^M. E., Phys. ^{Rev. 162} (1967) ^480.

[5] SCALAPINO, ^D. J., IMRY, ^{Y. and} PINCUS, P., Phys. ^{Rev. B 11} (1975) ^2042.

[6] MANNEVILLE, P., ^J. Physique ³⁶ (1975) ^701.

[7] PRANGE, R. E., Phys. ^{Rev. A 9} (1974) ^1711.

[8] BARI0160I0106, ^{S. and} UZELAC, K., ^J. Physique ³⁶ (1975) ^325.

0160AUB,

K., BARI0160I0106, ^{S. and} FRIEDEL, J., ^{to be} published.

[9] ^It should be noted that the kB ^T prefactor ^{of F was taken} equal ^to kB To in reference [13], ^{while we} keep it variable.

[10] WILSON, K. G. and KOGUT, J., Phys. Rep. 12C (1974) ^75.

TOULOUSE, ^{G. and} PFEUTY, P., Introduction au groupe de rénor- malisation et à ses applications, ^{Grenoble 1975.}

[11] HANKEY, A. and STANLEY, H. E., Phys. ^{Rev. B 6} (1972) ^3515.

[12] ^{See for} example GROVER, ^M. K., Phys. ^{Lett. 44A} (1973) ²⁵³ and references therein.

[13] SCALAPINO, ^D. J., SEARS, M. and FERRELL, ^R. A., Phys. ^Rev.

B 6 (1972) ^3409.

[14] The paper by KRUMHANSL, ^{J. A. and} SCHRIEFFER, ^J. R., Phys.

Rev. B 11 (1975) 3535, published while this _paper was in revision contains this same WKB procedure.

[15] ^DE GENNES, ^P. G., Solid State Commun. 1 (1963) ^132.

[16] PFEUTY, ^{P. and} ELLIOT, R. J., ^J. Phys. ^{C 4} (1971) ^2370.

[17] LANDAU, ^{L. D. and} LIFCHITZ, E. M., Mécanique Quantique (Moscow) 1966, Pb. No. 3, § ^50.