9 February 1998

PHYSICS LETTERS A

ELSEVIER Physics Letters A 23X ( 1998) 265-270

The interface tension of the three-dimensional Ising model in two-loop order

Peter Hoppe, Gernot Miinster ’ Instinct ,fir Theoretische Physik I, Univer.&if Miinster. WilheLn-Kletnm-Strussr 9. D-48149 Miinstet Germrny

Received 7 October 1997: revised manuscript received 24 November 1997; accepted for publication 24 November 1997

Communicated by CR. Doering

Abstract

In liquid mixtures and other binary systems at low temperatures the pure phases may coexist, separated by an interface. The interface tension vanishes according to u = a(,( 1 - T/T,)” as the temperature T approaches the critical point from

below. Similarly the correlation length diverges as 5 = f- ( I -T/T,)-” in the low temperature region. For three-dimensional systems the dimensionless product R_ = aof? is universal. We calculate its value in the framework of field theory in d = 3 dimensions by means of a saddle-point expansion around the kink solution including two-loop corrections. The result

R- = 0.1065(9), where the error is mainly due to the uncertainty in the renormalized coupling constant, is compatible with

experimental data and Monte Carlo calculations. @ 1998 Elsevier Science B.V.

PACS: 05.7O.Jk; 64.60.Fr; I I. IO.Kk; 68.35.Rh

Ke~ywods: Critical phenomena; Field theory; Interfaces; Amplitude ratios

For a statistical system near a critical point, various measurable quantities X have a singular behavior as a function of the temperature T like

x N X()P, (1)

with a critical exponent E, where

system, depending on the microscopic details of the

Hamiltonian.

The critical exponents of a universality class are

not independent of each other but obey a number of scaling and hyperscaling relations. For example, in

the low temperature region the exponents fi, y, and Y belonging to the magnetization M, the susceptibility

T - Tc

I I

x and the correlation length ( are related by

t= T, ’

(2) 2P+y-dv=O. (3)

and T, is the critical temperature. For a given observ- where d = 3 is the number of dimensions. Therefore, able X the critical exponent is a universal quantity in the combination and assumes the same value for systems belonging to the same universality class. The critical amplitude Xa, 3x

UR= *’ (4) however, is not universal and varies from system to

the exponents cancel and UR is expected to approach

’ E-mail: munstegauni-muensterke. a finite value,

0375.9601/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved

PII SO375960 I (97)009 13-4

266 P. Hqp. G. Miinster/Physics Letters A 238 (1998) 265-270

when the critical point is approached from below. It

is a another consequence of the scaling hypothesis that such combinations of critical amplitudes are also universal [ 1,2]. They are generally called amplitude

ratios. For a review, see Ref. [ 31. In the last decades much interest has focused on

critical indices, whose values are known from vari-

ous methods rather accurately by now. From a phe- nomenological point of view the amplitude ratios are, however, at least as interesting as the indices. They

are well accessible experimentally and their numerical values are often more characteristic for the universal- ity classes as the variations between different classes

are larger. In this Letter we consider an amplitude ratio related

to the interface tension in the universality class of the three-dimensional Ising model. In various binary sys- tems at temperatures T below T,, interfaces (domain

walls) may be present, separating coexisting phases. The interface tension r is the free energy per unit area

of the interfaces. As T increases towards T,, the re- duced interface tension

where k is Boltzmann’s constant, vanishes according to the scaling law

o- N qt+.

Widom’s scaling law [4,5],

(7)

p= (d- l)V, (8)

relates the universal critical exponent ,X to the criti- cal exponent v of the correlation length 5. Associated with this law is the universal dimensionless product of critical amplitudes

R- = a&‘. (9)

In this Letter we consider the correlation length as defined by means of the second moment of the cor- relation function [ 61, in contrast to the “true” (expo- nential) correlation length. Numerically they differ by less than 2% [7].

The amplitude a0 has been studied experimentally (see Refs. [ 3,8] ) as well as theoretically (see e.g.

Refs. [ 9-1 l] ). The presently most accurate Monte Carlo result has been obtained by Hasenbusch and Pinn [ 121.

In the theoretical toolbox we also have the three- dimensional Euclidean ti4-theory, which is believed to be in the same universality class as binary systems and the Ising model. Therefore it should describe the uni-

versal properties of these systems correctly. The scalar field 4(x) represents the local order parameter, which

in the case of binary fluid mixtures is proportional to the difference of the concentrations of the two fluids.

On the classical level, an interface in a system of

cylindrical geometry is represented by a classical so- lution of the field equations. It is a saddle point of the Hamiltonian. Taking thermal fluctuations into account

amounts to performing an expansion around the sad- dle point in the functional integral.

In the field theoretical framework the interface ten- sion, in particular the universal ratio R-, has been in-

vestigated by means of the e-expansion in 4 -E dimen-

sions [ 13,141 and directly in three dimensions [ 151. In both approaches the calculations were done in the one-loop approximation (see below). Whereas the re- sults from the e-expansion are afflicted by convergence problems and show large deviations from experimental and Monte Carlo results, the three-dimensional field theory leads to relatively small one-loop corrections and more reasonable numbers. However, higher-loop

corrections may spoil this situation. Therefore, in or- der to get a better impression of the numerical con-

vergence and to obtain more precise estimates, it is

highly desirable to know the two-loop contribution to R_. In this Letter we present the result of a two-loop calculation of the universal amplitude ratio R_ in the framework of three-dimensional 4J-theory.

The Hamiitonian X, which is called an action in the context of Euclidean field theory, in the broken symmetry phase is written in terms of the bare field

$0 as

‘FI= Cd3x, s

where the double-well potential

(10)

?? Hoppe. G. Miinster/Phyics 1.etter.s A 238 (1998) 265-270 267

has its minima at

(11)

q5 =* 0 1 ‘(I - _* d 3 go

(12)

The parameters are detined such that the value of the potential at its minima is zero and ma is the bare mass.

The renormalized mass

t?lR = I/( (13)

is the inverse of the second moment correlation length. It is defined together with the wave function renor-

malization ZR through the small momentum behavior

of the propagator,

G(p)-’ = 1

ZR (1112, + p2 + O(p4)}. ( 14)

The renormalized vacuum expectation value of the field is

t’R = z, -II’,, . (15)

where LJ is the expectation value of the field ~$0. For

the dimensionless renormalized coupling we adopt the

definition

UR=gR=3!!!E (16) IllK ,,; ’

which in the language of statistical mechanics corre- sponds to Eq. (4).

The basic idea behind the calculation of the inter-

face tension is its relation to the energy splitting due to tunneling in a finite volume. We refer to Refs. [ 16,151 for details. In a rectangular box with cross-section L’ the degeneracy of the ground state of the transfer ma- trix is lifted by an “energy” splitting EaCl. This gap

depends on L according to

Eo,, = C exp{ -aL’}, (17)

where (7 is the (reduced) interface tension [ 17-

19.15].

The energy splitting Eaa can be calculated in a semi-

classical approximation, which amounts to a saddle- point expansion around the classical kink solution

tanh T(,r, - a). (18)

of 44-theory, where a is a free parameter specifying the location of the kink. The classical energy of a kink is

XC = 2ZL’. (19)

The kink interpolates between the two field values at the minima of the potential and represents an interface

separating regions with different local mean values of the field.

In the two-loop approximation the functional inte-

gral

Z+_= e s

-HIhI D&) (20 )

(a factor of 1 /kT has been absorbed in ?-f) with appro- priate boundary conditions is calculated by expanding the energy H [ 41 around the kink solution 4(: up to or-

der go and evaluating the integral by the saddle-point method. For details, see Ref. [ 201. An analogous cal- culation for the case of the anharmonic oscillator in quantum mechanics has been performed in Ref. [ 2 11.

The zero mode belonging to the translations of the center of the kink, i.e. to shifts in a, is treated by the method of collective coordinates and leads to a non- trivial Jacobian J. For the energy splitting one obtains

where

M = -a,ap + in; - 5 cash-’ m0

2 0 -p ( >

(22)

is the operator of quadratic fluctuations around dC, and

268 P Hoppe, G. Mtinster/Physics Letters A 238 (1998) 265-270

MO = -J,tV + rni. (23)

The prime in det’ indicates a determinant without zero modes. The two-loop contributions are displayed as

Feynman diagrams. The propagators are

(28)

-2 G(x, Y),

--- g Go(x, Y) 9 (24)

where the propagator in the kink background,

G(XtY) = cwf’)-‘lY)~ (25)

is the Green function of M without zero mode and

GO(X*Y) = (Qf,‘lY) (26)

is the usual scalar propagator. Remember, however,

that the propagators refer to a system with finite cross-

and to write the kernel exp( -tM’) in the spectral rep-

resentation. The calculations have been done analyti-

cally as far as possible. In the later stages some infi- nite sums and low-dimensional integrations have been

done numerically. Details of the calculation can be found in Ref. [ 201.

The most difficult piece, of course, was the true two-loop diagram

_ _- section L2.

The vertices are

It contains ultraviolet divergences, which we isolated by means of dimensional regularization in d = 3 - E

dimensions. After some tedious calculations (we warn the curious reader) we obtained the two-loop contri-

bution as a function of L up to terms of order Lo. Whereas individual diagrams produce terms of the form L*(log(moL))* and L*log(moL), they cancel

in the total sum. The leading L-dependence is then proportional to L* as is required by the finiteness of the interface tension.

, / ^ ____< = \ - JGZ, \

The last vertex comes from the Jacobian J. The spectrum of the fluctuation operator M is

known exactly. Owing to this, the determinants can be evaluated analytically with the help of heat kernel and zeta-function techniques [ 16,151. They yield the one-loop contribution to the interface tension.

Much more involved is the calculation of the two- loop contributions. Although we have an expression for G(x, y) (covering one and a half page) it turned out to be advantageous to use the Schwinger represen- tation

The ultraviolet divergence is removed by renormal- ization as usual. Expressing the bare parameters mo and go in terms of their renormalized counterparts mR and gR in the two-loop approximation indeed cancels the pole in l/c. The final result for the interface ten- sion at L = cK) is

~=~{l+ull~+u2,(~~2+o(u:)}. (29)

with

Cl[ = 33 + ; log3) - $ = -0.2002602,

and

(30)

c21 = -0.0076( 8). (31)

Using ( 13) the desired amplitude ratio R- is obtained by evaluating the function

at the fixed point value UR = ui, i.e.

I? Hoppe. G. Miinster/Phyk Letten A 238 (1998) 265-270 169

’ f, 1E

Fig. I. R_ ins a function u;6. The solid line is the average of the

three PadC approximants. The thin lines indicate an error estimate

of 0.001.

R_ = J’(‘“i) = ; R

X {

l +rr,,~+~~~(~)2+O(U;$)}. (33)

The most recent results for ~4; are

II; = 14.3( I ) (34)

from Monte Carlo calculations [7] and z$ = 14.2

from three-dimensional field theory [ 221. Earlier es- timates were 14.73( 14) from the low temperature se-

ries [23] and 15.1( 1.3) used in Ref. [ 151. For these numbers the two-loop contribution to R_ is about l%,

while the one-loop contribution is about 24%. Al- though the apparent numerical convergence is surpris- ingly good, we have also applied PadC and Pad&Bore1 approximations to the quadratic polynomial appearing in f( UR) in order to get R-. The dependence of R_ on ui is illustrated in Fig. 1, where the average of the three PadC approximants and the corresponding error band is displayed.

Table I shows the results of the PadC and PadC- Bore] approximations evaluated at three of the values for r& mentioned above. The quoted errors are due to

the error of ~1. The average of all approximants at ui = 14.3 is

R_ = O.l08( 2). The [ 0,2] approximants appear to be off the rest. Leaving them out yields R- = 0.1065 ( 1). Taking the error of z& into account we obtain

R- = 0.1065(9). (35)

Table I Results for R- from PadC and Pad&Bore1 approximations evalu-

ated at three values for 14:

G H_

f f’l I. I I .flN?l fl I.1 I.I’B .fIO.?I.PB

IS.1 0.0991(2) 0.0989(2) 0.1011(1) 0.0991(3) 0.10454(S)

14.73 0.102.5(2) 0.1024(2) 0.1044( 1) 0.102.5(3) 0 10774(S)

14.3 0.1066(2) 0.1064(2) 0.1084(l) 0.1066(3) 0.11166(S)

Since we do not know the size of the higher-loop con- tributions the quoted error mainly reflects the spread due to the uncertainty of uz.

For comparison the Monte Carlo calculations 01‘ Hasenbusch and Pinn [ 121 yield

R- = 0.1040(8). (36)

In view of the remarks made above we consider the

results as being compatible. Experimentally, the universal amplitude combi-

nation R+ = croft, where f’+ is the amplitude of the correlation length in the high temperature phase, has been measured for various binary sys- tems; see Ref. [ 31. In order to compare with R_ the universal ratio _f+/f- has to be employed. Using

f+/f- = 1.95(2) from recent Monte Carlo calcu- lations [7] or f+/f- = 1.99(2) from field theory (see Ref. [22] and the remark in the conclusions of

Ref. [7] ) we obtain for R+ the numbers 0.40( 1 ) and (0) .42( 1 ), respectively. This compares well with the recent experimental result of 0.41(4 ) for the

cyclohexane-aniline mixture [ 8). Previous experi-

mental results are summarized in Ref. [ 8 ] as 0.37 (3 )

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