The 2D effective field theory of interfaces derived from 3D field theory

9 Pages·0.1 MB·English

by  Paolo Provero

DFTT 8/95
hep-th/9501104
January 1995
The 2D effective field theory of interfaces
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9 derived from 3D field theory
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Paolo Provero ∗ and Stefano Vinti †
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Dipartimento di Fisica Teorica
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Universita` di Torino
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0 Via P. Giuria 1
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10125 Torino, Italia
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- Abstract
p
e The one–loop determinant computed around the kink solution in
h 4
the3Dφ theory,incylindricalgeometry,allowsonetoobtainthepar-
:
v titionfunctionoftheinterfaceseparatingcoexistingphases. Thequan-
i
X tum fluctuations of the interface around its equilibrium position are
described by a c = 1 two–dimensional conformal field theory, namely
r
a a 2D free massless scalar field living on the interface. In this way
the capillary wave model conjecture for the interface free energy in its
gaussian approximationis proved.
∗e–mail: [e-mail]
†e–mail: [e-mail]
1 Introduction
Thephysics of interfaces separating coexisting phases in3D systems is dom-
inated by long–wavelength, low–energy fluctuations; it is therefore natural
to describe the interface fluctuations in terms of a 2D effective theory. A
common assumption is to take the interface free energy to be proportional
to the area of the interface: this is the well known capillary wave model
(CWM) which is believed to describe the interface physics [1, 2]. More re-
cently, the CWM predictions have been made explicit one order beyond the
gaussian approximation and verified by means of numerical simulations on
3D spin systems to high accuracy [3, 4].
Despite these results, the CWM is an ad hoc 2D effective theory: it is
not known in general how to derive it from the original 3D hamiltonians,
except in the zero–temperature limit (see e.g. [5] and references therein).
In this paper we provide an analytical derivation of the 2D effective
theory of interfaces in the framework of 3D Euclidean φ4 theory, which
is known to describe the scaling region of the Ising model (for a general
review see for instance [6]). Our result reproduces the predictions of the
CWM in its gaussian approximation: the partition function of an interface
isproportionaltothepartitionfunctionofa2D,c= 1conformalfieldtheory
(CFT), namely a free scalar field living on the interface.
Thisresultcanbethoughtofasanewinstanceofdimensionalreduction:
the relevant degrees of freedom of a physical system are described by an
effective theory of lower dimensionality.
To be more precise, we consider the 3D φ4 theory in a cylindrical ge-
ometry with two of the three space–time dimensions having finite lengths
L , L and periodic boundary conditions. Using ζ–function regularization,
1 2
we compute, in one–loop approximation, the energy–gap E(L ,L ) due to
1 2
tunneling: in the dilute–gas approximation, this quantity is proportional to
the partition function of an interface.
The paper is organized as follows: in Sec. 2 we establish our notations
and review theexpressionof E(L ,L )interms of afunctionaldeterminant,
1 2
regularized using the ζ–function method. In Sec. 3 we evaluate the determi-
nant: our main result is the expression (34) for E(L ,L ). Sec. 4 is devoted
1 2
to some concluding remarks.
1
2 The interface partition function
Consider the 3D field theory defined by the action
1
S[φ] = d3x ∂ φ∂ φ+V(φ) (1)
µ µ
2
Z (cid:20) (cid:21)
where
g 2
V(φ) = φ2 v2 (2)
4! −
(cid:16) (cid:17)
in Euclidean space–time with finite size in the ”spatial” directions x (i =
i
1,2) but infinite in the ”time” direction x . We put periodic boundary
0
conditions on the finite sizes:
φ(x ,x +L ,x ) = φ(x ,x ,x +L )= φ(x ,x ,x ) . (3)
0 1 1 2 0 1 2 2 0 1 2
The potential V has two degenerate minima in φ = v and a maximum in
±
φ = 0.
A solution of the equations of motion connecting the two minima is the
kink
m
φ (x) = vtanh (x a) (4)
cl 0
2 −
(cid:20) (cid:21)
where
1/2
gv2
m = , (5)
3 !
and its action is
2m3
S S[φ ] = L L . (6)
c cl 1 2
≡ g
The existence, in finite volume, of classical solutions connecting the two de-
generate minima of the potential, and hence of a non–vanishing tunneling
probability between the two minima, has the effect of removing the double
degeneracy of the vacuum which, in infinite volume, is due to the sponta-
neous breaking of the Z symmetry φ φ. The energy splitting is given,
2
→ −
in one–loop approximation, by (see e.g. [7])
S 1/2 det′M −1/2
E(L ,L ) = 2e−Sc c (7)
1 2
(cid:18)2π(cid:19) (cid:12)(cid:12)detM0(cid:12)(cid:12)
(cid:12) (cid:12)
where M is the operator (cid:12) (cid:12)
(cid:12) (cid:12)
∂ ∂
M = +V′′(φ (x)) . (8)
cl
−∂x ∂x
µ µ
2
Here det′ indicates the determinant without the zero mode, which is due to
the freedom in choosing the kink location a, and gives rise, when treated
with the collective coordinates method, to the prefactor (S /2π)1/2. M is
c 0
the free–field fluctuation operator
M = ∂ ∂ +m2 . (9)
0 µ µ
−
Thecomputation of the energy splitting (7) for the symmetriccase L =
1
L wasdoneinRef.[8]. Wewillseethatthegeneralization ofthecalculation
2
to asymmetric geometries allows one to recognize the interface partition
function as the partition function of a 2D CFT.
We use ζ–function regularization to compute the ratio of determinants
appearing in Eq.(7). It is useful to express the operators M and M as
0
M = Q(x ) ∂ ∂ (i = 1,2) (10)
0 i i
−
M = Q (x ) ∂ ∂ (11)
0 0 0 i i
−
where
3 1
Q(x ) = ∂2+m2 m2 (12)
0 − 0 − 2 cosh2 m(x a)
2 0−
Q0(x0)= −∂02+m2 . (cid:2) (cid:3) (13)
Theregularized ratio ofdeterminants appearinginEq. (7) isthen expressed
as
det′M d
= exp [ζ (s) ζ (s)] (14)
detM − ds M − M0
0 (cid:26) (cid:12)s=0(cid:27)
(cid:12)
where the ζ–function of an operator A with eigenvalu(cid:12)es a is defined as
(cid:12) n
ζ (s) = a−s (15)
A n
n
X
The spectra of the operators Q, Q and ∂ ∂ are known, and the relevant
0 i i
−
ζ–function is
ζ (s) ζ (s) = ′(λ )−s+ λ + 3m2 −s
M − M0 n1,n2 n1,n2 4
n1,n2 n1,n2(cid:18) (cid:19)
X X
+∞ −s
+ dp g(p) λ +p2+m2 (16)
n1,n2
nX1,n2Z−∞ (cid:16) (cid:17)
3
where the primed sum runs over (n ,n ) = (0,0). Here λ are the eigen-
1 2 6 n1n2
values of the two–dimensional operator ∂ ∂ with periodic boundary con-
i i
−
ditions on the rectangle of sides L , L :
1 2
n2 n2
λ = 4π2 1 + 2 n ,n Z . (17)
n1n2 L21 L22! 1 2 ∈
g(p) is the difference between the spectral densities of Q and Q :
0
m 2 1
g(p) = + . (18)
−2π p2+m2 p2+ m2!
4
3 Evaluation of the determinant
To complete our calculation we have to evaluate the ζ–function (16). Fol-
lowing Refs. [8, 9] we write
ζ (s) ζ (s) ζ (s)+ζ (s) (19)
M − M0 ≡ 1 2
where
ζ (s) = ′λ−s (20)
1 n1,n2
n1,n2
X
−s
3m2
ζ (s) = λ +
2 nX1,n2 n1n2 4 !
+∞ −s
+  dp g(p) λ +p2+m2 (21)
n1n2
Z−∞ (cid:16) (cid:17) (cid:27)
The term ζ can be recognized to be the ζ–function of a massless, 2D free
1
scalar field on the rectangle of sides L , L with periodic boundary condi-
1 2
tions, i.e. on a torus [10]. From 2D CFT we know that its derivative in
s = 0 is [10]
dζ
1 = 2log L η(τ) 2 (22)
1
ds − | |
(cid:12)s=0
(cid:12) h i
where (cid:12)
(cid:12) L
1
τ i (23)
≡ L
2
is modular parameter of the torus and η(τ) is the Dedekind function. When
combined with the prefactor (S /2π)1/2 coming from the zero mode in Eq.
c
4
(7), this term produces precisely the modular invariant partition function of
the c= 1 CFT defined by a free massless scalar field.
To evaluate ζ (s), we proceed like in Refs. [8, 9]: we write
2
1 ∞ 3m2
ζ (s) = dt ts−1 exp λ + t
2 Γ(s)n1n2Z0 ( "− n1n2 4 ! #
X
+∞
+ dp g(p)exp λ +p2+m2 t (24)
− n1n2
Z−∞ h (cid:16) (cid:17) i(cid:27)
and, introducing the Jacobi theta function
A(x) = exp πn2x , (25)
−
Xn (cid:16) (cid:17)
we have
1 ∞ 4πt 4πt
ζ (s) = dt ts−1A A F(m,t) (26)
2 Γ(s) L2 L2
Z0 (cid:18) 1 (cid:19) (cid:18) 2 (cid:19)
3 +∞
F(m,t) = exp m2t + dp g(p)exp p2+m2 t .(27)
−4 −
(cid:18) (cid:19) Z−∞ h (cid:16) (cid:17) i
Using Poisson’s summation formula A(x) is seen to satisfy
A(x) = x−1/2A(1/x) (28)
which we use to express ζ as
2
(a) (b)
ζ (s)= ζ (s)+ζ (s) (29)
2 2 2
where
L L 1 ∞
ζ(a)(s) = 1 2 dt ts−2F(m,t) (30)
2 4π Γ(s)
Z0
L L 1 ∞ L2 L2
ζ(b)(s) = 1 2 dt ts−2 A 1 A 2 1 F(m,t)(31)
2 4π Γ(s) Z0 " 4πt! 4πt!− #
(b)
The term ζ is exponentially suppressed for large L , L [9] and will there-
2 1 2
(a)
fore be neglected in what follows. ζ is then computed straightforwardly:
2
L L 1 3 (1−s) 3 Γ(1/2)Γ(s 1/2)
ζ(a)(s) = 1 2 m2(1−s) −
2 4π s−1 ((cid:18)4(cid:19) − 2π Γ(s)
3 +∞ (q2+1)−s
dq (32)
−8π −∞ q2+1/4 )
Z
5
so that
dζ(a) 3m2L L 1
2 = 1 2 1+ log3 . (33)
ds (cid:12) − 4π 4
(cid:12)s=0 (cid:18) (cid:19)
(cid:12)
(a) (cid:12)
Therefore the ζ term(cid:12) provides simply a quantum correction to the inter-
2
face tension.
Substituting (22) and (33) in (14) and (7) we finally obtain
C
E(L ,L ) = exp( σL L ) (34)
1 2 [Im(τ)]1/2 η(τ) 2 − 1 2
| |
where
1/2
2 m3
C = (35)
√π g !
2m3 3g 1
σ = 1+ 1+ log3 . (36)
− g 16πm 4
(cid:20) (cid:18) (cid:19)(cid:21)
In Ref. [8] the energy gap was computed in the symmetric case L = L ,
1 2
in which the τ–dependent contribution reduces to a constant. Notice that
in [8] the energy gap is expressed in terms of the physical mass m (in-
phys
verse of the correlation length) and the renormalized coupling u g /m
R R R
≡
where the renormalized parameters g and m are defined according to a
R R
particular renormalization scheme. However it is important to keep in mind
that, at one–loop, the renormalized parameters differ from the bare ones by
finite quantities: the one–loop Feynman diagrams in 3D φ4 are finite after
dimensional continuation. The formulae needed to make contact between
our result and the one quoted in Ref. [8] are
g 31u
u= u 1+ R + (u2) (37)
m ≡ R 128π O R
(cid:18) (cid:19)
u
m2 = m2 1+ R ( 4+3log3)+ (u2) . (38)
phys 16π − O R
(cid:20) (cid:21)
4 Conclusions
The effective, long–wavelength 2D theory of interface fluctuations in 3D φ4
theory has been derived from first principles by analytical methods. The
interface partition function turns out to be proportional to the partition
function of a free massless 2D scalar field living on the interface. In this way
6
one is able to obtain a 2D conformal invariant field theory by dimensional
reduction of 3D field theory.
This result is in agreement with the predictions of the capillary wave
model of interfaces, which assumes an interface free energy proportional to
the interface area. Indeed, the capillary wave model in its gaussian approx-
imation predicts exactly the functional form (34) for the interface partition
function [11, 3].
The predictions of the capillary wave model were tested against Monte
Carlo simulations of spin systems in Refs. [3, 4]. In particular in [4] the
modelwas successfully verified beyond thegaussian approximation: itwould
be interesting to investigate whether the CWM contributions beyond the
gaussian one can be derived in a field–theoretic framework.
Acknowledgements
We are grateful to M. Caselle, F. Gliozzi and R. Levi for useful discus-
sions.
7
References
[1] F.P. Buff, R.A. Lovett and F.H. Stillinger Jr., Phys. Rev. Lett. 15
(1965) 621.
[2] M.E. Fisher and V. Privman, J. Stat. Phys. 33 (1983) 385.
M.P. Gelfand and M.E. Fisher, Physica A 166 (1990) 1.
[3] M. Caselle, F. Gliozzi and S. Vinti, Phys. Lett. B 302 (1993) 74.
[4] P. Provero and S. Vinti, Physica A 211 (1994) 436.
M.Caselle, R. Fiore, F. Gliozzi, M. Hasenbusch, K.Pinnand S.Vinti,
Nucl. Phys. B 432 (1994) 590.
[5] R.K.P. Zia, Nucl. Phys. B 251[FS13] (1985) 676.
[6] G. Parisi, Statistical Field Theory, (Addison–Wesley, 1988)
[7] S. Coleman, Aspects of Symmetry (Cambridge, 1985) Chap. 7
[8] G. Mu¨nster, Nucl. Phys. B 340 (1990) 559.
[9] G. Mu¨nster, Nucl. Phys. B 324 (1989) 630.
[10] C. Itzykson and J.B. Zuber, Nucl. Phys. B 275 (1986) 580.
[11] B. Bunk, Int. J. Mod. Phys. C 3 (1992) 889.
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