Interfacial roughening in field theory

0.19 MB·English

by  Michael H. Köpf

MS-TP-08-01
Interfacial roughening in field theory
∗
Michael H. K¨opf and Gernot Mu¨nster
8
0
January 23, 2008
0
2

n
a
Abstract
J
3 In the rough phase, the width of interfaces separating different
2 phases of statistical systems increases logarithmically with the sys-
tem size. This phenomenon is commonly described in terms of the

] capillary wave model, which deals with fluctuating, infinitely thin
h
membranes,requiringadhoccut-offsinmomentumspace. Weinves-
c
tigatetheinterfacerougheningfromfirstprinciplesintheframework
e
m of the Landau-Ginzburgmodel, that is renormalized field theory,in
the one-loop approximation. The interface profile and width are
-
t calculated analytically, resulting in finite expressions with definite
a
coefficients. They are valid in the scaling region and depend on the
t
s known renormalized coupling constant.
.
t
a
m KEY WORDS: Interfaces, field theory
-
d
1 Introduction
n
o
c Interface roughening is a phenomenon which has attracted interest of ex-
[
perimental and theoretical investigators, see e.g. [1, 2, 3, 4, 5, 6, 7], since

1 its discovery [8]. It is displayed by interfaces, separating different coexist-
v ing phases or substances of a system of statistical physics, in a range of
8 temperatures T < T < T between the roughening temperature T and
R c R
9
the criticaltemperature T . Rougheningmanifests itself ina characteristic
5 c
dependence of the interface width on the system size. For an interface of
3
. diameter L the width increases logarithmically with L in the roughphase,
1
whereas it remains constant of the order of the correlation length ξ for
0
8 temperatures below TR.
0 Thiseffectiscommonlydescribedtheoreticallyintermsofthecapillary
: wave model or drumhead model [8]. In this model the interface is repre-
v
i sented in an idealized way by an infinitely thin fluctuating membrane, so
X
∗Institut fu¨r Theoretische Physik, Universita¨t Mu¨nster, Wilhelm-Klemm-Str. 9,
r
a D-48149Mu¨nster,Germany;e-mail: [email protected]
1
thattheinstantaneousmicroscopicinterfaceprofileisasharpstepfunction
betweenthe twophases. Nevertheless,inthe thermalaveragethecapillary
wave fluctuations produce a continuous density profile with a finite width
w, whichcanbe shownto be givenby anintegraloverallwave-numbersof
the fluctuations, which is essentially of the form
1 kmax
w2 = dk kD−4, (1)
2πσ Z
kmin
whereDisthenumberofdimensionsofspaceandσistheinterfacetension.
A natural lower limit on the wave numbers is given by the system size,
const.
k = . (2)
min
L
In order to avoid the divergence of the integral, an upper cut-off k has
max
to be introduced. As there should be no waves with wavelength smaller
thantheintrinsicwidthofthephysicalinterface,the uppercut-offistaken
to be of the order of the inverse correlation length. In the case of D = 3,
considered here, one obtains
1 L
w2 = ln (3)
2πσ cξ
with an unknown constant c. The logarithmic increase with L is due to
the contribution of capillary waves with long wavelengths near the system
size L.
Complementary to the capillary wave model is the mean field descrip-
tion of interfaces. In mean field theory and its field theoretic refinements,
interfaces possess an intrinsic continuous profile with a well-definedwidth,
which is proportional to the bulk correlation length and does not depend
on the system size.
Mean field and capillary wave theory can be combined in the “convo-
lution approximation”[9, 10]. In this picture the intrinsic profile describes
the interface on a microscopic scale of the order of the correlation length,
whilecapillarywavetheorydescribesthemacroscopicinterfacefluctuations
of wavelengths much larger than the correlation length. The intrinsic pro-
file is thus centered around a two-dimensional surface subject to capillary
wave fluctuations. In the convolution approximation the square of the re-
sulting total interface width is obtained as a sum of the intrinsic part and
the capillary wave contribution,
1 L
w2 =c ξ2+ ln . (4)
1
2πσ c ξ
2
Thedescriptionofroughinterfacesbymeansofthecapillarywavemodel
and the convolution approximation is unsatisfactory for different reasons.
2
First of all, it has so far not been possible to define the concept of an
intrinsic interface profile and width unambiguously outside a given theory.
In experiments or Monte Carlo simulations of systems with interfaces, the
observedinterfaceprofileandwidtharethetotalones,includingtheeffects
of the intrinsic structure as well as of the capillary waves, and there is no
clear way to separate the intrinsic structure from the effects of capillary
waves. Secondly, the models sketched above contain ad hoc constants,
whose numerical values are arbitrary and cannot be fixed unambiguously
within the models.
In this article we investigate the profile and width of rough interfaces
in a coherent approach from first principles. Statistical systems with co-
existing phases, separatedby interfaces, are described in the framework of
the fieldtheoreticversionoftheLandau-Ginzburgmodel,including fluctu-
ations on all length scales. No cut-off on wave-numbers is introduced. For
explicit calculations we employ the one-loop approximation. It should be
noted,however,thatanextensiontoarbitraryhigherloopordersispossible
inprinciple. Theinterfaceprofile,resultingfromthecalculation,showsthe
expected logarithmic broadening with the system size L. We obtain ana-
lyticalresultsforthenumericalcoefficients,whicharefixedunambiguously
in this approach.
Interfaces have been studied before in the framework of field theory by
other authors. In [11, 12] the profile is calculated to first order in the ǫ-
expansion,whereD =4 ǫandanextrapolationtoǫ=1isnecessary. The
−
ǫ-expansion is an expansion around the four-dimensional case. As can be
seen from Eq. (1), in four dimensions the contribution of long-wavelength
modes converges and no roughening is present. This has the consequence
thatwithintheǫ-expansion,evenafterextrapolationtoD =3dimensions,
roughening effects do not show up, as is well known. The calculation of
[11] is extended to include the effects of an external field in [13].
Our calculations are performed in D = 3 physical dimensions in con-
trast to the ǫ-expansion. The three-dimensional approach is based on a
systematic expansion in a dimensionless coupling [14, 15]. Ultraviolet di-
vergences are treated by dimensional regularization (D = 3 ǫ), which
−
does not vitiate the fact that the results for physical quantities strictly re-
fer to D = 3 dimensions. This is also seen explicitly by the fact that the
calculation reveals the typical roughening effects. Renormalization of the
three-dimensional field theory is performed in the scheme used in [16] to
two-loop order, employing the results of [17, 18].
A three-dimensional study has previously been done in [19], where the
interfaceprofile isconsideredinD =3 dimensionsatone-looporderin the
presence of anexternal gravitationalfield. A functional form of the profile
is given, including capillary wave effects. The dependence on the system
size is, however, not considered. We shall compare our results with the
3
ones of [19] below.
2 Interfaces in field theory
Intheframeworkoffieldtheory,thesystemunderconsideration,possessing
interfaces, is described by an order parameter field φ(x) representing the
difference between the concentrations of the two coexisting phases. The
physicsofthesystemisgovernedbytheLandau-GinzburgHamiltonian[20]
H[φ]= d3x (φ(x)) (5)
Z H
with the Hamiltonian density
1
(φ)= ∂ φ∂ φ+V (φ). (6)
µ µ 0
H 2
In the situation with interfaces the potential is of the double-well type,
V (φ)= g0 φ2 v2 2. (7)
0 4! − 0
(cid:0) (cid:1)
Meanfieldtheoryamountstotheclassicalapproximationwherefluctu-
ations are neglected. The minima of the potential then correspond to the
two homogeneous phases. The mean field correlation length ξ is defined
0
through the second moment of the correlation function in the mean field
approximation. It is given by the second derivative of the potential in its
minima:
3
ξ2 =(V′′(v ))−1 = . (8)
0 0 0 g v2
0 0
With the bare mass m , defined by
0
1
m = , (9)
0
ξ
0
the Hamiltonian density can be written as
1 m2 g 3m4
(φ)= ∂ φ∂ φ 0φ2+ 0φ4+ 0 . (10)
µ µ
H 2 − 4 4! 8 g
0
Thesimplestdescriptionofinterfacesisalsobasedonmeanfieldtheory
[21]. Inthisapproximationthe interfaceprofileisgivenbyminimizationof
the Hamiltonian H with boundary conditions appropriatefor an interface.
The corresponding field equation
δH
=0 (11)
δφ(x)
4
leads to the differential equation
∆φ V′(φ)=0. (12)
− 0
If we choose the interface to be perpendicular to the z-axis, we find the
typical hyperbolic tangent profile [22]
z z
φ(z0)(z)=v tanh − 0 . (13)
0 0 (cid:18) 2ξ (cid:19)
0
Its width is proportional to the mean field correlation length ξ . The
0
parameter z specifies the location of the interface.
0
Essential for a field theoretic treatment, as being considered in this
article, are corrections to mean field theory coming from fluctuations of
the order parameter field. They can be calculated systematically in renor-
malized perturbation theory. The fluctuations result in different modi-
fications of the mean field result, as will be considered in detail below.
First of all, higher order corrections change the form of the profile from
the tanh-function to a different function. Secondly, renormalization of the
parameters v and ξ becomes necessary and, as a result, the mean field
0 0
correlationlengthξ isreplacedbythe physicalcorrelationlengthξ,which
0
diverges near the critical point with a characteristic exponent ν. Finally,
long-wavelength fluctuations lead to the roughening phenomenon, which
impliesabroadeningofthe interface,suchthatitswidthdepends logarith-
mically on the system size and diverges in the limit of an infinite system.
The partition function for the system with an interface can be written
as a functional integral of the form
Z = ϕ exp( H[φ +ϕ]), (14)
0
ZD −
whereφ (z)isaclassicalinterfacesolutionasgivenaboveandϕ(x)denotes
0
the fluctuations aroundit. The Hamiltoniandensity, expressedin terms of
ϕ, reads
1 g g
(φ +ϕ)= (φ )+ ϕ(x)Kϕ(x)+ 0φ (x)ϕ3(x)+ 0ϕ4(x). (15)
0 0 0
H H 2 3! 4!
Here the operator K is given by
m2 g
K = ∆ 0 + 0φ2(x), (16)
− − 2 2 0
where ∆ is the Laplacean.
In the loop expansion the quadratic terms in are treated by means
H
of Gaussianfunctional integrals,andthe higher orderterms aretakeninto
account by Taylor expansions.
5
The spectrum of K is known analytically [23]. We have to employ it
for our calculation and give details below. At this point we would like to
draw the attention to the fact that K has a single zero mode
Kψ(x)=0. (17)
Thezeromodeofthefluctuationoperatorisdirectlyrelatedtotranslations
of the interface, as parameterized by the parameter z . For every value of
0
thisparameter,thefunctionφ(z0)isasolutionoftheclassicalfieldequation.
0
This implies that
dφ(z0)(z)
ψ(z)= 0 (18)
dz
0
is a zero mode of K.
Theexistenceofazeromoderequirestotreatthecorrespondingfluctua-
tions,whichareproportionaltoψ,separatelyfromtheremainingGaussian
integrals in the functional integrals. This is done by the method of collec-
tive coordinates [24]. The collective coordinate in question is z . In the
0
Gaussian integral it is set to an arbitrary value, which we choose to be
z = 0, and the fluctuations are restricted to the space of functions
0 ⊥
N
orthogonalto the zero mode ψ:
d3xϕ(x)ψ(x) =0. (19)
Z
When expectation values in the presence of an interface are calculated,
integration over z would imply averaging over all translations of the in-
0
terface, leading to translationally invariantresults. In case of the interface
profile, however, this is obviously not appropriate, since one is interested
in the profile function relative to the position of the interface. Therefore
integration over z has to be omitted, leaving us with Gaussian integrals
0
over . So the interface profile is given by
⊥
N
φ (x)=φ (x)+φ (x) (20)
c 0 f
with
1
φ (x)= ϕ(x) = ϕϕ(x) exp( H[φ +ϕ]). (21)
f h i Z′ Z D − 0
N⊥
3 The profile equation
For functional integrals over the fluctuation field ϕ Feynman rules
⊥
∈ N
can be set up analogously to the usual case. The propagator and vertices
can be read off the Hamiltonian (15). The propagatoris the inverse of the
fluctuation operator restricted to :
⊥
N
K′ =K . (22)
|N⊥
6
There are three-point and four-point vertices, given by
= g φ (z), = g .
0 0 0
− −
The fluctuation part the interface profile gets contributions from all
orders of the loop expansion:
φ (x)=φ (x)+φ (x)+... (23)
f 1 2
In the one-loop approximation, which we employ, the Feynman diagram
contributing to the profile function
leads to
g
φ (x)= 0 d3x′K′−1(x,x′)K′−1(x′,x′)φ (x′). (24)
1 0
− 2 Z
Here the kernel of the inverse operator K′−1 enters. It would be possible
to calculate φ from this expression. It is, however, more convenient to
1
obtain it as a solution of a differential equation. Acting with the operator
K on Eq. (24), we obtain the profile equation
g
Kφ (x)+ 0K′−1(x,x)φ (x)=0. (25)
1 0
2
In order to solve this equation we need the explicit form of K′−1(x,x),
which is discussed below.
Analternativederivationoftheprofileequationisbasedontheso-called
effective actionΓ[Φ], which is a functionalof a fieldΦ(x). Γ[Φ]is obtained
by Legendre transformationfrom the free energy in the presence of a non-
constant external field. For constant Φ the effective action reduces to the
Gibbs potential. For a definition and discussion see e.g. [25]. Calculating
Γ[Φ] in the one-loop approximation and finding the interface profile φ(x)
as a stationary point of Γ,
δΓ
=0, (26)
δΦ(x)
again leads to Eq. (25).
7
4 Solution of the profile equation
The inverse of the fluctuation operator K′ at coinciding arguments, which
enters the profile equation, can be obtained by means of the spectral rep-
resentation. K is the sum of the negative two-dimensional Laplacean and
a one-dimensional Schr¨odinger operator K˜,
K = ∆(2)+K˜ , (27)
−
where
3m2 m
K˜ = ∂2+m2 0 sech2 0z . (28)
− z 0− 2 (cid:16) 2 (cid:17)
The negative Laplacean on the L L square has eigenvalues
×
2π
k2 with ~k = ~n, ~n Z2, (29)
L ∈
and corresponding eigenfunctions
ϕ~n(~x)=L−1 ei2Lπ~n·~x, ~x [0,L]2. (30)
∈
The spectrum ofK˜ is knownexactly [23]. It consistsoftwo discrete eigen-
values
3m m
ω(0) =0, ψ (z)= 0 sech2 0z , (31)
0 r 8 2
(cid:16) (cid:17)
3 3m m m
ω(1) = m2, ψ (z)= 0 tanh 0z sech 0z , (32)
4 0 1 r 4 2 2
(cid:16) (cid:17) (cid:16) (cid:17)
and a continuum
ω =m2+p2 with p R, (33)
p 0 ∈
m2 3 m m
ψ (z)= eipz 2p2+ 0 m2tanh2 0z +3im ptanh 0z
ωp Np (cid:20) 2 − 2 0 (cid:16) 2 (cid:17) 0 (cid:16) 2 (cid:17)(cid:21)
(34)
with the normalization factor
Np =(2π(4p4+5m20p2+m40))−12. (35)
The spectrum of K is thus given by
4π2
λ = n2+ω, Ψ (x)=ϕ (~x)ψ (z), (36)
~nω L2 ~nω ~n ω
where ω runs through the eigenvalues of K˜. The zero mode, discussed
above, is represented by Ψ .
~00
8
In terms of the spectrum we write
1
K′−1(x,x)= ψ (x)ψ∗(x) . (37)
Z λ λ λ
X
λ
Inserting the explicit expressions, we obtain
m m
K′−1(x,x)=C +(C C +C )sech4 0z +(C +C )sech2 0z ,
0 1 2 4 2 3
− (cid:16) 2 (cid:17) (cid:16) 2 (cid:17)
(38)
where the coefficients C are
i
1 1
C = dp , (39)
0 2π Z 4π2n2+(m2+p2)L2
X~n 0
3m 1
0
C = , (40)
1 8 4π2n2
~nX6=~0
3m 1
0
C = , (41)
2 4 4π2n2+ 3m2L2
X~n 4 0
m2+p2
C = 3m2 dp 2 0 , (42)
3 − 0Z Np 4π2n2+(m2+p2)L2
X~n 0
9 1
C = m4 dp 2 . (43)
4 4 0Z Np 4π2n2+(m2+p2)L2
X~n 0
These expressions are divergent and have to be regularized, as discussed
below.
Withthe explicitformofK′−1(x,x)athand,the solutionofthe profile
equation is found as
g v m
0 0 0
φ (z)= C tanh z
1 2m20(cid:26) 0 (cid:16) 2 (cid:17)
2 m
0
(C C +C )tanh z
1 2 4
−(cid:20)3 − (cid:16) 2 (cid:17)
m m
(C +C +C ) 0z sech2 0z . (44)
0 2 3
− 2 (cid:21) (cid:16) 2 (cid:17)(cid:27)
Written in this way, the expression for the profile contains the divergent
coefficientsC aswellasthebareparametersg ,m andv . Inordertoar-
i 0 0 0
rive at a finite expressionin terms of physicalparameters,renormalization
has to be performed.
The divergences have to be treated in some regularizationscheme. We
choose to employ dimensional regularization in D = 3 ǫ dimensions.
−
9
It should be noted that this does not amount to an ǫ-expansion, since
after renormalization ǫ is sent to zero, whereas in the ǫ-expansion one
has D = 4 ǫ and the results have to be extrapolated to ǫ = 1. So
−
our use of dimensional regularization does not vitiate the fact that the
results for physical quantities strictly refer to D = 3 dimensions. Using
other regularization schemes, like Pauli-Villars, would lead to the same
final results.
We adopt the renormalization scheme used in [16] to one-loop order.
The renormalizedmassm =1/ξ isequalto theinversecorrelationlength
R
ξ, which in turn is defined through the second moment of the correlation
function. Thefieldφanditsexpectationvaluevarerenormalizedaccording
to
1 1
φ (x)= φ(x), v = v, (45)
R R
√Z √Z
R R
where Z is the usual field renormalization factor. The renormalized cou-
R
pling is specified as in [26] through
3m2
g = R . (46)
R v2
R
In addition we define a dimensionless renormalized coupling according to
g
R
u = . (47)
R m4−D
R
Employing the relations given in [17, 18], the bare quantities m and g
0 0
are expressed in terms of their renormalized counterparts.
The coefficients C are evaluated in the same scheme. Leaving out the
i
lengthy details, we quote the results
m
0
C = (48)
0
−4π
3m
0
C +C = ln3 (49)
2 3
16π
3m
0
C C +C = ( α+ln(m L)) (50)
1 2 4 0
− 16π −
with
3Γ2(1/4)
α=ln γ 1.832, (51)
(cid:18) 2√π (cid:19)− ≈
where γ 0.577 is Euler’s constant.
≈
ThecoefficientsC containadditionaltermsdecayingexponentiallyfast
i
with L, which we neglect here.
Inserting everything into the expression for the interface profile yields
the renormalized interface profile φ (z) depending on the parameters m
R R
10