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Coarse Graining the φ4 Model: Landau-GinzburgPotentials from
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Landau-Ginzburg Potentials from Computer Simulations',
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Ferroelectrics, 354:225–237, 2007Copyright © Taylor &
Francis Group, LLCISSN: 0015-0193 print / 1521-0464 onlineDOI:
10.1080/00150190701454982
Coarse Graining the φ4 Model: Landau-GinzburgPotentials from
Computer Simulations
A. TRÖSTER∗ AND C. DELLAGO
Faculty of Physics, University of Vienna, Boltzmanngasse 5,
A-1090Vienna, Austria
We discuss the problem of how to calculate Landau and
Landau-Ginzburg free energiesfor lattice spin models from computer
simulations. In setting up a proper simulationalgorithm, emphasis
is placed on the coarse grained nature of these potentials,
whichmust be take into account by any suitable simulation approach.
The development oftheory and simulation results is reviewed and the
results of a novel Monte Carlo al-gorithm using Fourier amplitudes
are presented, which partly confirm and sharpen theassumptions on
the temperature behavior of the Landau-Ginzburg coefficients made
inthe literature.
PACS Numbers: 05.10.Ln; 05.70.Ce; 64.60.-i; 05.10.Cc
Keywords: Monte Carlo simulations; coarse graining;
Landau-Ginzburg model
Introduction
Up to the present days, the importance of Landau and
Landau-Ginzburg theory as one ofthe central paradigmas in both
qualitative as well as quantitative approaches to
describecontinuous or close-to-continuous phase transitions can
hardly be overestimated. In fact, itis probably impossible to find
any textbook on phase transitions that does not include anextensive
discussion of its main concepts. The following short account is
neither followingthe actual historical development nor aiming at
completeness. Rather, we give an introduc-tion to Landau-Ginzburg
theory with a bias on the problems we address in the remainingpart
of the paper.
A central notion of the Landau approach is the so-called order
parameter m, a quantityof generally tensorial type, whose emergence
and transformation properties at the phasetransition reflect the
spontaneous symmetry breaking accompanying the transition. Its
de-tailed behavior below the transition is governed by a certain
thermodynamic potential, theso-called Landau free energy F(m), for
which m plays the role of a variational parameter,such that upon
minimization with respect to m the physical properties of the
system canbe calculated. While an explicit calculation of this
potential starting from a given micro-scopic model turns out to be
an extremely difficult task for all but the most trivial cases,in
the vicinity of a continuous phase transition, however, the
presupposed continuity of thetransition suggests to replace the
full but unknown potential F by its expansion in powersof the order
parameter components truncated at low order. The general structure
of the
Received in final form March 25, 2007.∗Corresponding author.
E-mail: [email protected]
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symmetry-allowed terms in such a truncated expansion, which is
generally referred to as aLandau potential, can be deduced from
group representation theory [1, 2]. What remainsto be determined is
the temperature behavior of the expansion coefficients for these
terms.Both too far from and too close to the phase transition, this
also represents a formidabletask—for different reasons, as we will
discuss below. However, in an “intermediate” tem-perature range,
excluding the very vicinity of the phase transition as well as the
far highand low temperature domains, Landau made quite reasonable
assumptions concerning thegeneral behavior of these coefficients
for “generic” types of transitions. Without going toomuch into
details, we note that Landau, assuming that the coefficients are
analytic functionsof temperature, concluded that for a generic
second order phase transition the coefficientof the part of the
potential which is quadratic in the order parameter components
behaveslinearly with temperature and changes sign at the transition
temperature, while the higherorder expansion coefficients can be
taken to remain constant. In particular, for the simplestcase of a
scalar order parameter m representing the breaking of a Z2 (Ising)
symmetry in asystem of volume V , and for which the Landau free
energy function takes the form
F(m) = V(
A22
m2 + A44
m4 + A66
m6 + · · ·)
(1)
Landau argued that near the transition A2 ∼ A(0)2 (T − T0)
should behave linearly withT − T0 while the coefficients A4, A6, .
. . can be taken to be roughly constant around T0.Analyticity thus
gives rise to a kind of universal parametrization accompanyied by
the setα = 0, β = 1/2, γ = 1, δ = 3 of universal values for the
critical exponents, should one takethe expansion seriously even at
the very transition point. Moreover, this parametrization,which is
mostly implied when one speaks of “Landau theory,” represents a
simple andready-to-use approach which turned out to be extremely
sucessful in providing both aqualitative as well as (in many cases)
quantitative description of second order and close-to-second order
phase transitions outside the narrow critical teperature range. In
particular,Landau theory makes predicitions on the temperature
dependence of the order parameterand its susceptibility (the
so-called Curie-Weiss law), the possible domain structure inthe low
temperature phase and coupling effects to other degrees of freedom.
Over thelast decades, these predictions have been verified by
comparison to experimental data incountless cases.
As is well known, however, it soon turned out that despite its
heuristic success, theLandau description fails in what it was
actually designed to account for, namely in describingthe close
vicinity of a critical point. Actually, both analytically solvable
models like the 2dIsing model as well as renormalization group
calculations of critical phenomena are knownto display a behavior
dramatically different from the simple Landau predictions. This
fact isalso reflected by the apparent large deviations of the
actual values of the critical exponentsfrom the above “classical”
Landau predictions. The basic reason for this failure of
Landautheory is the neglect of spatial fluctuations which is
already implicit in the assumptionof a spatially homogeneous order
parameter[3]. To incorporate the effects of such
spatialfluctuations, the order parameter must be regarded as a
tensor field rather than a mereconstant. The Landau potential is
then replaced by an integral over a potential density. Itremains to
account for the free energy costs of spatial variations of the
order parametercomponents. Since such variations imply nonzero
partial derivatives of the order parametercomponents, the potential
density must therefore be augmented by low order symmetryinvariants
constructed from the order parameter components and its partial
derivatives. Tolowest order, this usually amounts to adding a
so-called gradient correction, which consistsof all quadratic
invariants built from the first partial derivatives of the order
parameter
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Coarse Graining the φ4 Model 227
components, to the potential density. The above reasoning for
our simple scalar examplethus suggest a free energy functional of
the form
F[m] =∫
Vd3x
[D(∇m)2(x) + A2
2m2(x) + A4
4m4(x) + A6
6m6(x) + · · ·
](2)
where the gradient coefficient D must again be assumed to be
constant around T0. ThisLandau-Ginzburg functional has proved to be
a valuable tool for both qualitatively as wellas quantitatively
incorporating the effects of small fluctuation into Landau theory.
Close tothe critical temperature Tc of a second order phase
transition, however, the system’s behav-ior is dominated by
fluctuation effects, and it is thus impossible to treat them as
“small”.Accordingly, it comes as no surprise that this ansatz also
fails in the critical region, as onecan show that it gives again
rise to the classical values for the exponents α, β, γ, δ andalso
pins down the exponents ν and η to their classical values ν = 1/2,
η = 0. The originof this problem is well understood. In fact, it
turns out that Landau-Ginzburg theory itselfprovides a
semi-quantitative self-consistency condition for its own validity
[4, 5], known asthe Levanyuk-Ginzburg criterion. To date, several
different derivations of this criterion havebeen given. Without
going into details, here we only note that they all rest on a
comparison ofthe magnitude of the Landau predictions of observables
like the specific heat or the order pa-rameter susceptibility to
the corresponding contributions due to the lowest order
fluctuationcorrections. Attempts to actually penetrate the usually
narrow but conceptually highly inter-esting region that the
Ginzburg criterion excludes and eventually led to the construction
ofrenormalization group theory by Wilson and others. In this
development, Landau-Ginzburgtheory continued to play a major role,
as the LG free energy turned out to provide a kindof “minimal
Hamiltonian” for the field-theoretic analysis of the problem. In
fact, we wouldlike to stress that only in the context of the
renormalization group one is actually allowed(and, as an in-depth
analysis reveals, even forced [6–8]) to drop “spurious” terms
likehigher order powers of the order parameter, higher order
gradient contributions, additionalT -dependences of coefficients
and so on, from the LG functional. However, the “relevance”or
“marginality” of the remaining terms and the irrelevance of the
dropped couplings maybe completely different for systems of
different dimensionalities and symmetries.
The Philosophy of Coarse Graining
Despite the beauty of renormalization group theory, it turns out
that for most phase transi-tions in solid state physics, e.g. for
structural phase transitions, the critical region—if it existsat
all—is extremely narrow and consequently it is difficult to study
their critical behaviorexperimentally. On the other hand, many
workers sucessfully apply Landau and Landau-Ginzburg theory to
quantitatively explain solid state data. An analysis of the range
of validityof the set of assumptions on which Landau and
Landau-Ginzburg theory is based upon istherefore of vital interest
to anyone applying these concepts outside the critical region.
There are group-theoretic arguments that allow to determine the
minimal set of in-variants (and thus the maximum total power of the
order parameter components) that haveto be included in the Landau
potential to provide a unique characterization of the
phasetransition [9], but the sign of the corresponding coupling
coefficients is not determined bysymmetry. Therefore one may be
forced or tempted to introduce a particular coupling ofstill higher
order for stability or accuracy reasons or propose a “nonstandard”
temperaturevariation of a particular coupling coefficient to
quantitatively explain a certain experimentalfinding. Nevertheless,
most Landau practioners tend to feel uncomfortable with such a
useof the theory and giving a physically sound justification of
introducing such “deformations”of the theory is usually considered
a delicate subject. Yet, the question of this noncritical
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“relevance” or “irrelevance” of a particular coupling or
coefficient behavior is not the onlyhidden difficulty in using the
Landau-Ginzburg approach as a noncritical description.
Another fact that is frequently obscured in simple presentations
of the theory is thecoarse-grained nature of the Landau and
Landau-Ginzburg approaches. This becomes clearby looking at the
example of lattice spin systems. In these systems, even at
intermediate tem-peratures the individual microscopic spins may
show violent spatial variations. In contrast,the macroscopic
magnetization which typically plays the role of the order parameter
field,is regarded as smooth. The passage from microscopic
individual spins to mesoscopic andmacroscopic smooth fields
necessarily involves a certain spatial averaging procedure.
Suchcoarse graining approaches, in which a microscopically rapidly
oscillating quantity, whoseshort range details are irrelevant to
the actual problem, is replaced by a comparatively slowlyvarying
continuous macroscopic quantity using some smooth spatial averaging
procedurewhich is defined on a much larger length scale, have
proved to be useful in diverse branchesof physics. Examples include
e.g. the derivation of Maxwell’s equation for a macroscopicsystem
[10] in classsical electrodynamics or that of the Navier-Stokes
equation in hydrody-namics. Quite analogously, in the case of
lattice spin models, one must realize that what wemean by a smooth
order parameter field is actually the local magnetization averaged
overvolumes that are considerably larger than the unit cells of the
underlying lattice. An orderparameter field is thus implicitly
defined with respect to some chosen coarse graining lengthl, which,
in the case of a cubic lattice of lattice constant a, should
satisfy a � l � L , whereL3 is the total volume of the system, in
such a way that variations on length scales between aand l should
be averaged out [11]. Switching to a Fourier representation, this
in turn impliesthat fluctuations with wave numbers between a cutoff
= 2π/ l and the Brillouin zoneboundary 2π/a should be averaged out,
leaving us with an effective Hamiltonian for theremaining long
wavelength degrees of freedom – which is nothing but the
Landau-Ginzburgfree energy functional (we assume for simplicity
that the critical wavevector of the transitionis kc = 0). Thus, a
Landau-Ginzburg functional is (explicitly or implicitly) always
definedwith respect to a certain cutoff, which should in principle
be supplied together with thecorresponding set of its coupling
coefficients discussed above.
How do we choose the “correct” scales l or for a given problem?
In principle anyscale will do, as we will see that further
averaging over the remaining degrees of freedomalways should allow
to reconstruct the full partition function (at least if it could be
done in anexact way). In practice, the choice of scale depends on
the situation we want to study. Firstof all, in a numerical attempt
to describe experimental data, it is clear that for most
practicalcases the cutoff must not be chosen too large, since this
would require a knowledge ofthe lattice dispersion way beyond the
parabolic approximation corresponding to the simplegradient
correction in real space. However, for the study of phase
transitions, this is not asevere restricition since the anomalies
appearing in such transitions result from long rangecorrelations,
i.e. the behavior of the system at small k-vectors.
In contrast to the above reasoning, a situation in which one
prefers an “intrinsic” choicefor the coarse graining context occurs
in the study of metastable states well below a phasetransition and
their decay[12]. For example, consider an Ising system with fixed
value ofthe total “magnetization”
∑x sx (such systems are e.g. realized by a binary alloy or a
lattice
of fixed total density). If the system is cooled below its
critical point, phase separation willset in, and an interface
between the two appearing domains will form. As we will
discussbelow, this behavior, which sets in as soon as the
correlation length ξ becomes much smallerthan the length scale L of
the total system, is due to the extensive growth of the bulk
freeenergy gain, which starts to outgrow the subextensive free
energy costs of the domain wall.
Now suppose that we divide the configuration space of our system
into cubic boxes ofsize l. Let us further impose a coarse graining
constraint on the system, which we formulate
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Coarse Graining the φ4 Model 229
as follows: We shall require that, in addition to the fixed
global value of the magnetization,only such microstates are
admissible for which all the cell magnetizations are equal to
thisglobal magnetization. In other words, we shall constrain our
system to be homogenous onthe box level. How will this constrained
system behave when the temperature is loweredstarting from Tc? Well
below Tc, the correlation length ξ (T ) will finally drop below
thelength scale l of the boxes. Then, by the same reasoning as in
the case of the global system,phase separation will set in inside
the boxes. On the other hand, this little thought experimentreveals
that imposing such a coarse graining constraint for a box size of
the order of l ∼ ξ (T ),phase separation is effectively supressed
and the decay of the metastable homogeneous stateof the system into
an inhomogeneous stable state is inhibited. Therefore, the study of
suchcoarse-graining potentials is of great interest for
understanding the problems posed bymetastable states [13–15].
Review of Previous Simulation Results
The physics of structural phase transitions is typically
governed by the competition of severalenergy scales of different
physical origin and accompanying length scale. On the one hand,one
can frequently define a so-called on-site potential with (possible
multiple) energy barri-ers separating local energy minima. On the
other hand, there is also an energy scale attachedto the coupling
of these local degrees of freedom between different neighboring
lattice sites.Apart from the interaction range and the detailed
structure of the onsite potential, the ratio ofthese scales, which
is known as the displaciveness or displacive degree can thus be
regardedas the main non-universal parameter characterizing such
systems. In particular, systems inwhich the neighbor couplings
dominate over the heights of the local on-site energy wells,
aretermed displacive, while the opposite state of affairs
characterizes order-disorder systems.
Probably the simplest model system to capture these features is
the so-called φ4-model[16]. Unfortunately, despite its apparent
simplicity, the literature reveals a considerablediversity in the
conventions for parametrizing the model (of which the reader will
only haveto absorb a small part below). In what follows we will
work on a 3-dimensional simplecubic lattice with lattice sites x, y
and periodic boundary conditions. For simplicity wechoose units in
which the lattice constant a = 1 and denote the corresponding basic
cubictranslation vectors as a1 = (1, 0, 0), a2 = (0, 1, 0) and a3 =
(0, 0, 1). In this work we studythe class of generalized φ4-models
defined by a lattice Hamiltonian
H[{s(x), H (x)}] = 12
∑x,y
C(x − y)[s(x) − s(y)]2
+∑
x
[A22
s2(x) + A44
s4(x) − H (x)s(x)]
(3)
which combines an on-site fourth order double well potential and
a short-ranged site-siteinteraction for real-valued spins s(x) in
an external field H (x). We can make the abovemodel look a little
more Ising-like by rearranging the site-site interaction according
to
H[{s(x), H (x)}] = −12
∑x,y
J (x − y)s(x)s(y) +∑
x
[Ã22
s2(x) + A44
s4(x) − H (x)s(x)]
(4)
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where J (x, y) = 2C(x, y) is reminiscent of an Ising type
interaction coefficient, and theonsite potential is “renormalized”
as follows. For the lattice interaction J (x − y) it willbe assumed
that it is short-ranged, such that in particular the
∑y |J (x − y)| < ∞ is fi-
nite, guaranteeing the existence of the model’s thermodynamic
limit [17]. Guarded by thisassumption, it was therefore possible to
introduce the parameters
J∞ :=∑
y
J (x − y), Ã2 := A2 − 6J∞ (5)
in the above formula. As we will not be interested in absolute
energy scales, we will set theBoltzmann constant kB = 1 and −A2 =
A4 = 1 from now on.
In the special case where J (x − y) corresponds to a nearest
neighbor interaction it isconvenient to define a function
γ (x) :={
1, ±x ∈ {a1, a2, a3}0, else
(6)
such that the coordination number γ of the cubic lattice is
equal to γ = ∑x γ (x) = 6, andthe lattice interaction J (x − y)
corresponding to a cubic nearest neighbor (NN) couplingcan be
written as J (x − y) ≡ Jγ (x − y) = 2Cγ (x − y) and J∞ = γ J = 2γ C
. A positivecoupling constant J > 0 then favors parallel
alignment of the spins s(x).
The model’s displacive degree is governed by J∞ (indeed it can
be shown that forhomogeneous external field the coupling J (x − y)
enters only through J∞ in the model’smean field free energy). The
system would then be termed as displacive for J, J∞ � 1,whereas for
J, J∞ � 1 it resembles an order-disorder system. In fact it is
clear that in theorder/disorder limit the above model actually
becomes equivalent to an Ising model. Mostreal structural phase
transitions are situated in the crossover regime between displacive
andorder-disorder limits. Representing the simplest caricature of
an order/disorder—displacivecrossover parametrization, φ4 models
have consequently received a lot of attention. His-torically,
analytic calculations [16, 18] were followed by simulations using
Molecular Dy-namics[19] and Monte Carlo [20–24] computer
simulations. The basic approach in thesesimulations was to compute
the zero field (H = 0) order parameter probability
distributionfunction
P(m) = 1Z (0)
∫Ds δ
(m − 1
N
∑x
s(x))
e−βH[{s(x),0}] (7)
with∫Ds := ∫ ∞−∞ ∏x ds(x), and identify
F(m) ≡ − 1β
log [Z (0)P(m)] (8)
with the Landau free energy of the system. Here, of course,
Z [{H (x)}] =∫
Ds e−βH[{s(x),0}] (9)
denotes the unconstrained partition function of the model (see
Ref. [25] for details). Inthese computer simulations, which were
aimed at determining the Landau free energy fordifferent
temperatures, the intrinsic coarse-grained character of the Landau
free energy wasthus simply ignored and the Landau free energy in
fact identified with the (Helmholtz)free energy of the system in
the simulation, for an ensemble where the magnetization
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Coarse Graining the φ4 Model 231
in controlled. In the light of the preceeding discussion, this
approach must therfore beconsidered as problematic. A detailed
critical review was given in Ref. [25]. Here we onlysummarize its
weaknesses as follows.
It is well known that for stability reasons the Helmholtz free
energy of the infinitesystem should satisfy convexity conditions.
Below Tc, this implies that any concave partof the free energy
should be replaced by its convex envelope, an operation which is
knownas Maxwell’s construction. Physically, this is of course
caused by phase separation in theinfinite system. As we have also
discussed at length above, in the finite system of volumeL3 at
fixed global magnetization, phase separation sets in as soon as the
correlation lengthdrops well below L and the surface free energy
“costs” accompanying the formation of adomain wall are reduced
below a critical value. Mathematically, phase separation is
signaledby a “flat” central part of the simulated free energy as a
function of the magnetization m,since no free erergy costs are
caused by a slight displacement of the planar domain wallfrom its
position for m = 0. But this in turn means that any power series
expansion of thefree energy below Tc is useless—it is just
identically zero. In other words: as soon as phaseseparation sets
in, the free energy becomes nonanalytic. The tricky point, however,
is, thatin simulating a system of the above φ4 type, which—for the
purpose of a simulation—is necessarily finite—these facts may be
obscured at sufficiently high displacive degree,since large values
of J imply large energetic costs for the formation of domain
walls.Also, the total simulation time of a Monte Carlo simulation
is necessarily limited. At lowtemperatures a straightforward Monte
Carlo simulation therefore will fail to explore thepotential for
unlikely values of the magnetization but will stay trapped in the
vicinity ofone of its minima. In particular, information on the
height of the central barrier separatingthe regions of positive and
negative magnetization is unavailable from such simulations.
Toovercome these difficulties, in Ref. [25] thermal Wang-Landau
simulations [26–28] werecarried out, since this approach allows to
explore the potential over the whole magnetizationregion of
interest. In accordance with the expectations outlined above, the
free energy stillappears to be a nice analytic double well for
finite displacive systems. For order/disordersystems, however, the
simulations clearly confirmed the qualitative picture sketched
above,not only revealing the onset of the Maxwell construction but
also signs of phase separationthrough the formation of cylindrical
and spherical nuclei. The situation can be summarizedby noting that
a simulation evaluating the probability distribution (7) is
actually incapableof determining a meaningful “Landau” free energy.
The interested reader is referred toRef. [25] for more details.
We close this section with a brief account on the simulations of
coarse-grained Isingsystems. Motivated by the main objective to
understand the physics of metastable states, theone-to-one
correspondence of the Ising model with the lattice gas model led to
Monte Carloattempts to—approximately—simulate coarse grained free
energies of Ising-like systems,which can be traced back from the
early 1980’s [29–31] to nowaday’s active research (seeeg. Ref. [32]
and references therein). Motivated by the cell subdivision
construction wehave outlined above, a real space Metropolis
algorithm which is compatible with a cellsubdivision construction
of the type we have outlined above seems to be a natural approachto
follow. The basic idea is as follows. The system is initialized in
a state in which allsubcells of the system have the same value of
their “cell magnetization.” In constructingthe random walk, one
must then design each Monte Carlo move in the configuration spaceof
the system in such a “coordinated” way that after each step this
property of “homogenityof the cell magnetization” is preserved.
Despite its simplicity, it soon turns out that this idea has
several severe drawbacks. Firstof all, we note that quite trivially
such an approach will suffer from severe constraints arising
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from the restricted possible choices for dividing a finite
lattice of manageable size L intocommensurable subcells. Also, each
admissible trial move necessarily involves the flippingof a large
number of spins (at least as many as there are subcells in the
system), and thus leadsto generally large energy differences of
successive configurations. But this in turn implieslow Monte Carlo
acceptance rates, which renders an algorithm of the above type
practicallyuseless even at modestly low temperatures. Moreover, its
seems to be impossible to includegradient corrections in such a
simulation, let alone to compute the full k-dispersion of
theresulting coarse grained free energy.
Coarse Grained Potentials from Simulations
The above obstacles indicate that a satisfying solution of the
problem of how to computeLandau-Ginzburg free energies might call
for a fundamentally different approach. Indeed,the basic idea of
coarse graining is that in studying the systen the focus is put on
the“long wavelength” degrees of freedom, i.e. a kind of
hydrodynamic limit. This way ofthinking suggests that we should
trade the direct space/spin representation of the systemfor a
wavevector/amplitude description, i.e. formulate the whole problem
in Fourier space.While the technical details of constructing such
an algorithm are considerable [33, 34], wewill sketch the main idea
below.
In the proposed simulation algorithm, the real and imaginary
parts of the amplitudesof the discrete Fourier modes
s̃(k) := N−1/2∑
x
s(x)eikx (10)
for each configuration of spins s(x) of our φ4 model on the
lattice should play the role ofMonte Carlo variables, and Monte
Carlo moves consist in performing shifts of individualamplitudes.
In particular, amplitudes belonging to “fast” variations of the
spin configurationare labeled by “large” k-vectors, i.e. k-vectors
residing near the Brillouin zone boundary,while those describing
the “slow” variations have small wave vectors near the zone
center.Averaging out the fast variations of the system thus means
performing a partial trace of thepartition function over the modes
s̃(k), whose k-vectors have components ki which are, say,larger
than some given cutoff value = 2πl/L , l = 1, . . . , L/2. This
partial trace thendefines an effective Hamiltonian for the
“surviving” slow modes, which we will denote asη̃(k) in the
following. By symmetry considerations this Hamiltonian must have
the structure[35]
β H (L ,l)[η] = 12
∑k
u(L ,l)2 (k)η̃(k)η̃(k∗)
+ 14N
∑k1,···,k4
u(L ,l)4 (k1, · · · , k4)η̃(k1) · · · η̃(k4)�( 4∑
i=1ki
)
+ 16N 2
∑k1,···,k6
u(L ,l)6 (k1, · · · , k6)η̃(k1) · · · η̃(k6)�( 6∑
i=1ki
)+ · · · (11)
where the lattice delta function � (k) evaluates to 1 if k is a
reciprocal vec-tor and to zero otherwise. It then remains to
extract the coefficients u(L ,l)2 (k), u
(L ,l)4
(k1, . . . , k4), u(L ,l)6 (k1, . . . , k6), . . . from the
simulation result. This is still a hard problem as
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Coarse Graining the φ4 Model 233
far as their general k-dependence is concerned. However, to
compute a “classical” Landau-Ginzburg potential only the
homogeneous coefficients u(L ,l)4 (0, . . . , 0), u
(L ,l)6 (0, . . . , 0) and
the full quadratic dispersion term u(L ,l)2 (k) are needed. To
achieve this, we perform simu-lations at “special” configurations
of the surviving slow modes. To compute the homoge-neous
coefficients u(L ,l)2 (0), u
(L ,l)4 (0, . . . , 0), u
(L ,l)6 (0, . . . , 0), one performs a series of Wang-
Landau simulations [25–28], summing over the fast modes while
setting all the slow modesexcept the “central” homogeneous mode
η̃(0) identically to zero. The logarithm of the re-sulting
probability distribution P0(η̃0) for this mode is then fitted to a
polynomial, whichallows to extract the above coefficients as fit
parameters. Generalizing this idea to k = 0,we consider
configurations where only the real part r = η̃(±k) of one of the
slow modesis allowed to be different from zero. Just like for the
“ordinary” zero mode, we can thencompute the corresponding
probability distribution, which we denote by Pk(r ). This func-tion
is again fitted to a polynomial in the variable r , whose curvature
at zero gives thecorresponding value u(L ,l)2 (k). Doing this for
various k-vectors thus allows to reconstructthe full quadratic
dispersion term.
To test the algorithm sketched above, we have performed a series
of simulations fordifferent temperatures, cutoffs and corresponding
possible choices of k-vectors, takingan order/disorder system with
nearest neighbor coupling parameter C = 0.1 at a systemsize N = L3
= 123 as our first example. As we have just seen, the above method
shouldin principle allow to determine the deviations of the
dispersion from a pure “gradientcorrection“ ∝ k2. Consider e.g. the
case of a nearest neighbor interaction using the notationintroduced
above. For the Fourier transform of this lattice interaction we
obtain the concreteform
− J̃ (k) = −∑x∈
J (x)e−ikx = −J3∑
μ=1(e−ikaμ + eikaμ ) = −2J
3∑μ=1
cos kμ (12)
of the (negative) Fourier transform of the nearest neighbor
lattice interaction. Using thetrigonometric identity cos x = 1 − 2
sin2 x2 , we rewrite this as
− J̃ (k) = −2J3∑
μ=1
(1 − 2 sin2 kμ
2
)= −J∞ + 4J
3∑μ=1
sin2kμ2
(13)
which obviously includes a considerable extension of the simple
lowest order gradientcorrection − J̃ (k) = −J∞+ Jk2 +O(k4μ). This
formula for the “bare” NN lattice interactionsuggests to try using
the rather restrictive ansatz
u(L ,l)2 (k) ≡ u(L ,l)2 (0) + 4β D03∑
i=1sin2(ki/2) (14)
as a fitting function for u(L ,l)2 (k). Note that as far as the
k-dependence is concerned, theabove ansatz contains only a single
free parameter D0 multiplying an otherwise fixed k-dependent
dispersion term. Despite the rigidity of this parametrization, the
quality of thecorresponeding fits to the curvatures for the
potentials Pk(r ) (which in turn originate fromfits as explained
above!) turned out to be excellent, as is obvious from Fig. 1.
Moreover,apart from statistical fluctuations around the “bare”
value D0 ≡ J = 2C , the parameter D0shows no pronounced temperature
variation over a large temperature interval, as can be seenfrom
inspecting Fig. 2. In other words, as far as the k-dependence of
the quadratic termof our simulated Landau-Ginzburg Hamiltonian is
concerned, we conclude that—except
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234 A. Tröster and C. Dellago
Figure 1. Fits of u(L ,l)2 (k) to ansatz (14) for various
temperatures at C = 0.1, L = 12 and coarsegraining length l = 4
along the 100 direction for vectors k = 2πmi/L , mi = 0, 1, 2,
3.
maybe for the close vicinity of the critical temperature—the
presence of anhamonicitydoes not lead to significant
temperature-dependent deviations of the dispersion term
(whichrepresents a generalized gradient correction in the sense of
Landau-Ginzburg theory) fromthe “bare” lattice dispersion.
In contrast to the gradient correction, it is already clear from
mean field considera-tions that the coefficients of the pure
“Landau” contribution to the potential should not beexpected to
strictly follow the temperature behavior predicted by Landau in
general butonly approximately for a small temperature interval. For
instance, in Ref. [25] we listed thewell-known fact that all
polynomial order parameter expansion coefficients of the the
mean
Figure 2. Values for fit parameter D0 obtained from fits at
various coarse graining lengths l in the100-direction. The observed
fluctuations are largest for l = 1 due to the trivial fact that for
l = 1,apart from k = 0, only one k-vector along 100 is
available.
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Coarse Graining the φ4 Model 235
Figure 3. T -dependence of coefficients A(12,l)2 , A(12,l)4 ,
A
(12,l)6 for L = 12, C = 0.1. Symbols: l = 0
(stars), l = 1 (full boxes), l = 2 (full triangles), l = 3 (open
boxes), l = 4 (open triangles). Lineindicates mean field
results.
field potential of the Ising model (which is at the
order/disorder limit C → 0 of the presentmodel) are linear
functions of temperature. On the other hand, for displacive systems
thecorresponding mean field potentials should conform to the
standard 2–4 form and indeedexhibt a linear T -dependence of the
quadratic and a constant fourth order expansion coef-ficient. To
test for this qualitative behavior and also check that our findings
concerning theT -independence of gradient corrections carry over to
more displacive systems, we extendedour simulations to the
corresponding systems with parameter values C = 1.0, 10.0.
Thesesimulations in fact confirm the above qualitative statements
as follows. For the gradientcorrections the observed behavior was
similar to that found for C = 0.1. As concerns theLandau
coefficients A2, A4 and A6, which are defined as expansion
parameters of the effec-tive free energy with respect to the
magnetization per site m = √N η̃(0) similar to Eqn. 1,the above
expectations can be compared to the results of the simulations in
Figs. 3–5.In passing, we note that, as expected from general
entropy arguments, the “coarse grainedcritical temperature” T (L
,l) at which the quadratic coefficient A(L ,l)2 reverses its sign
andthe character of the homogeneous part of the coarse-grained
potential thus changes fromsingle to double well, increases with
increasing coarse graining length l, which, as we havestressed in
the introductory part of the paper, must be regarded as a parameter
definingthe “experimental window” that we have to choose for the
problem we want to describe.In particular, since the coarse grained
potential for l = L/2 is just trivially identical tothe onsite
potential which is always double-welled, T (L ,l) must necessarily
diverge for thisvalue of l. In understanding this variation of T (L
,l), it is important to realize that in oursimulation algorithm
itself, in principle there is no approximation involved, as we
onlyperform a partial summation of the partition function, which
can still be completed toobtain the “full” one, no matter which
value we chose for l. It is only in the fitting procedureleading to
the Landau-Ginzburg form of the potential, that approximations in
the form oftruncations at certain powers of the mode amplitudes and
possibly certain maximum powersof k-vector components are
introduced. In fact, performing additional simulations, in
which
Figure 4. T -dependence of coefficients A(12,l)2 , A(12,l)4 ,
A
(12,l)6 for L = 12, C = 1.0. Symbols similar
to that of Fig. 3. Line indicates mean field results.
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236 A. Tröster and C. Dellago
Figure 5. T -dependence of coefficients A(12,l)2 , A(12,l)4 ,
A
(12,l)6 for L = 12, C = 10.0. Symbols similar
to that of Fig. 3. Line indicates mean field results. At the
moment the reason for the large fluctuationsis not well
understood.
we averaged out the remaining slow modes for this approximate
effective Landau-GinzburgHamiltonian, we have checked that the full
partition function can indeed be reconstructedfrom just the
Landau-Ginzburg approximation with satisfying accuracy.
Discussion and Outlook
We are just starting out to explore the possible applications of
our new Fourier Monte Carloalgorithm in combination with or without
the accompanying coarse graining prescriptionwe have presented in
this work. For instance, it is certainly of great interest to study
the φ4
model at the order/disorder limit, as this limit in fact
resembles the Ising model and—via itsisomorphism with the latttice
gas model—therefore allows to study discretized versions ofphase
transitions in many interesting classical systems. In fact, in such
a formulation evenquite complicated site-site interactions are not
expected to pose any particular problems, asthey will only enter in
the Hamiltonian through the dispersion K̃ (k), which can be
tabulatedfor all k-vectors once and for all at the initialization
of the simulation. The increase inthe fluctuations of the results
for larger values of displacive degree are up to now not
wellunderstood. Other problems which are currently still under
investistgation are the simulationof coarse grained potentials
where the bulk correlation length plays the role of the
coarsegraining length, and the finite-size scaling properties of
our coarse-graining algorithm. Wealso plan to extend our
calculations to compressible lattices. Another possible
applicationis the computation of the renormalization group flow of
coupling constants in the contextof Wilson’s momentum shell
formulation.
References
1. L. D. Landau, E. M. Lishitz, and L. D. Pitaevskii,
Statistical Physics Part I, Butterworth andHeinemann, Oxford,
2001.
2. O.V. Kovalev, Representations of the crystallographic space
groups, Gordon and Breach SciencePubl., Yverdon (1983).
3. To simplify the following discussion, we only consider
systems for which the phase transitiontakes place for k = 0, such
that the equilibrium state of the ordered phase is constant in
space.
4. A. P. Levanyuk, Soviet Physics JETP 36, 571 (1959).5. V. L.
Ginzburg, Soviet Physics-Solid State 2, 1824 (1960).6. J.
Zinn-Justin, Quantum Field Theory and Critical Phenomena, Fourth
Edition, Claredon Press,
Oxford (2002).7. A. N. Vasilév, The Field Theoretic
Renormalization Group in Critical Behavior Theory and
Stochastic Dynamics, Chapmann & Hall/CRC, Boca Raton
(2004).8. D. Amit and V. Martin-Mayor, Field Theory, the
Renormalization Group and Critical Phenomena,
Third Edition, World Scientific (2005).
-
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nloa
ded
By:
[Trö
ster
, A.]
At:
12:1
9 31
Aug
ust 2
007
Coarse Graining the φ4 Model 237
9. J.-C. and P. Tolédano, The Landau Theory of Phase
Transitions, World Scientific Lecture Notesin Physics Vol. 3, World
Scientific, Singapore (1987).
10. N. W. Ashcroft, and N. D. Mermin, Solid State Physics,
Saunders College, Fort Worth (1976).11. K. Binder, in Materials
Science and Technology, Vol. 5, Ed. P. Haasen, 143 (1991).12. J. S.
Langer, Physica 73, 61 (1974).13. J. S. Langer, M. Bar-on, and H.
D. Miller, Phys. Rev. A 11, 1417 (1975).14. K. Kawasaki, T. Imeda,
and J. D. Gunton, in Perspectives in Statistical Physics, Ed. H. J.
Raveché,
North Holland Publishing (1981).15. K. Binder, Rep. Prog. Phys.
50, 783 (1987).16. A. D. Bruce and R. A. Cowley, Structural Phase
Transitions, Taylor and Francis Ltd., London
(1981).17. C.J. Thompson, Classical equilibrium statistical
mechanics, Claredon Press, Oxford (1988).18. E. Eisenriegler, Phys.
Rev. B 9, 1029 (1974).19. A. P. Giddy, M. T. Dove, and V. Heine, J.
Phys.: Condens. Matter 1, 8327 (1989).20. A. Milchev, D. W.
Heermann, and K. Binder, J. Stat. Phys. 44, 749 (1986).21. A. N.
Rubtsov, J. Hlinka, and T. Janssen, Phys. Rev. E, 61, 126
(2000).22. G. H. F. van Raaij, K. H. van Bemmel, and T. Janssen,
Phys. Rev. B 62, 3752 (2000).23. J. M. Perez-Mato, S. Ivantchev, A.
Garcia, and I. Etxebarria, Ferroelectrics 236, 93 (2000).24. T.
Radescu, I. Etxebarria, and J. M. Perez-Mato, J. Phys.: Condens.
Matter 7, 585 (1995).25. A. Tröster, C. Dellago, and W. Schranz,
Phys. Rev. B 72, 094103 (2005).26. F. Wang and D. P. Landau, Phys.
Rev. Lett. 86, 2050 (2001).27. F. Wang and D. P. Landau, Phys. Rev.
E 64, 056101 (2001).28. F. Calvo, Mol. Phys. 100, 3421 (2002).29.
K. Binder, Phys. Rev. Lett. 47, 693 (1981).30. H. Furukawa and K.
Binder, Phys. Rev. A 26, 556 (1982).31. K. Kaski, K. Binder, and J.
D. Gunton, J. Phys. A: Math. Gen. A 16, L623 (1983).32. M. E.
Gracheva, J. M. Rickman, and J. D. Gunton, JCP 113, 3525 (2000).33.
A. Tröster, submitted to Phys. Rev. Lett. B, in print (2007).34.
A. Tröster and C. Dellago, in preparation (2007).35. K. G. Wilson
and J. Kogut, Physics Reports 12C, 75 (1974).
