Critical behavior of interfaces in disordered Potts ferromagnets : statistics of free-energy, energy and interfacial adsorption

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by  Cecile Monthus

Critical behavior of interfaces in disordered Potts ferromagnets :
statistics of free-energy, energy and interfacial adsorption
C´ecile Monthus and Thomas Garel
Service de Physique Th´eorique, CEA/DSM/SPhT
Unit´e de recherche associ´ee au CNRS
91191 Gif-sur-Yvette cedex, France
A convenient way to study phase transitions of finite spins systems of linear size L is to fix
boundary conditions that impose the presence of a system-size interface. In this paper, we study
8 thestatistical properties of such an interface in a disordered Potts ferromagnet in dimension d=2
0 within Migdal-Kadanoff real space renormalization. We first focus on the interface free-energy
0 and energy to measure the singularities of the average and random contributions, as well as the
2 corresponding histograms, both in the low-temperature phase and at criticality. We then consider
n the critical behavior of the interfacial adsorption of non-boundary states. Our main conclusion is
a that all singularities involve the correlation length ξav(T) ∼ (Tc−T)−ν appearing in the average
J free-energy F ∼ (L/ξav(T))ds of the interface of dimension ds = d−1, except for the free-energy
5 width ∆F ∼ (L/ξvar(T))θ that involves the droplet exponent θ and another correlation length
1 ξvar(T) which diverges more rapidly than ξav(T). We compare with the spin-glass transition in
d = 3, where ξvar(T) is the ’true’ correlation length, and where the interface energy presents

] unconventionalscalingwithachaoscriticalexponentζc >1/ν [NifleandHilhorst,Phys. Rev. Lett.
n 68, 2992 (1992)]. The common feature is that in both cases, the characteristic length scale L (T)
ch
n
associated with the chaotic nature of the low-temperature phase, diverges more slowly than the
-
s correlation length.
i
d
.
t I. INTRODUCTION
a
m
A. Critical properties of interfaces in pure systems
-
d
n
The critical points of statistical physics models are usually discussed in terms of bulk properties. However, it is
o
also interesting to study how a critical system reacts when boundary conditions impose the presence of a system-size
c
[ interface. For instance in an Ising ferromagnetdefined on a cube of volume Ld, on may impose the (+) phase on the

left boundary and the ( ) phase on the right boundary, with periodic boundary conditions in the other directions.
2 −
Thestudyofinterfacesbetweencoexistingphasesnearcriticalityhasalonghistory[1]. Themostimportantproperty
v
8 is that the free-energy Finter associated to the interface is proportional below Tc to its area Lds with ds =d 1
−
7
8 Pure Ferromagnets: Finter(L,T <Tc) f0(T)Lds +... (1)
≃
2
1. where the so-called interfacial tension f0(T) vanishes at criticality as a power-law [1]
1
f (T) (T T)µ (2)
07 0 T→≃Tc− c−
:
v Finite-sizescalingimpliesthatnearcriticality,theinterfacefree-energyFinter shouldonlydependontheratioL/ξ(T)
i between the system linear size L and the correlationlength ξ(T) (T T)−ν
X ∝ c−
ar Finter(L,T <T ) L ds +... (3)
c
∝ ξ(T)
(cid:18) (cid:19)
This is similar to the requirement that the singular part of the bulk free-energy should scale as Fbulk (L/ξ(T))d
(T T)νdLd. The identification with the definition Fbulk (T T)2−αLd in terms of the specific he∼at exponent∼α
c c
− ∼ −
yields the hyperscaling relation 2 α = νd. So the exponent µ of the interfacial tension f (T) satisfies the Widom
0
−
relation [1]
µ=νd =ν(d 1) (4)
s
−
Exactlyat criticality,the free-energybecomes oforderFinter(L,T ) O(1), whereasthe energy andentropygrowas
c
∼
Einter(L,T ) L1/ν. Beyondthesethermodynamicproperties,theinterestintointerfaceswasrecentlyrevivedbythe
c
∼
discovery [2] that some two dimensional critical interfaces are fractal curves which can be constructed via Stochastic
2
Loewner Evolutions (SLEs) reviewed in [3]. Accordingly, the fractal dimensions of spin cluster boundaries of various
two-dimensional spin models have been recently measured via Monte Carlo simulations in [4, 5].
Whenever the system under study presents more than two phases, such as the Potts model considered in the this
paper, a system-size interface between states 1 and 2 tends to produce a net adsorption of any non-boundary state,
calledstate 3 here. This phenomenonofinterfacialadsorptionhas been muchstudied in variouspure models [6] with
the followingconclusions. The excessofstate’3’due to the presenceofa (1:2)interfacewith respectto the case(1:1)
with no interface, defined as
N (<δ > <δ > ) (5)
nb ≡ σi,3 1:2 − σi,3 1:1
i
X
is proportional to the area Lds of the interface for T <Tc
N (L,T <T ) w (T)Lds (6)
nb c 0
∼
Finite-size scaling argument yields that the coefficient w (T) diverges at criticality as [6]
0
w (T) (T T)β−ν (7)
0 c
T→∝Tc− −
where ν is the correlation length introduced above, and where β is the order parameter exponent. This means that
at criticality, the adsorption of non-boundary states N scales as the global order parameter M =(T T)βLd.
nb c
−
Nnb(L,Tc) M(L,Tc) Ld−βν (8)
∼ ∼
B. Properties of interfaces below Tc in disordered systems
In the field of disordered systems such as spin-glasses where the order parameter of the low-temperature phase is
morecomplicatedthaninpuresystems,itturnsoutthatthepropertiesofinterfacesareveryconvenienttocharacterize
the low-temperature phase via a so-called droplet exponent θ [7, 8]
Spin glass: Finter(L,T <T )=Υ(T)Lθu +... (9)
c F
−
where Υ(T) is a generalized ’stiffness’ modulus and where u is a random variable of order O(1). The exponent θ is
F
expected to satisfy the bound θ (d 1)/2 [7] The interface is expected to have a non-trivial fractal dimension d
s
≤ −
with d 1 d d in the whole low-temperature phase [7]. This fractal dimension d governs the energy and the
s s
− ≤ ≤
entropy of the interface [7]
Spin glass: Einter(L,T <Tc) =σ(T)Ld2suE +... (10)
−
TSinter(L,T <Tc) =σ(T)Ld2suE+
One actually expects the strict inequality θ < ds, so that the optimized free-energy of Eq 9 is a near cancellation of
2
much larger energy and entropy contributions of Eq. 11. This is at the originof the sensitivity of disorderedsystems
to temperature changes or disorder changes, called ’chaos’ in this context [7, 8, 9, 10, 11] : roughly speaking, the
chaosexponentζ = ds θ governsthelengthscaleL∗ ǫ−1/ζ abovewhichasmallperturbationǫinthe temperature
2 − ∼
or in the disorder will change the state of the system.
For non-frustrated disordered systems such as ferromagnetic spin models in dimension d, the interface below T is
c
expected to be described by a directed manifold of dimension d = (d 1) in a random medium. In particular, in
s
−
two-dimensional disordered ferromagnets, the one-dimensional interface is described by the directed polymer model
[12]. Forthismodel,adropletscalingtheoryhasbeendeveloped[13]indirectcorrespondencewiththedroplettheory
of spin-glasses [7] summarized above. In particular, the free-energy of the interface reads
Random Ferromagnets: Finter(L,T <T )=f (T)Lds +Υ(T)Lθ(d)u +... (11)
c 0 F
with a droplet exponent θ which is exactly known to be θ(d = 2) = 1/3 on the two-dimensional lattice [14, 15, 16].
The energy and the entropy of the interface reads [13, 17]
Random Ferromagnets: Einter(L,T <Tc) =e0(T)Lds +σ(T)Ld2suE +... (12)
TSinter(L,T <Tc) =Ts0(T)Lds +σ(T)Ld2suE +...
where the fluctuating termhas againa biggerexponent ds >θ that the fluctuating termofthe free-energyof Eq. 11.
2
As a consequence, the interface is again very sensitive to temperature or disorder changes with the chaos exponent
ζ = d /2 θ. In particular in dimension d = 2, where the interface is a directed polymer in dimension 1+1, the
s
−
chaos exponent is exactly known ζ =1/2 1/3=1/6 [13, 18, 19, 20].
−
3
C. Properties of interfaces at criticality in disordered systems
At criticality, the interface free-energy is expected to be a random variable u of order O(1)
Fc
Finter(L,T )=u +... (13)
c Fc
For the spin-glass case, the interface free-energy of Eq. 9 is expected to scale as (L/ξ(T))θ in terms of the diverging
correlation length ξ(T) (T T)−ν, so that the critical exponent governing the vanishing of Υ(T) is [7]
c
∼ −
Υ(T) (T T)νθ (14)
c
∼ −
which is the analog of Widom scaling relation for ferromagnets (Eq. 4).
For the energy of the interface, two possibilities have been described in the literature :
(i)inthefirstscenariodescribedin[7],thecriticalbehaviorfollowstheusualfinite-sizescalingformsintermsofthe
divergingcorrelationlengthξ(T) (T T)−ν. Moreprecisely, the singularpartof the energyor entropyis assumed
c
∼ −
to be of order 1/(T T) on the scale ξ(T), so that the coefficient σ(T) in Eq. 11 presents the following singularity
c
−
ds
1 1 2
SG with ′Conventional′ critical point: σ(T) (15)
T→≃Tc− Tc−T (cid:18)ξ(T)(cid:19)
Equivalently, one then obtains the following ’conventional random critical’ behavior exactly at criticality [7]
SG with ′Conventional′ critical point: Einter(L,Tc)=Lν1uEc +... (16)
where u is a random variable of order O(1). In our recent study of the directed polymer delocalization transitions
Ec
onhierarchicallattices withb=5 [21], we havefound thatthe energyandentropy aregovernedby the ’conventional’
critical behaviors of Eqs 15 and 16.
(ii)howeverin[10,11],ithasbeenfoundthatanewexponentζ calledthe ’criticalchaosexponent’cangovernthe
c
response to disorderperturbations of spin-glassesat criticality,providedthe inequalityζ >1/ν is satisfied. We refer
c
to [10, 11] for a detailed description of these chaos properties. Here, we will only mention an important consequence
for the interface : it has been argued in [10, 11] that this new exponent ζ should govern the scaling of the interface
c
energy at criticality
SG with′Chaos critical exponent′ : Einter(L,T )=Lζcu +... (17)
c Ec
in contrast with Eq. 16. As a final remark on spin-glasses, let us mention that in d =2 where there is no spin-glass
phase(T =0),recentstudies havesuggestedthatzero-temperatureinterfacesareactuallydescribedbySLE[22, 23].
c
For random ferromagnetic spin models, one expects ’conventionalscaling’ as in Eq. 16 for the energy where u is
Ec
a randomvariable of order O(1). More generally,in the presence of relevant disorder,there is a lack of self-averaging
in all singular contributions of thermodynamic observables in the sense that the leading term remains distributed
[24]. Note that for the Potts model with q 3 states, the interface becomes a non-directed branching object at
≥
criticality. Some authors have studied the relevance of branching within a solid-on-solid approximation where the
’directed’ character of the low-temperature phase is kept [25, 26]. However, the ’directed’ character is not expected
to hold at criticality for at least two reasons : first, this ’directed’ character does not hold at criticality already for
pure ferromagnets,and second, in two dimensions, the directed polymer is alwaysin its disordereddominated phase,
whereas ferromagnets undergo a phase transition where the disorder relevance of the Harris criterion depends on q
(see [26] for a more detailed discussion).
The aim of this paper is to study numerically the critical behavior of some two-dimensional random Potts ferro-
magnet in the presence of relevant disorder. We have chosen to work on the diamond hierarchical lattice of effective
dimension d = 2, where large length scales can be studied via exact renormalization, and with the Potts model
eff
with q =8 states so that disorderis relevantaccordingto the Harriscriterion(see Appendix A). We presentdetailed
resultsonthestatisticsoftheinterfacefree-energy,energy,entropyandinterfacialadsorptionofnon-boundarystates.
D. Organization of the paper
The paper is organized as follows. In Section II, we recall the exact renormalization equations for the diamond
hierarchicallattice, that are used to study numerically the disordered Potts model with q =8 states on the diamond
hierarchicallatticeofeffectivedimensiond =2. Wethendescribeournumericalresultsontheinterfacefree-energy
eff
statistics (Section III), on the interface energy and entropy statistics (Section IV), and on the interfacial adsorption
of non-boundary states (Section V). In Section VI, we discuss the similarities and differences with the spin-glass
transition in effective dimension d = 3. Finally we give our conclusions in Section VII. Appendix A contains a
eff
reminder on the pure Potts model on hierarchical lattices.
4
II. RENORMALIZATION EQUATIONS FOR SPIN MODELS ON HIERARCHICAL LATTICES
A. Reminder on the diamond hierarchical lattices
A A A
(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)
1(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)2 (cid:0)(cid:0)(cid:1)(cid:1)b (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)
(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)
B B B
n=0 n=1 n=2
FIG. 1: Hierarchical construction of the diamond lattice of branching ratio b.
Among real-spacerenormalizationprocedures [27], Migdal-Kadanoffblock renormalizations [28] play a special role
becausetheycanbeconsideredintwoways,eitherasapproximaterenormalizationproceduresonhypercubiclattices,
or as exact renormalization procedures on certain hierarchical lattices [29, 30]. One of the most studied hierarchical
lattice is the diamond lattice which is constructed recursively from a single link called here generation n = 0 (see
Figure 1): generation n = 1 consists of b branches, each branch containing 2 bonds in series ; generation n = 2 is
obtained by applying the same transformationto each bond of the generationn=1. At generationn, the length L
n
between the two extreme sites A and B is L =2n, and the total number of bonds is
n
ln(2b)
B =(2b)n =Ldeff(b) with d (b)= (18)
n n eff
ln2
where d (b) represents some effective dimensionality.
eff
B. Spin models on hierarchical lattices
On this diamond lattice, various disordered models have been studied, such as the diluted Ising model [31], ferro-
magnetic random Potts model [32, 33, 34] and spin-glasses [10, 11, 35, 36, 37, 38]. The random ferromagnetic Ising
Hamiltonian reads
H = J S S (19)
Ising i,j i j
−
<i,j>
X
where the spins take the values S = 1 and where the couplings J are positive random variables (the spin-glass
i i,j
±
Hamiltonian corresponds to the case of couplings of random sign). The random ferromagnetic Potts Hamiltonian is
a generalization where the variable σ can take q different values.
i
H = 2J δ (20)
Potts − i,j σi,σj
<i,j>
X
(We choose (2J) to recover the Ising case for q =2)
C. Renormalization equation for the interface free-energy
The free-energy cost Finter of creating an interface between the two end-points A and B of the diamond lattice of
L
Fig. 1 is defined by
Finter F+− F++ (21)
≡ −
5
where F++ = TlnZ++ and F+− = TlnZ+− are the free-energies corresponding respectively to the same color at
− −
both ends or to two different colors at both ends. The renormalization equation are simpler to write in terms of the
ratio of the two partitions functions Z++ and Z+−
Z
x eβFinter = ++ (22)
≡ Z+−
and one obtains [32, 33, 34]
b x(i1)x(i2)+(q 1)
n n
x = − (23)
n+1 iY=1 x(ni1)+x(ni2)+(q−2)!
D. Renormalization equation for the interface energy
The energy cost for creating an interface between the two ends at distance L is defined similarly by
d d Z
Einter =E+− E++ = (lnZ+− lnZ++)= ln ++ (24)
− −dβ − dβ Z+−
The renormalizationequation reads in terms of the variable E Einter and x introduced above (Eq 22)
≡
b x(i1)(x(i2) 1)(x(i2)+q 1)E(i1)+x(i2)(x(i1) 1)(x(i1)+q 1)E(i2)
n n
E = − − − − (25)
n+1 (x(i1)+x(i2)+q 2)(x(i1)x(i2)+q 1)
i=1 − −
X
E. Renormalization equations for the order parameter and the interfacial adsorption
To study the order parameter and the interfacial adsorption, let us introduce the notation
Ma,b = <δ > (26)
n σi,1 a:b
i
X
for the number of spins in state ’1’ on a hierarchicallattice at generationn when the two end points A and B of Fig.
1 arerespectivelyinstates’a’and’b’. Using symmetries,onefinally obtains closedrenormalizationsfor the following
five variables
b x x (M(1,1)+M(1,1) 1)+(q 1)(M(1,2)+M(2,1))
M(1,1) = n,i1 n,i2 n,i1 n,i2 − − n,i1 n,i2 (27)
n+1 x x +q 1
i=1 n,i1 n,i2 −
X
b x (M(1,1)+M(1,2) 1)+x (M(1,2)+M(2,2))+(q 2)(M(1,2)+M(2,3))
M(1,2) = n,i1 n,i1 n,i2 − n,i2 n,i1 n,i2 − n,i1 n,i2
n+1 x +x +q 2
i=1 n,i1 n,i2 −
X
b x (M(2,1)+M(1,1) 1)+x (M(2,2)+M(2,1))+(q 2)(M(2,3)+M(2,1))
M(2,1) = n,i2 n,i1 n,i2 − n,i1 n,i1 n,i2 − n,i1 n,i2
n+1 x +x +(q 2)
i=1 n,i2 n,i1 −
X
b (M(2,1)+M(1,2) 1)+x x (M(2,2)+M(2,2))+(q 2)(M(2,3)+M(2,3))
M(2,2) = n,i1 n,i2 − n,i1 n,i2 n,i1 n,i2 − n,i1 n,i2
n+1 x x +q 1
i=1 n,i1 n,i2 −
X
b (M(2,1)+M(1,2) 1)+x (M(2,2)+M(2,3))+x (M(2,3)+M(2,2))+(q 3)(M(2,3)+M(2,3))
M(2,3) = n,i1 n,i2 − n,i1 n,i1 n,i2 n,i2 n,i1 n,i2 − n,i1 n,i2
n+1 x +x +q 2
i=1 n,i1 n,i2 −
X
The order parameter can then be defined as
M M(1,1) M(2,2) (28)
≡ −
whereas the net absorption of non-boundary states of Eq. 5 reads
N M(2,3) M(2,2) (29)
nb
≡ −
6
F. Numerical ’pool’ method
Thenumericalresultspresentedbelowhavebeenobtainedwiththeso-called’pool-method’whichisveryoftenused
for disorderedsystems onhierarchicallattices : the idea is to representthe probabilitydistribution P (F ,E )of the
n n n
interface free-energy F and energy E at generation n, by a pool of N realizations (F(1),E(1)),..,(F(N),E(N))) .
n n n n n n
{ }
(i) (i)
Thepoolatgeneration(n+1)isthenobtainedasfollows: eachnewrealization(F ,E )isobtainedbychoosing
n+1 n+1
(2b)realizationsatrandomfromthe poolofgenerationn andby applyingthe renormalizationequationsgivenin Eq.
23 and in Eq. 25.
The initial distribution of couplings was chosen to be
2
PPotts(J)=θ(J 0)Je−J2 (30)
≥
for the ferromagnetic Potts case, and Gaussian for the spin-glass case
1 2
PSG(J)= e−J2 (31)
√2π
At generation n = 0 made of a single link (see Fig. 1), the free-energy and the energy of the interface coincide and
read in terms of the random coupling J drawn with either Eq 31 or Eq. 30
i
F(i) =E(i) =2J (32)
n=0 n=0 i
The numerical results presented below have been obtained with a pool of size N = 4.107 which is iterated up to
n=60 or n=80 generations. In the following sections, we study the random ferromagnetic Potts q =8 on diamond
lattice of effective dimension d =2 corresponding to a branching ratio b=2.
eff
III. STATISTICS OF THE INTERFACE FREE-ENERGY
Asrecalledintheintroduction,theinterfacefree-energyisexpectedtofollowthescalingbehaviorofEq11belowT
c
andtobecomearandomvariableoforderO(1)atT (Eq13). Inthissection,wepresentnumericalresultsconcerning
c
the singularities of the average and random contributions, as well as histograms, both in the low-temperature phase
and at criticality.
A. Flow of the average value and width of the interface free-energy
50 15
ln F 40 ln ∆ F
10
30
20
T=0.1 5 T=0.1
10
0 0
−10
−5
−20 T=T + ε
c T=T + ε
−30 −10 c
T=2 (b)
−40 (a) T=2
−500 10 20 30 40 −150 10 20 30 40
ln L Ln L
FIG. 2: (Color online) Flows of the average value and of the width of the interface free-energy for many temperatures : (a)
log-log plot of the average value F(L) of the free-energy distribution as a function of L. (b) log-log plot of the width ∆F(L)
of the free-energy distribution as a function of L.
7
The flows of the averagefree-energy F(L) and of the free-energy width ∆F(L) are shown on Fig. 2 for many tem-
peratures. Oneclearlyseesontheselog-logplotsthe twoattractivefixedpointsseparatedbythe criticaltemperature
T . The value of T obtainedby the pool method depends onthe pool, i.e. on the discrete sampling with N values of
c c
thecontinuousprobabilitydistribution. ItisexpectedtoconvergetowardsthethermodynamiccriticaltemperatureT
c
onlyinthelimitN . Nevertheless,foreachgivenpool,theflowoffree-energyallowsaveryprecisedetermination
of this pool-depend→ent∞critical temperature, for instance in the case considered 1.21685522<Tpool <1.21685523.
c
For T > T , both the average free-energy and the free-energy width decay exponentially in L. For T < T , the
c c
average free-energy grows asymptotically with the interface dimension d =d 1=1 (see Eq 11)
s eff
−
L ds
F(L) +... with d =1 (33)
s
≃ ξ (T)
(cid:18) av (cid:19)
where ξ (T) is the correlation length that diverges as T T−. The free-energy width grows asymptotically with
av → c
the droplet exponent θ (see Eq 11)
θ(b)
L
∆F(L) with θ(b=2) 0.299 (34)
≃ ξ (T) ≃
(cid:18) var (cid:19)
whereξ (T)isthe associatedcorrelationlengththatdivergesasT T−. Notethatθ(b=2) 0.299is the droplet
var → c ≃
exponent of the corresponding directed polymer model [39, 40].
B. Divergence of the correlation lengths ξav(T) and ξvar(T)
30 30
(a)
ln ξ
20 20 var
ln ξ
av
10
10
ln ξ (b)
var
ln ξ
av
0
0 −20 −15 −10 −5 0
0 0.5 1 1.5
ln |T −T|
T c
FIG. 3: Divergence of the correlation lengths ξav(T) ((cid:3)) and ξvar(T) ((cid:13)) (a) lnξav(T) and lnξvar(T) as a function of T (b)
lnξav(T) and lnξvar(T) as a function of ln|Tc−T|: theasymptotic slopes are of order νav ≃1.07 and νvar ≃1.34
The correlation lengths ξ (T) and ξ (T) as measured from the free-energy average value (Eq 33) and from the
av var
free-energy width (Eq 34 ) are shown on Fig. 3 (a). The log-log plot shown on Fig. 3 (b) indicates power-law
divergences with two distinct correlation length exponents
ξ (T) (T T)−νav with ν 1.07
av c av
T→∝Tc − ≃
ξ (T) (T T)−νvar with ν 1.34 (35)
var c var
T→∝Tc − ≃
In conclusion, our numericalresults point towards the following singular behavior for the interface free-energy (see
Eq. 11)
L ds L θ
Finter(L,T <T ) + u +... (36)
c F
T→∝Tc−(cid:18)ξav(T)(cid:19) (cid:18)ξvar(T)(cid:19)
8
where the averagecontribution and the randomcontribution involve two correlationlengths ξ (T) and ξ (T) that
av var
diverge with distinct exponents (Eq 35). The presence of these two distinct correlation length exponents in the
interface free-energy was a surprise for us, and we are not aware of any discussion of this possibility in the literature.
The ’true’ correlation length is expected to be ξ (T) that appears in the extensive non-random contribution to the
av
interface free-energy. However, the presence of another length scale ξ (T) that diverges with a larger exponent
var
remains to be better understood.
C. Histogram of the interface free-energy below Tc
0
ln Π
0
−2
−4
−6
−8
−10
−12
−7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7
x
F
FIG. 4: Statistics of the interface free-energy in the low-temperature phase (here T = 0.5) : Log representation of the
distribution Π0(xF) of therescaled variable xF = F∆−FF, which present asymmetric tails (see Eq. 38)
In the low-temperature phase, the the interface free-energy is expected to follow the behavior of Eq 11, where u
F
is a random behavior of order O(1). We show on Fig. 4 the probability distribution Π (x ) of the rescaled variable
0 F
xF = F∆−FF in log-scale to see the tails. The two tails exponents (η−,η+) defined by
lnΠ (x ) x η± (37)
0 F F
xF→≃±∞−| |
are compatible with the relations proposed in our previous work [39] with d =1, d=2 and θ 0.299 (see Eq 34)
s
≃
d 1
s
η− = = 1.43
d θ 1 θ ∼
s
− −
d 2
η = = 2.85 (38)
+
d θ 1 θ ∼
s
− −
D. Histogram of the interface free-energy at criticality
At criticality, the interface free-energy is expected to become a random variable of order O(1) (Eq. 13) : we show
itsprobabilitydistributiononFig. 5(a). Toseethetails,weshowinlog-scalethedistributionoftherescaledvariable
x = F−F on Fig. 5 (b).
F ∆F
IV. STATISTICS OF THE INTERFACE ENERGY
In this section, we present the numerical results concerning the statistics of the interface energy.
9
1.5 0
ln Π
P c
c
−2
1 −4
−6
0.5 −8 (b)
(a) −10
0 −12
0 0.5 1 1.5 2 2.5 3 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7
x
F F
FIG. 5: Histogram of the interface free-energy at criticality (a) Unrescaled probability distribution Pc(F) of the interface
free-energy F at Tc (b) Log representation of thedistribution Πc(xF) of the rescaled variable xF = F∆−FF
A. Average and width of the interface energy
As recalled in the introduction, the interface energy is expected to follow the scaling behavior of Eq 12 below
T . The extensive non-random part e (T) is directly related to the corresponding non-random part f (T) of the
c 0 0
free-energy of Eq. 11 via the usual thermodynamic relation
df (T)
0
e (T)=f (T) T (39)
0 0
− dT
As a consequence, the singularity found previously for f (T) (Eq 33)
0
1 ds
f (T) (T T)dsνav (40)
0 c
≃(cid:18)ξav(T)(cid:19) T→∝Tc− −
determines the singularity of e (T) near T
0 c
e (T) (T T)dsνav−1 (41)
0 c
T→∝Tc− −
25 20
ln ∆ E
20 15
15 10
T=T + ε T=0.5 ln ∆ E ln ∆ S
10 c 5
ln ∆F
5 0
(a)
(b)
00 10 20 30 40 −50 5 10 15
ln L ln L
FIG. 6: (Color online) Flow of the width ∆E(L) of the energy distribution as L grows (a) ln∆E(L) as a function of lnL
for many temperatures between T = 0.5 and T = Tc+ǫ = 1.21685523 (b) Comparison of ln∆E(L) ((cid:3)), ln∆S(L) (♦) and
ln∆F(L) ((cid:13)) as a function of lnLat criticality.
10
We now consider the random contribution to the interface energy in Eq. 12. The flow of the width ∆E(L) as L
grows is shown on Fig. 6 for many temperatures. For T < T , this width grows asymptotically with the exponent
c
d /2=1/2 as expected (see Eq 12)
s
∆E(L) Ld2s =L12 (42)
≃
Exactly at criticality, the energy width (and entropy width) grows as a power-law (see Fig 6 b)
∆E(L) Lyc with y 0.92 (43)
c
≃ ≃
This value for y is in agreement with the value 1/ν 0.93 (see Eq. 35).
c av
≃
30 30
25 25
20 20
ln ξ
var
15 ln ξ 15
var
10 10 ln ξav
ln ξ
av 5
5
ln ξ
E, S
0 ln ξE, S 0 (b)
(a)
−5
−5 −17 −15 −13 −11 −9 −7 −5 −3
1.2 1.205 1.21 1.215 1.22
ln |T −T|
T c
FIG. 7: (Color online) Correlation lengths ξE(T) (∗) and ξS(T) (♦) as measured from thebehavior of theenergy and entropy
widths (Eq. 45) (a) lnξE(T) and lnξS(T) as a function of T, as compared to lnξav(T) ((cid:3)) and lnξvar(T) ((cid:13)) (b) lnξE(T)
and lnξS(T) as a function of ln|Tc−T|, as compared to lnξav(T) and lnξvar(T)
We now define a correlation length ξ (T) for T <T via the finite-size scaling form
E c
L
∆E(L,T) LycΦ (44)
≃ ξ (T)
(cid:18) E (cid:19)
Inthe regimeL ξ (T), oneshouldrecoverthe L-dependence ofthe low-temperaturephase ofEq42, so the scaling
E
function Φ(x) sh≫ould present the asymptotic behavior Φ(x) x1/2−yc yielding the temperature dependence of the
∼
prefactor
1 1/2−yc
∆E(L,T) L (45)
L≫≃ξE(T)(cid:18)ξE(T)(cid:19)
One similarly may define a correlationlength ξ (T) from the finite-size scaling of the entropy width.
S
As shown on Fig 7 b, the log-log plot presents some curvature, so that the asymptotic slope ν defined by
E
ξ (T) (T T)−νE (46)
E c
≃ −
is difficult to measure precisely. However, the slope ν is close to the value ν 1.07 of Eq. 35.
E av
≃
In conclusion, our numerical results point towards the following singular behavior for the interface energy (see Eq.
12)
1 L ds L d2s
Einter(L,T <T ) + u +... (47)
c E
T→∝Tc− Tc−T "(cid:18)ξav(T)(cid:19) (cid:18)ξav(T)(cid:19) #
i.e. both the average contribution and the random contribution involve the same correlation length ξ (T). This
av
result seems natural within the Fisher-Huse droplet theory [7, 13] where the interface energy is a sum of random
terms that follow some CentralLimit asymptotic behavior. This picture is confirmed by the Gaussiandistribution of
the random variable u that we now consider.
E