Critical behavior of interfaces in disordered Potts
statistics of free-energy, energy and interfacial adsorption
A convenient way to study phase transitions of finite spins systems of linear size is to fix boundary conditions that impose the presence of a system-size interface. In this paper, we study the statistical properties of such an interface in a disordered Potts ferromagnet in dimension within Migdal-Kadanoff real space renormalization. We first focus on the interface free-energy and energy to measure the singularities of the average and random contributions, as well as the corresponding histograms, both in the low-temperature phase and at criticality. We then consider the critical behavior of the interfacial adsorption of non-boundary states. Our main conclusion is that all singularities involve the correlation length appearing in the average free-energy of the interface of dimension , except for the free-energy width that involves the droplet exponent and another correlation length which diverges more rapidly than . We compare with the spin-glass transition in , where is the ’true’ correlation length, and where the interface energy presents unconventional scaling with a chaos critical exponent [Nifle and Hilhorst, Phys. Rev. Lett. 68, 2992 (1992)]. The common feature is that in both cases, the characteristic length scale associated with the chaotic nature of the low-temperature phase, diverges more slowly than the correlation length.
i.1 Critical properties of interfaces in pure systems
The critical points of statistical physics models are usually discussed in terms of bulk properties. However, it is also interesting to study how a critical system reacts when boundary conditions impose the presence of a system-size interface. For instance in an Ising ferromagnet defined on a cube of volume , on may impose the phase on the left boundary and the phase on the right boundary, with periodic boundary conditions in the other directions. The study of interfaces between coexisting phases near criticality has a long history . The most important property is that the free-energy associated to the interface is proportional below to its area with
where the so-called interfacial tension vanishes at criticality as a power-law 
Finite-size scaling implies that near criticality, the interface free-energy should only depend on the ratio between the system linear size and the correlation length
This is similar to the requirement that the singular part of the bulk free-energy should scale as . The identification with the definition in terms of the specific heat exponent yields the hyperscaling relation . So the exponent of the interfacial tension satisfies the Widom relation 
Exactly at criticality, the free-energy becomes of order , whereas the energy and entropy grow as . Beyond these thermodynamic properties, the interest into interfaces was recently revived by the discovery  that some two dimensional critical interfaces are fractal curves which can be constructed via Stochastic Loewner Evolutions (SLEs) reviewed in . 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, called state 3 here. This phenomenon of interfacial adsorption has been much studied in various pure models  with the following conclusions. The excess of state ’3’ due to the presence of a (1:2) interface with respect to the case (1:1) with no interface, defined as
is proportional to the area of the interface for
Finite-size scaling argument yields that the coefficient diverges at criticality as 
where is the correlation length introduced above, and where is the order parameter exponent. This means that at criticality, the adsorption of non-boundary states scales as the global order parameter .
i.2 Properties of interfaces below in disordered systems
In the field of disordered systems such as spin-glasses where the order parameter of the low-temperature phase is more complicated than in pure systems, it turns out that the properties of interfaces are very convenient to characterize the low-temperature phase via a so-called droplet exponent [7, 8]
where is a generalized ’stiffness’ modulus and where is a random variable of order . The exponent is expected to satisfy the bound  The interface is expected to have a non-trivial fractal dimension with in the whole low-temperature phase . This fractal dimension governs the energy and the entropy of the interface 
One actually expects the strict inequality , so that the optimized free-energy of Eq 9 is a near cancellation of much larger energy and entropy contributions of Eq. I.2. This is at the origin of the sensitivity of disordered systems to temperature changes or disorder changes, called ’chaos’ in this context [7, 8, 9, 10, 11] : roughly speaking, the chaos exponent governs the length scale above which a small perturbation in the temperature or in the disorder will change the state of the system.
For non-frustrated disordered systems such as ferromagnetic spin models in dimension , the interface below is expected to be described by a directed manifold of dimension in a random medium. In particular, in two-dimensional disordered ferromagnets, the one-dimensional interface is described by the directed polymer model . For this model, a droplet scaling theory has been developed  in direct correspondence with the droplet theory of spin-glasses  summarized above. In particular, the free-energy of the interface reads
where the fluctuating term has again a bigger exponent that the fluctuating term of the free-energy of Eq. 11. As a consequence, the interface is again very sensitive to temperature or disorder changes with the chaos exponent . In particular in dimension , where the interface is a directed polymer in dimension , the chaos exponent is exactly known [13, 18, 19, 20].
i.3 Properties of interfaces at criticality in disordered systems
At criticality, the interface free-energy is expected to be a random variable of order
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) in the first scenario described in , the critical behavior follows the usual finite-size scaling forms in terms of the diverging correlation length . More precisely, the singular part of the energy or entropy is assumed to be of order on the scale , so that the coefficient in Eq. I.2 presents the following singularity
Equivalently, one then obtains the following ’conventional random critical’ behavior exactly at criticality 
where is a random variable of order . In our recent study of the directed polymer delocalization transitions on hierarchical lattices with , we have found that the energy and entropy are governed by the ’conventional’ critical behaviors of Eqs 15 and 16.
(ii) however in [10, 11], it has been found that a new exponent called the ’critical chaos exponent’ can govern the response to disorder perturbations of spin-glasses at criticality, provided the inequality is satisfied. We refer 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 should govern the scaling of the interface energy at criticality
in contrast with Eq. 16. As a final remark on spin-glasses, let us mention that in where there is no spin-glass phase (), recent studies have suggested that zero-temperature interfaces are actually described by SLE [22, 23].
For random ferromagnetic spin models, one expects ’conventional scaling’ as in Eq. 16 for the energy where is a random variable of order . 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 . Note that for the Potts model with 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 always in its disordered dominated phase, whereas ferromagnets undergo a phase transition where the disorder relevance of the Harris criterion depends on (see  for a more detailed discussion).
The aim of this paper is to study numerically the critical behavior of some two-dimensional random Potts ferromagnet in the presence of relevant disorder. We have chosen to work on the diamond hierarchical lattice of effective dimension , where large length scales can be studied via exact renormalization, and with the Potts model with states so that disorder is relevant according to the Harris criterion (see Appendix A). We present detailed results on the statistics of the interface free-energy, energy, entropy and interfacial adsorption of non-boundary states.
i.4 Organization of the paper
The paper is organized as follows. In Section II, we recall the exact renormalization equations for the diamond hierarchical lattice, that are used to study numerically the disordered Potts model with states on the diamond hierarchical lattice of effective dimension . We then describe our numerical results on the interface free-energy 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 . Finally we give our conclusions in Section VII. Appendix A contains a reminder on the pure Potts model on hierarchical lattices.
Ii Renormalization equations for spin models on hierarchical lattices
ii.1 Reminder on the diamond hierarchical lattices
Among real-space renormalization procedures , Migdal-Kadanoff block renormalizations  play a special role because they can be considered in two ways, either as approximate renormalization procedures on hypercubic lattices, 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 (see Figure 1): generation consists of branches, each branch containing bonds in series ; generation is obtained by applying the same transformation to each bond of the generation . At generation , the length between the two extreme sites and is , and the total number of bonds is
where represents some effective dimensionality.
ii.2 Spin models on hierarchical lattices
On this diamond lattice, various disordered models have been studied, such as the diluted Ising model , ferromagnetic random Potts model [32, 33, 34] and spin-glasses [35, 36, 37, 38, 10, 11]. The random ferromagnetic Ising Hamiltonian reads
where the spins take the values and where the couplings are positive random variables (the spin-glass Hamiltonian corresponds to the case of couplings of random sign). The random ferromagnetic Potts Hamiltonian is a generalization where the variable can take different values.
(We choose to recover the Ising case for )
ii.3 Renormalization equation for the interface free-energy
The free-energy cost of creating an interface between the two end-points and of the diamond lattice of Fig. 1 is defined by
where and 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 and
ii.4 Renormalization equation for the interface energy
The energy cost for creating an interface between the two ends at distance is defined similarly by
The renormalization equation reads in terms of the variable and introduced above (Eq 22)
ii.5 Renormalization equations for the order parameter and the interfacial adsorption
To study the order parameter and the interfacial adsorption, let us introduce the notation
for the number of spins in state ’1’ on a hierarchical lattice at generation when the two end points and of Fig. 1 are respectively in states ’a’ and ’b’. Using symmetries, one finally obtains closed renormalizations for the following five variables
The order parameter can then be defined as
whereas the net absorption of non-boundary states of Eq. 5 reads
ii.6 Numerical ’pool’ method
The numerical results presented below have been obtained with the so-called ’pool-method’ which is very often used for disordered systems on hierarchical lattices : the idea is to represent the probability distribution of the interface free-energy and energy at generation , by a pool of realizations . The pool at generation is then obtained as follows : each new realization is obtained by choosing realizations at random from the pool of generation and by applying the renormalization equations given in Eq. 23 and in Eq. 25.
The initial distribution of couplings was chosen to be
for the ferromagnetic Potts case, and Gaussian for the spin-glass case
The numerical results presented below have been obtained with a pool of size which is iterated up to or generations. In the following sections, we study the random ferromagnetic Potts on diamond lattice of effective dimension corresponding to a branching ratio .
Iii Statistics of the interface free-energy
As recalled in the introduction, the interface free-energy is expected to follow the scaling behavior of Eq 11 below and to become a random variable of order at (Eq 13). In this section, we present numerical results concerning the singularities of the average and random contributions, as well as histograms, both in the low-temperature phase and at criticality.
iii.1 Flow of the average value and width of the interface free-energy
The flows of the average free-energy and of the free-energy width are shown on Fig. 2 for many temperatures. One clearly sees on these log-log plots the two attractive fixed points separated by the critical temperature . The value of obtained by the pool method depends on the pool, i.e. on the discrete sampling with values of the continuous probability distribution. It is expected to converge towards the thermodynamic critical temperature only in the limit . Nevertheless, for each given pool, the flow of free-energy allows a very precise determination of this pool-dependent critical temperature, for instance in the case considered .
For , both the average free-energy and the free-energy width decay exponentially in . For , the average free-energy grows asymptotically with the interface dimension (see Eq 11)
where is the correlation length that diverges as . The free-energy width grows asymptotically with the droplet exponent (see Eq 11)
iii.2 Divergence of the correlation lengths and
The correlation lengths and as measured from the free-energy average value (Eq 33) and from the 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
In conclusion, our numerical results point towards the following singular behavior for the interface free-energy (see Eq. 11)
where the average contribution and the random contribution involve two correlation lengths and that 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 that appears in the extensive non-random contribution to the interface free-energy. However, the presence of another length scale that diverges with a larger exponent remains to be better understood.
iii.3 Histogram of the interface free-energy below
In the low-temperature phase, the the interface free-energy is expected to follow the behavior of Eq 11, where is a random behavior of order . We show on Fig. 4 the probability distribution of the rescaled variable in log-scale to see the tails. The two tails exponents defined by
iii.4 Histogram of the interface free-energy at criticality
Iv Statistics of the interface energy
In this section, we present the numerical results concerning the statistics of the interface energy.
iv.1 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 . The extensive non-random part is directly related to the corresponding non-random part of the free-energy of Eq. 11 via the usual thermodynamic relation
As a consequence, the singularity found previously for (Eq 33)
determines the singularity of near
We now consider the random contribution to the interface energy in Eq. 12. The flow of the width as grows is shown on Fig. 6 for many temperatures. For , this width grows asymptotically with the exponent as expected (see Eq 12)
Exactly at criticality, the energy width (and entropy width) grows as a power-law (see Fig 6 b)
This value for is in agreement with the value (see Eq. 35).
We now define a correlation length for via the finite-size scaling form
In the regime , one should recover the -dependence of the low-temperature phase of Eq 42, so the scaling function should present the asymptotic behavior yielding the temperature dependence of the prefactor
One similarly may define a correlation length from the finite-size scaling of the entropy width.
As shown on Fig 7 b, the log-log plot presents some curvature, so that the asymptotic slope defined by
is difficult to measure precisely. However, the slope is close to the value of Eq. 35.
In conclusion, our numerical results point towards the following singular behavior for the interface energy (see Eq. 12)
i.e. both the average contribution and the random contribution involve the same correlation length . This result seems natural within the Fisher-Huse droplet theory [7, 13] where the interface energy is a sum of random terms that follow some Central Limit asymptotic behavior. This picture is confirmed by the Gaussian distribution of the random variable that we now consider.
iv.2 Histogram of the interface energy below
In the low-temperature phase, the the interface energy is expected to follow the behavior of Eq 12, where is a random variable of order which is expected to be Gaussian distributed within the droplet theory [7, 13] : this is in agreement with our numerical histogram of the rescaled variable shown on Fig 8.
iv.3 Histogram of the interface energy at criticality
We show on Fig. 9 our numerical results for the histogram of the interface energy at criticality : the unrescaled distribution for generation is shown on Fig. 9 (a), whereas the distribution of the rescaled variable is shown in log-scale on Fig. 9 (b). The fast decay of the tails show that the scaling of the interface energy at criticality is well measured via its variance (see Eq. 43).
V Statistics of the order parameter and of the interfacial adsorption
v.1 Statistics of the order parameter
Below , we find that the order parameter follows the scaling form
The coefficient of the extensive non-random term vanishes at criticality with the exponent
Exactly at criticality, the order parameter is expected to follow the behavior of Eq. 8, up to a random variable of order
in agreement with the scaling relation of Eq 50.
v.2 Statistics of the interfacial adsorption
The coefficient of the extensive non-random term is expected to diverge at criticality as in the pure case (Eq. 7). The coefficient of the random term is expected to diverge to yield the same finite-size scaling as Eq. 8 exactly at criticality, so that
where is a random variable of order . We measure (see Fig. 10)
again in agreement with the scaling relation of Eq 53.
The conclusion of this section is that our numerical results concerning the order parameter and the interfacial adsorption are consistent with ’conventional’ scaling in terms of the correlation length .
Vi Comparison with the spin-glass on diamond lattice of effective dimension
The Migdal-Kadanoff renormalizations with a branching ratio , which corresponds to an effective dimension (Eq 18), have been much used to study spin-glasses [35, 36, 37, 38, 10, 11] (for the case corresponding to an effective dimension there is no spin-glass phase). As recalled in the introduction, a new ’chaos exponent’ has been introduced in [10, 11] to describe chaos properties at criticality, and this exponent was argued to govern the energy at criticality (Eq 17). In the following, we confirm this scenario by directly measuring the statistical properties of the interface energy. We also discuss the similarities and differences with the random ferromagnetic case discussed above.
vi.1 Flow of the width of the interface free-energy
In the spin-glass case, there is no non-random leading term (Eq 9) in contrast to the random ferromagnetic case ( Eq 11). As a consequence, we only have to consider here the flow of the free-energy width shown on Fig. 11. For , the free-energy width decays exponentially in . For , the free-energy width grows asymptotically with the droplet exponent (see Eq. 9)
Again, the critical temperature obtained by this pool method depends on the pool, i.e. on the discrete sampling with values of the continuous probability distribution. It is expected to converge towards the thermodynamic critical temperature only in the limit . Nevertheless, for each given pool, the flow of free-energy width allows a very precise determination of this pool-dependent critical temperature, for instance in the case considered . This value is in agreement with previous measures for a Gaussian initial condition [35, 38, 10].