Numerical Study on a Crossing Probability for the Four-State Potts Model: Logarithmic Correction to the Finite-Size Scaling

TL;DR

This study numerically investigates the crossing probability in the critical four-state Potts model, confirming the SLE description of interfaces and revealing a logarithmic correction to finite-size scaling.

Contribution

It provides the first numerical confirmation of the SLE description of spin cluster interfaces and identifies a logarithmic correction to the finite-size scaling of crossing probability.

Findings

Interface described by SLE with κ=4

Crossing probability exhibits a 1/log(L M) correction

Numerical results agree with analytical predictions

Abstract

A crossing probability for the critical four-state Potts model on an $L\times M$ rectangle on a square lattice is numerically studied. The crossing probability here denotes the probability that spin clusters cross from one side of the boundary to the other. First, by employing a Monte Carlo method, we calculate the fractal dimension of a spin cluster interface with a fluctuating boundary condition. By comparison of the fractal dimension with that of the Schramm-Loewner evolution (SLE), we numerically confirm that the interface can be described by the SLE with $\kappa=4$, as predicted in the scaling limit. Then, we compute the crossing probability of this spin cluster interface for various system sizes and aspect ratios. Furthermore, comparing with the analytical results for the scaling limit, which have been previously obtained by a combination of the SLE and conformal field theory, we numerically find that the crossing probability exhibits a logarithmic correction $\sim 1/\log(L M)$ to the finite-size scaling.

Tables

Table 1. Table 1: The values of κ 𝜅 \kappa and c 𝑐 c calculated by ( 3 ) and ( 5 ) for the q 𝑞 q -state Potts model. The relations for q = 1 𝑞 1 q=1 (critical percolation) [ 28 , 29 ] and for q = 2 𝑞 2 q=2 (Ising) [ 30 ] have been rigorously proven.

$q$	$c$	$\kappa$
$1$	$0$	$8/3,6$
$2$	$1/2$	$3,16/3$
$3$	$4/5$	$10/3,24/5$
$4$	$1$	$4$

Equations

$$d{g}_t(z)=\frac{2dt}{g_t(z)-\sqrt{\kappa }{B}_t},g_0(z)=z\in \mathbb{H},$$

$$d_{\mathrm{f}}=\min \left(2,1+\frac{\kappa }{8}\right).$$

$$c=\frac{(3\kappa -8)(6-\kappa )}{2\kappa },$$

$$Z=\sum _{\{{\sigma }_j\in Q\}}\exp \left[J\sum _{⟨j,k⟩}{\delta }_{{\sigma }_j,{\sigma }_k}\right],Q=\{1,2,\dots ,q\},$$

$$c=1-\frac{6}{s(s+1)},s=-1+\frac{\pi }{\mathrm{arcsec}(2/\sqrt{q})}.$$

$$d{g}_t(z)=\sum _{j=1}^N\frac{2dt}{g_t(z)-X_t^{(j)}},d{X}_t^{(j)}=\sqrt{\kappa }d{B}_t^{(j)}+d{F}_t^{(j)},$$

$$d{F}_t^{(j)}=\kappa dt({\partial }_{X_t^{(j)}}\log Z_t)+\sum _{k\ne j}\frac{2dt}{X_t^{(j)}-X_t^{(k)}}.$$

$$Z_t=⟨\psi (\infty ){\psi }_{2,1}(X_t^{(1)})\cdots {\psi }_{2,1}(X_t^{(N)})⟩,$$

$$Z_0=⟨{\psi }_{2,1}(\infty ){\psi }_{2,1}(0){\psi }_{2,1}(x){\psi }_{2,1}(1)⟩=Z_{{\mathrm{C}}_1}(x)+Z_{{\mathrm{C}}_2}(x).$$

$$\mathbf{P}[{\mathrm{C}}_1]=\frac{Z_{{\mathrm{C}}_1}(x)}{Z_{{\mathrm{C}}_1}(x)+Z_{{\mathrm{C}}_2}(x)},\mathbf{P}[{\mathrm{C}}_2]=\frac{Z_{{\mathrm{C}}_2}(x)}{Z_{{\mathrm{C}}_1}(x)+Z_{{\mathrm{C}}_2}(x)},$$

$$Z_{{\mathrm{C}}_1}(x)=(1-x{)}^{2/\kappa }{x}^{2/\kappa }{}_2{F}_1\left(\frac{4}{\kappa },\frac{12-\kappa }{\kappa };\frac{8}{\kappa };1-x\right),$$

$$Z_{{\mathrm{C}}_2}(x)=Z_{{\mathrm{C}}_1}(1-x).$$

$$r=\frac{K(1-k^2)}{2K(k^2)},k=\frac{1-\sqrt{x}}{1+\sqrt{x}},$$

$$P(r)=\mathbf{P}[{\mathrm{C}}_1]$$

$$P(r)=1-x(0<x<1),$$

$$l\sim a{\left(\frac{L}{a}\right)}^{d_{\mathrm{f}}}(L/a\gg 1),$$

Full text

1. Introduction  The geometrical description of critical phenomena has renewed interest in the theoretical study of phase transitions. Various theoretical tools have been developed, particularly in the two-dimensional (2D) case where the structure of critical phenomena is expected to be strongly constrained by an infinite-dimensional conformal symmetry.

In particular, the Schramm-Loewner evolution (SLE) [1, 2, 3, 4, 5] describes the random fractals arising in 2D critical phenomena as a growth process defined by a stochastic evolution of conformal maps: the SLE generates a random curve on a planar domain from a 1D Brownian motion on the boundary. For instance, the SLE defined on the complex upper half-plane $\mathbb{H}$, which is conventionally called the chordal SLE, is described by the evolution of the following conformal map:

[TABLE]

where $B_{t}$ is the 1D standard Brownian motion living on $\mathbb{R}$, i.e., its expectation value and variance are, respectively, given by ${\bf E}[dB_{t}]=0$ and ${\bf E}[dB_{t}dB_{t}]=dt$. The tip $\gamma_{t}$ of the random curve evolves as $\gamma_{t}=\lim_{\epsilon\to+0}g^{-1}_{t}(\sqrt{\kappa}B_{t}+i\epsilon)$. Namely, the SLE is able to give a direct description of non-local geometrical objects which are difficult to describe within traditional approaches. The SLE has only one parameter $\kappa$ associated with the diffusion constant of the Brownian motion. This parameter $\kappa$ qualitatively and quantitatively characterizes the random curves generated by the SLE. For example, the fractal dimension $d_{\rm f}$ of the SLE curve is expressed as [6, 7]:

[TABLE]

On the other hand, the universal properties of 2D critical systems are explained by conformal field theory (CFT) [8, 9]. In contrast to the SLE, CFT determines correlation functions among local operators. The SLE with $\kappa$ is conjecturally related to CFT [10, 11] via

[TABLE]

where $c$ is the central charge characterizing the universality class of 2D critical systems. By combining the SLE with CFT, we can systematically analyze geometrical objects for 2D critical statistical mechanics models.

Crossing probabilities in statistical mechanics models, which are mainly discussed in this letter, are one of these non-local geometrical objects. The crossing probability in this letter denotes the probability that a cluster composed by local variables such as spins connects two disjoint segments on the boundary of a simply connected planar domain. Thanks to the conformal invariance expected in the critical systems, this problem can be mapped to that of the upper half-plane $\mathbb{H}$. For the statistical models with cluster boundaries described by the SLE with $\kappa>4$, the crossing probability can be computed by use of the single SLE curve (1). The result contains the one for critical percolation ($\kappa=6$; $c=0$), which is well-known as the Cardy formula [12], and for the Fortuin-Kasteleyn (FK) clusters in the Ising model ($\kappa=16/3$; $c=1/2$). On the other hand, the crossing probability for the interfaces characterized by the SLE with $\kappa\leq 4$, which includes the spin cluster boundaries of the Ising model ($\kappa=3$; $c=1/2$), may be evaluated by considering the multiple (three) SLE curves as constructed in [13].

There still exists, however, a non-trivial problem: what kinds of cluster boundaries in the scaling limit of the lattice model do SLE curves actually correspond to? This problem becomes serious for the systems possessing multiple (typically more than two) local variables, such as the three- or four-state Potts models (see, for instance, [14] for a treatment of the three-state Potts model), where various kinds of cluster boundaries can be defined.

In this letter, using a Monte Carlo method, we numerically compute the crossing probability for certain spin clusters of the four-state Potts model on an $L\times M$ rectangle on a square lattice. In the scaling limit, the universal behavior of the four-state Potts model is classified by the $c=1$ CFT. Intuitively, from (3), appropriately defined cluster boundaries become the SLE curves with $\kappa=4$ in the scaling limit. Indeed, it is known that the FK clusters of the model become the SLE with $\kappa=4$ (see [2, 4, 5] for example). We hypothesize that properly defined spin cluster boundaries are also described by the SLE with $\kappa=4$. First, we numerically evaluate the fractal dimension of a spin cluster interface with a ‘fluctuating’ boundary condition. Comparing the result with that obtained by the SLE, we numerically confirm that the above hypothesis is correct for the interface with the fluctuating boundary condition. Second, from high-precision Monte Carlo data, we find that the spin cluster interface under the fluctuating boundary condition gives the crossing probability described by the multiple version of the SLE. Moreover, comparing the results with those obtained in the scaling limit [13], we numerically show that the crossing probability exhibits a logarithmic correction $\sim 1/\log(LM)$ to the finite-size scaling.

2. Potts Model  We shall study the Potts model on a rectangle on a square lattice. The $q$-state Potts model is an interacting spin model with the spin variables taking on the values $1,2,\dots,q$. The partition function $Z$ of the system is given by

[TABLE]

where $J>0$ (i.e., ferromagnetic interaction) and $\langle j,k\rangle$ denotes adjacent sites on the square lattice. For $q\to 1$ and $q=2$, the model is equivalent to the bond percolation model and the Ising model, respectively. It is well known that the Potts model exhibits a second-order phase transition for $q\leq 4$, while for $q>4$ the transition is of first order. Note that, for $q\leq 4$, the scaling behavior is governed by CFT [27] with

[TABLE]

In Table 1, the values of $\kappa$ and $c$ calculated by (3) and (5) for the $q$-state Potts model are summarized (see [2, 4, 5] and references therein for some examples of the correspondence between SLE and interfaces in the scaling limit of some lattice models including the FK clusters of the Potts model). The relations $\kappa=8/3,6$ for $q=1$ (critical percolation) [28, 29] and $\kappa=3,16/3$ for $q=2$ (Ising) [30] have been rigorously proven. The border value $q=4$ (corresponding to $c=1$ and $\kappa=4$) is the most crucial: though the model undergoes a second-order transition, the power-law behavior of local operators is expected to be modified by logarithmic factors from marginally irrelevant operators [15, 16, 17, 18, 19]. For our purpose, hereafter we set $J=J_{\rm c}:=\log(\sqrt{q}+1)$ where $J_{\rm c}$ denotes the critical temperature.

3. Multiple SLE and Crossing Probabilities  The SLEs can be generalized to multiple versions generating $N$ random curves evolving from $N$ points on the boundary as in [13, 22, 23, 24, 25, 26]. For the chordal case, the multiple SLE is given by the following form [13]:

[TABLE]

where $\{B_{t}^{(j)}\}$ is an $\mathbb{R}^{N}$-valued Brownian motion whose expectation value and variance are given by ${\bf E}[dB^{(j)}_{t}]=0$ and ${\bf E}[dB^{(j)}_{t}dB^{(k)}_{t}]=\delta_{jk}dt$, respectively. Reflecting interactions among curves, the driving term $X_{t}^{(j)}$ has an additional drift term $dF_{t}^{(j)}$ given by

[TABLE]

Here $Z_{t}$ is a correlation function of boundary condition changing (bcc) operators:

[TABLE]

where the $\psi_{2,1}(X_{t}^{(j)})$ are bcc operators inserted at the positions $z=X_{t}^{(j)}$ and are degenerate at level two: the conformal weight $h$ of $\psi_{2,1}$ is given by $h=h_{2,1}=(6-\kappa)/(2\kappa)$. Also $\psi(\infty)$ is a bcc operator inserted at $z=\infty$. The tip $\gamma_{t}^{(j)}$ of the $j$th curve is given by $\gamma_{t}^{(j)}=\lim_{\epsilon\to+0}g_{t}^{-1}(X_{t}^{(j)}+i\epsilon)$. Most importantly, the conformal weight of $\psi(\infty)$ (denoted by $h_{\infty}$) characterizes the topological configurations of the SLE curves [13]. This indicates that some non-local geometrical properties can be determined by the expectation value of products of local operators, which can be exactly computed by CFT.

For instance, for $N=3$ and $h_{\infty}=h_{2,1}$, the two topologically inequivalent configurations are allowed. See Fig. 1 (a) for these configurations where the three curves are assumed to start from the points $X_{t=0}^{(1)}=0$, $X_{t=0}^{(2)}=x$ ($0<x<1$) and $X_{t=0}^{(3)}=1$. Each configuration is characterized by the correlation function

[TABLE]

Namely, the probability ${\bf P}[{\rm C}_{1}]$ (resp. ${\bf P}[{\rm C}_{2}]$) of the occurrence of the configuration $\rm{C}_{1}$ (resp. $\rm{C}_{2}$) as in the left (resp. right) panel of Fig. 1 (a) is expressed as

[TABLE]

where up to an overall factor $Z_{\rm C_{1}}(x)$ and $Z_{\rm C_{2}}(x)$ are, respectively, given by [13]:

[TABLE]

By the Schwarz-Christoffel transformation, the two configurations ${\rm C_{1}}$ and ${\rm C_{2}}$ defined on $\mathbb{H}$ can be mapped to those on a rectangle (see Fig. 1 (b)) with the aspect ratio $r$:

[TABLE]

where $K(k^{2})$ is the complete elliptic integral of the first kind with the modulus $k$ (see, for instance, Chapter 11 in [9] for a detailed derivation, and see also arguments in [20, 21] for more general polygons). The probability of the occurrence of the configuration ${\rm C^{\prime}_{1}}$ in the left panel of Fig. 1 (b) corresponds to the crossing probability that the spin clusters cross from the top edge to the bottom edge (see also Fig. 2 (a) or Fig. 3 (c) for a configuration of spin clusters corresponding to ${\rm C^{\prime}_{1}}$). In this letter, we discuss this crossing probability defined on the rectangular domain. By an assumption that the probabilistic measure is invariant under the conformal transformation, the crossing probability for the rectangle of aspect ratio $r$ is given by

[TABLE]

with (10), (11) and (12).

For the four-state Potts model ($\kappa=4$), the probability reduces to a simpler form ${\bf P}[{\rm C_{1}}]=1-x$ and ${\bf P}[{\rm C_{2}}]=x$, which can be directly obtained by setting $\kappa=4$ in (11). Thus, the analytical expression of the crossing probability in the scaling limit of the four-state Potts model explicitly reads

[TABLE]

where $x$ is a function of $r$ as given by (12).

4. Numerical Results  Now, let us numerically calculate the crossing probability of the four-state Potts model (4) on a rectangle on the square lattice.

First we shall detect the interface and the boundary condition giving the above $P(r)$ (14). In fact, several types of spin cluster boundaries can be defined in the Potts model for $q\geq 3$, which is in contrast to the Ising case ($q=2$) where there exists only one type of spin cluster boundary: the interfaces between clusters of spin type 1 and those of spin type 2. For the Ising model, we therefore can easily set up a boundary condition of the rectangle which is compatible with the multiple SLE with $\kappa=3$. Namely, the boundary spins on the top and bottom edges are set to type 1, and those on the left and right edges are fixed to type 2 and vice versa (see Fig. 2 for an example). Indeed, as in Fig. 2, the numerical results calculated by a Monte Carlo simulation (the Wolf algorithm) for the crossing probability agree well with the ones in the scaling limit, which is obtained by (13) with $\kappa=3$.

Let us go back to the case of the four-state Potts model ($q=4$; $\kappa=4$). The situation is quite different from the Ising case. For instance, if we fix the boundary spins as for the Ising model (see Fig. 3 (a)), the spin cluster boundaries starting from the corners no longer join any corners, due to the existence of clusters consisting of spins different from the ones on the boundaries. Namely, the spin cluster boundary starting from one corner splits into two different types of spin cluster interfaces, when it meets the other spin clusters in the bulk, which is not compatible with the SLE (see a snapshot in Fig. 3 (a)).

Alternatively, we adopt a ‘fluctuating’ boundary condition as defined in [14, 21]. The fluctuating boundary condition in our case is defined as follows. We divide the four types of spin $1,2,3,4$ into two parts, say $\{1,2\}$ and $\{3,4\}$ (see [21] for another choice of the fluctuating boundaries). On the top and bottom edges (resp. left and right edges), the type 1 and 2 spins (resp. the type 3 and 4 spins) are randomly assigned and vice versa. See Fig. 3 (b) as an example. Under this configuration, the spin cluster boundary is defined as the interface between the spin cluster consisting of $\{1,2\}$ and the one consisting of $\{3,4\}$. See Fig. 3 (c) and (d) where the type $\{1,2\}$ is colored yellow, while the type $\{3,4\}$ is colored purple. Consequently, we can construct the spin cluster boundary compatible with the SLE.

Indeed, by numerically analyzing the fractal dimension, we can confirm that this spin cluster boundary can be described by the SLE with $\kappa=4$. Here the fractal dimension $d_{\rm f}$ is evaluated in the following scaling law:

[TABLE]

where $l$ is the total length of the curve, and $L$ and $a$ are the system size and the lattice spacing, respectively. As shown in Fig. 4, using a Monte Carlo method (the Wolf algorithm) we numerically confirm that the fractal dimension of the above-defined spin cluster boundary is $d_{\rm f}=1.504\pm 0.020$, which is obtained by the least squares method. The result agrees well with the prediction from the SLE of $d_{\rm f}=3/2$ obtained by the substitution of $\kappa=4$ into the formula (2). Note that, in the scaling limit, the boundary spins $1$ and $2$ (resp. $3$ and $4$) are uniformly distributed on the top and bottom edges (the left and right edges) for the fluctuating boundary condition. Therefore, for our numerical calculations, we arrange the boundary spins alternately as $1212\cdots$ (resp. $3434\cdots$) on the top and bottom edges (resp. the left and right edges).

Next, we examine the crossing probability for the four-state Potts model on the rectangle with the fluctuating boundary condition. Figure 5 shows the numerical results of the crossing probability for various system sizes and aspect ratios. In comparison with the Ising model in Fig. 2 (c), the finite-size data converges much more slowly to that in the scaling limit given by (14). This slow convergence might be explained in terms of logarithmic corrections caused by the existence of marginally irrelevant operators, since the crossing probability is essentially governed by the correlation function of local operators (9). In fact, as shown in Fig. 6 (a) and (b), the crossing probability for the $L\times M$ rectangle exhibits a logarithmic correction $\sim 1/\log(LM)$ to the finite-size scaling. This logarithmic behavior can be more clearly confirmed by Fig. 6 (c) and (d), as both $\log|P(r)-c|$ are linearly dependent with $\log(\log LM)$. This behavior cannot be explained by power functions. Similar logarithmic behavior is also observed for various values of $r$. This behavior also might be analytically confirmed by analyzing the correlation function (9).

5. Concluding Remarks  In this letter, we have numerically investigated the crossing probability of the Potts model on the $L\times M$ rectangle for the fluctuating boundary condition. Comparing the fractal dimension with the prediction from the SLE, we have numerically confirmed that the spin cluster interfaces under the fluctuating boundary condition are described by the SLE with $\kappa=4$. We also have shown numerically that the crossing probability of this spin cluster boundary exhibits a logarithmic correction $\sim 1/\log(LM)$ to the finite-size scaling. This logarithmic correction might be explained by the existence of marginally irrelevant operators. Recently, similar logarithmic behavior has also been observed in other random geometries (four-point boundary connectivities of FK clusters) for the four-state Potts model [31] (see also [32] for the Ising model). It remains a crucial problem to show this logarithmic behavior analytically by analyzing the correlation function of the local operators.

Acknowledgment

The present work was partially supported by Grant-in-Aid for Scientific Research (C) No. 16K05468 from Japan Society for the Promotion of Science.