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arXiv:2411.11644 · Physical review. B./Physical review. B

Geometric percolation of spins and spin dipoles in the Ashkin-Teller model

Aikya Banerjee, Priyajit Jana, P. K. Mohanty

Published Jan 6, 2025• 3 citations
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Abstract

The Ashkin-Teller model is a two-layer lattice model where spins in each layer interact ferromagnetically with strength $J$, and the spin-dipoles (product of spins) interact with neighbors with strength $\ensuremath{\lambda}.$ The model exhibits simultaneous magnetic and electric transitions along a self-dual line on the $\ensuremath{\lambda}\text{\ensuremath{-}}J$ plane with continuously varying critical exponents. In this article, we investigate the percolation of geometric clusters of spins and spin-dipoles denoted, respectively, as magnetic and electric clusters. We find that the largest cluster in both cases becomes macroscopic in size and spans the lattice when interaction exceeds a critical threshold given by the same self-dual line where magnetic and electric transitions occur. The fractal dimension of the critical spanning clusters is related to order parameter exponent ${\ensuremath{\beta}}_{m,e}$ as ${D}_{m,e}=d\ensuremath{-}\frac{5}{12}\frac{{\ensuremath{\beta}}_{m,e}}{\ensuremath{\nu}},$ where $d=2$ is the spatial dimension and $\ensuremath{\nu}$ is the correlation length exponent. This relation determines all other percolation exponents and their variation with respect to $\ensuremath{\lambda}.$ We show that for magnetic percolation, the binder cumulant, as a function of ${\ensuremath{\xi}}_{2}/L$ with ${\ensuremath{\xi}}_{2}$ being the second-moment correlation length, remains invariant all along the critical line and matches with that of the spin-percolation in the usual Ising model. The function also remains invariant for the electric percolation, forming a new superuniversality class of percolation transition.

DOI
10.1103/physrevb.111.014403

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The Ashkin-Teller model is a two-layer lattice model where spins in each layer interact ferromagnetically with strength JJJ, and the spin-dipoles (product of spins) interact with neighbors with strength \ensuremathλ.\ensuremath{\lambda}.\ensuremathλ. The model exhibits simultaneous magnetic and electric transitions along a self-dual line on the \ensuremathλ\ensuremath-J\ensuremath{\lambda}\text{\ensuremath{-}}J\ensuremathλ\ensuremath-J plane with continuously varying critical exponents. In this article, we investigate the percolation of geometric clusters of spins and spin-dipoles denoted, respectively, as magnetic and electric clusters. We find that the largest cluster in both cases becomes macroscopic in size and spans the lattice when interaction exceeds a critical threshold given by the same self-dual line where magnetic and electric transitions occur. The fractal dimension of the critical spanning clusters is related to order parameter exponent \ensuremathβm,e{\ensuremath{\beta}}_{m,e}\ensuremathβm,e​ as Dm,e=d\ensuremath−512\ensuremathβm,e\ensuremathν,{D}_{m,e}=d\ensuremath{-}\frac{5}{12}\frac{{\ensuremath{\beta}}_{m,e}}{\ensuremath{\nu}},Dm,e​=d\ensuremath−125​\ensuremathν\ensuremathβm,e​​, where d=2d=2d=2 is the spatial dimension and \ensuremathν\ensuremath{\nu}\ensuremathν is the correlation length exponent. This relation determines all other percolation exponents and their variation with respect to \ensuremathλ.\ensuremath{\lambda}.\ensuremathλ. We show that for magnetic percolation, the binder cumulant, as a function of \ensuremathξ2/L{\ensuremath{\xi}}_{2}/L\ensuremathξ2​/L with \ensuremathξ2{\ensuremath{\xi}}_{2}\ensuremathξ2​ being the second-moment correlation length, remains invariant all along the critical line and matches with that of the spin-percolation in the usual Ising model. The function also remains invariant for the electric percolation, forming a new superuniversality class of percolation transition.
The fractal dimensions and the percolation exponents of geometrical spin clusters of like sign at criticality are obtained numerically for an Ising model with temperature dependent annealed bond dilution, also known as the thermalized bond Ising model (TBIM), in two dimensions. For this purpose, a modified Wolff single-cluster Monte Carlo simulation is used to generate equilibrium spin configurations on square lattices in the critical region. A tie-breaking rule is employed to identify nonintersecting spin cluster boundaries along the edges of the dual lattice. The values obtained for the fractal dimensions of the spanning geometrical clusters D c , and their interfaces D I , are in perfect agreement with those reported for the standard two-dimensional ferromagnetic Ising model. Furthermore, the variance of the winding angles results in a diffusivity κ = 3 for the two-dimensional thermalized bond Ising model, thus placing it in the universality class of the regular Ising model. A finite-size scaling analysis of the largest geometrical clusters results in a reliable estimation of the critical percolation exponents for the geometrical clusters in the limit of an infinite lattice size. The percolation exponents thus obtained are also found to be consistent with those reported for the regular Ising model. These consistencies are explained in terms of the Fisher renormalization relations, which express the thermodynamic critical exponents of systems with annealed bond dilution in terms of those of the regular model system.
Geometric representations provide a useful perspective on critical phenomena in the Ising model. In a recent study [Phys. Rev. E 112, 034118 (2025)], we found that the two-dimensional critical Ising model exhibits two consecutive percolation transitions for geometric spin clusters as the bond-occupation probability ppp between parallel spins increases. Here, through extensive Monte Carlo simulations, we show that this phenomenon does not persist in three dimensions, where we observe only a single percolation transition on critical Ising configurations. Further theoretical analysis of the Ising model on the complete graph also yields the same scenario. In addition, we study percolation on a two-dimensional layer embedded in the three-dimensional critical Ising model. For this layer system, we estimate the red-bond exponent yp=0.426(6)y_p = 0.426(6)yp​=0.426(6) and the fractal dimensions of the largest cluster, hull, and shortest path as df=1.8926(20)d_f = 1.8926(20)df​=1.8926(20), dhull=1.663(4)d_{\rm hull} = 1.663(4)dhull​=1.663(4), and dmin=1.080(10)d_{\rm min} = 1.080(10)dmin​=1.080(10), respectively. These values indicate a distinct universality class induced by coupling to out-of-plane critical correlations.
We consider geometrical or Ising clusters (i.e., domains of parallel spins) in the square lattice random-field Ising model by varying the strength of the Gaussian random field $\ensuremath{\Delta}$. In agreement with the conclusion of a previous investigation [Phys. Rev. E 63, 066109 (2001)], the geometrical correlation length, i.e., the average size of the clusters $\ensuremath{\xi}$ is finite for $\ensuremath{\Delta}>{\ensuremath{\Delta}}_{c}\ensuremath{\approx}1.65$ and divergent for $\ensuremath{\Delta}\ensuremath{\leqslant}{\ensuremath{\Delta}}_{c}$. The scaling function of the distribution of the mass of the clusters as well as the geometrical correlation function are found to involve the scaling exponents of critical percolation. On the other hand, the divergence of the correlation length, $\ensuremath{\xi}(\ensuremath{\Delta})\ensuremath{\sim}{(\ensuremath{\Delta}\ensuremath{-}{\ensuremath{\Delta}}_{c})}^{\ensuremath{-}\ensuremath{\nu}}$, with $\ensuremath{\nu}\ensuremath{\approx}2$, is related to that of tricritical percolation. It is verified numerically that critical geometrical correlations transform conformally.
In this paper we consider the Ising model (called the dynamical Ising model with dynamical temperature TTT) on the Ising-correlated percolation lattice. To construct this lattice, another Ising model (which is called the quenched Ising model with quench artificial temperature Tq{T}_{q}Tq​) is employed to model the correlations between imperfections. We argue that for each quenched temperature, there is a particular dynamical temperature Tc(Tq){T}_{c}({T}_{q})Tc​(Tq​) for which the dynamical Ising model becomes critical. We present some finite-size arguments (based on the moments of the magnetism and the energy as well as the spanning cluster probability) to extract the critical points and also show that they are compatible with the finite-size scaling of the singular point of the magnetic susceptibility. The model is thoroughly characterized in and out of these points. We find that the critical behaviors of the model change significantly with respect to the regular Ising model as well as the Ising model on the uncorrelated percolation lattice. It is shown that for the critical lattice, ${T}_{q}={T}_{c}^{\text{square}\phantom{\rule{4.pt}{0ex}}\text{Ising}\phantom{\rule{4.pt}{0ex}}\text{model}}\ensuremath{\approx}2.269$, the critical temperature for the dynamical Ising model in the thermodynamic limit is ${T}_{c}({T}_{q}={T}_{c}^{\text{square}\phantom{\rule{4.pt}{0ex}}\text{Ising}\phantom{\rule{4.pt}{0ex}}\text{model}})=1.94\ifmmode\pm\else\textpm\fi{}0.005$, and the fractal dimension of the exterior perimeter of geometrical spin clusters is ${D}_{f}^{T={T}_{c}({T}_{q}=2.269)}=1.408\ifmmode\pm\else\textpm\fi{}0.002$. Many quantities, such as the dynamical critical temperatures, all local and global critical exponents, and the fractal dimension of loops Df{D}_{f}Df​, scale with the quench temperature in a power-law fashion, with some critical exponents that are reported. Significantly we see that ${D}_{f}^{T={T}_{c}({T}_{q})}\ensuremath{-}{D}_{f}^{T={T}_{c}({T}_{q}=2.269)}\ensuremath{\sim}\frac{1}{\sqrt{\ensuremath{\zeta}({T}_{q})}}$ in which $\ensuremath{\zeta}({T}_{q})$ is the correlation length of the quenched Ising model at temperature Tq{T}_{q}Tq​.
Monte Carlo simulations in the three-dimensional (3D) Ashkin-Teller model on a cubic lattice are performed in the regions of the two-parameter space diagram where Ising-type phase transitions are expected. The scaling behavior of the Binder cumulant Q and the magnetic susceptibility in the critical region are exploited. In simulations the periodic boundary conditions and the Metropolis algorithm are used. Starting from Ising critical exponents and applying the finite-size-scaling analysis of the cumulant Q with nonlinear corrections, the accurate positions of the critical couplings on the continuous phase transition lines are calculated. For these couplings the critical exponent y(h) is calculated analyzing the magnetic susceptibility in the framework of the finite-size scaling with corrections. The values of y(h) agree with the 3D Ising model value along two lines determined by the order parameter [ssigma] .
Abstract The N -color Ashkin–Teller model corresponds to N Ising models coupled by four-spin interactions. We consider the two-dimensional case in presence of quenched disorder and use scale invariant scattering theory to determine all the solutions of the exact renormalization group fixed points equations. The weak disorder sector is characterized by a solution that, for any fixed N larger than 1, is a line of fixed points with Ising thermal exponents and continuously varying magnetic exponents. The number of fixed point solutions allowed by the symmetries of the model increases at strong disorder illustrating the growing dependence on the distributions of the two random couplings. The presence of some critical exponents which do not depend on the symmetry parameter N confirms this type of superuniversality as a peculiar feature of random criticality.