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arXiv:2412.19664

Non-reciprocal interactions preserve the universality class of Potts model

Soumya K. Saha, P. K. Mohanty

Published Dec 27, 2024• 2 citations
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Abstract

We study the $q$-state Potts model on a square lattice with directed nearest-neighbor spin-spin interactions that are inherently non-reciprocal. Both equilibrium and non-equilibrium dynamics are investigated. Analytically, we demonstrate that non-reciprocal interactions do not alter the critical exponents of the model under equilibrium dynamics. In contrast, numerical simulations with selfish non-equilibrium dynamics reveal distinctive behavior. For $q=2$ (non-reciprocal non-equilibrium Ising model), the critical exponents remain consistent with those of the equilibrium Ising universality class. However, for $q=3$ and $q=4$, the critical exponents vary continuously. Remarkably, a super-universal scaling function -- Binder cumulant as a function of $ξ_2/ξ_0$, where $ξ_2$ is the second moment correlation length and $ξ_0$ its maximum value -- remains identical to that of the equilibrium $q=3,4$ Potts models. These findings indicate that non-reciprocal Potts models belong to the superuniversality class of their respective equilibrium counterparts.

DOI
10.48550/arxiv.2412.19664

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We investigate the short-time critical dynamics of the 4-state Potts (FSP) model, Baxter-Wu (BW) model and Ising model with three-spin interactions (IMTSI) in one direction, that belong to the same universality class at equilibrium. The scaling behavior of the global persistence probability P(t)P(t)P(t) is studied in order to estimate the global persistence exponent θg\theta_gθg​ of the models. We also calculate the exponent x0x_0x0​, that represents the anomalous dimension of the initial. Although the exponents θg\theta_gθg​ and x0x_0x0​ for the FSP model are in complete agreement with the results of the IMTSI model, they are completely different of that for the BW model, showing that the BW model does not belong to the same dynamic universality class of the FSP model.
Critical scaling and universality in short-time dynamics for spin models on a twodimensional triangular lattice are investigated by using Monte Carlo simulation. Emphasis is placed on the dynamic evolution from fully ordered initial states to show that universal scaling exists already in the short-time regime in form of power-law behavior of the magnetization and Binder cumulant. The results measured for the dynamic and static critical exponents, θ, z, β and ν, confirm explicitly that the Potts models on the triangular lattice and square lattice belong to the same universality class. Our critical scaling analysis strongly suggests that the simulation for the dynamic relaxation can be used to determine numerically the universality.
We find that the ordinary universality does not hold for a frustrated spin-1 Ising model with nearest-neighbor bilinear interaction J1{\mathit{J}}_{1}J1​ and next-nearest-neighbor bilinear interaction J2{\mathit{J}}_{2}J2​ on the square lattice. We estimated critical indices \ensuremath{\nu}, \ensuremath{\eta}, and ys{\mathit{y}}_{\mathit{s}}ys​ by applying the Monte Carlo renormalization-group method to the model. The critical index \ensuremath{\nu} varies with J1{\mathit{J}}_{1}J1​/J2{\mathit{J}}_{2}J2​ in a region where the ground state is of the super-antiferromagnetic configuration. Although the behavior of \ensuremath{\nu} is analogous to that of the spin-1/2 Ising model as a function of J1{\mathit{J}}_{1}J1​/J2{\mathit{J}}_{2}J2​, the value of \ensuremath{\nu} appears to be smaller than that of the corresponding spin-1/2 Ising model. The value of \ensuremath{\eta} agrees with the value predicted by the Onsager universality class. The crossover exponent ys{\mathit{y}}_{\mathit{s}}ys​ agrees with the value estimated for the spin-1/2 Ising model.
We study the critical behavior of a two-dimensional Ising model with nonreciprocal vision-cone interactions, which explicitly violate reciprocity and detailed balance. Extensive Monte Carlo simulations and dynamic renormalization-group analysis show that the asymptotic critical exponents remain fully consistent with the equilibrium Ising universality class over a broad range of nonreciprocal coupling strengths λλλ. In contrast, dimensionless quantities such as the Binder cumulant and the correlation-length ratio display pronounced anisotropic nonequilibrium corrections and systematically deviate from their equilibrium Ising values. The renormalization-group flow further demonstrates that the nonreciprocal perturbation is irrelevant at the Wilson-Fisher fixed point while generating a finite shift of the critical temperature proportional to λ2λ^2λ2. Our results demonstrate the remarkable robustness of two-dimensional Ising criticality against this class of directional interactions.
We investigate the dynamical critical behavior of the two-dimensional three-state Potts model with single spin-flip dynamics in equilibrium. We focus on the mean-squared deviation of the magnetization M (MSD_{M}) as a function of time, as well as on the autocorrelation function of M. Our simulations reveal the existence of two crossover behaviors at times τ_{1}∼L^{z_{1}} and τ_{2}∼L^{z_{2}}, separating three dynamical regimes. MSD_{M} appears to shift from ordinary diffusion in the first regime, to anomalous diffusion in the second, and finally to be constant in the third regime. The magnetization autocorrelation function, on the other hand, is found to fluctuate between exponential decay, stretched-exponential decay, and then again exponential decay along these three regimes. This behavior is in agreement with the one reported recently for the two-dimensional Ising ferromagnet [Phys. Rev. E 108, 034118 (2023)2470-004510.1103/PhysRevE.108.034118], indicating that the existence of two dynamic critical exponents is not a peculiarity of the Ising model itself. A comparison of both MSD_{M} and the magnetization's autocorrelation function suggests that within our numerical accuracy the exponents z_{1} and z_{2} are shared between the Ising and three-state Potts models at least for the particular case of single spin-flip dynamics studied here, even though their equilibrium universality classes are clearly distinct. Continuity of MSD_{M} requires that α(z_{2}-z_{1})=γ/ν-z_{1}, in which α is the anomalous exponent in the intermediate regime. Since the ratio γ/ν is not shared between the two models, it follows that α is not shared either, an aspect well verified in our simulations. Finally, we also discuss the relevance of our main findings using another useful observable, namely the line magnetization M_{l}.
The critical properties of an antiferromagnetic Ising model on a square lattice with interactions of the next-to-nearest neighbors are investigated by a replica Monte-Carlo method. Using the finite-size scaling theory the static critical exponents of specific heat, ordering parameter, susceptibility, correlation radius as well as the Fisher exponent are calculated. An analysis of data is performed both with and without taking into account a correction to the finite-size scaling. It was found that in the model under consideration the second order phase transition is observed. It is shown that this model belongs to the new class of universality of critical behavior.