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arXiv:1807.05123 · Physical review. E

Critical dynamical exponent of the two-dimensional scalar ϕ4 model with local moves

Wei Zhong, G. T. Barkema, Debabrata Panja, R. C. Ball

Published Dec 19, 2018• 8 citations
arXivHTMLPDF

Abstract

We study the scalar one-component two-dimensional (2D) ${\ensuremath{\phi}}^{4}$ model by computer simulations, with local Metropolis moves. The equilibrium exponents of this model are well established, e.g., for the 2D ${\ensuremath{\phi}}^{4}$ model $\ensuremath{\gamma}=1.75$ and $\ensuremath{\nu}=1$. The model has also been conjectured to belong to the Ising universality class. However, the value of the critical dynamical exponent ${z}_{c}$ is not settled. In this paper, we obtain ${z}_{c}$ for the 2D ${\ensuremath{\phi}}^{4}$ model using two independent methods: (a) by calculating the relative terminal exponential decay time $\ensuremath{\tau}$ for the correlation function $\ensuremath{\langle}\mathrm{\ensuremath{\Phi}}(t)\mathrm{\ensuremath{\Phi}}(0)\ensuremath{\rangle}$, and thereafter fitting the data as $\ensuremath{\tau}\ensuremath{\sim}{L}^{{z}_{c}}$, where $L$ is the system size, and (b) by measuring the anomalous diffusion exponent for the order parameter, viz., the mean-square displacement $\ensuremath{\langle}\mathrm{\ensuremath{\Delta}}{\mathrm{\ensuremath{\Phi}}}^{2}(t)\ensuremath{\rangle}\ensuremath{\sim}{t}^{c}$ as $c=\ensuremath{\gamma}/(\ensuremath{\nu}{z}_{c})$, and from the numerically obtained value $c\ensuremath{\approx}0.80$, we calculate ${z}_{c}$. For different values of the coupling constant $\ensuremath{\lambda}$, we report that ${z}_{c}=2.17\ifmmode\pm\else\textpm\fi{}0.03$ and ${z}_{c}=2.19\ifmmode\pm\else\textpm\fi{}0.03$ for the two methods, respectively. Our results indicate that ${z}_{c}$ is independent of $\ensuremath{\lambda}$, and is likely identical to that for the 2D Ising model. Additionally, we demonstrate that the generalized Langevin equation formulation with a memory kernel, identical to those applicable for the Ising model and polymeric systems, consistently captures the observed anomalous diffusion behavior.

DOI
10.1103/physreve.98.062128

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We study the scalar one-component two-dimensional (2D) \ensuremathϕ4{\ensuremath{\phi}}^{4}\ensuremathϕ4 model by computer simulations, with local Metropolis moves. The equilibrium exponents of this model are well established, e.g., for the 2D \ensuremathϕ4{\ensuremath{\phi}}^{4}\ensuremathϕ4 model \ensuremathγ=1.75\ensuremath{\gamma}=1.75\ensuremathγ=1.75 and \ensuremathν=1\ensuremath{\nu}=1\ensuremathν=1. The model has also been conjectured to belong to the Ising universality class. However, the value of the critical dynamical exponent zc{z}_{c}zc​ is not settled. In this paper, we obtain zc{z}_{c}zc​ for the 2D \ensuremathϕ4{\ensuremath{\phi}}^{4}\ensuremathϕ4 model using two independent methods: (a) by calculating the relative terminal exponential decay time \ensuremathτ\ensuremath{\tau}\ensuremathτ for the correlation function \ensuremath⟨\ensuremathΦ(t)\ensuremathΦ(0)\ensuremath⟩\ensuremath{\langle}\mathrm{\ensuremath{\Phi}}(t)\mathrm{\ensuremath{\Phi}}(0)\ensuremath{\rangle}\ensuremath⟨\ensuremathΦ(t)\ensuremathΦ(0)\ensuremath⟩, and thereafter fitting the data as \ensuremathτ\ensuremath∼Lzc\ensuremath{\tau}\ensuremath{\sim}{L}^{{z}_{c}}\ensuremathτ\ensuremath∼Lzc​, where LLL is the system size, and (b) by measuring the anomalous diffusion exponent for the order parameter, viz., the mean-square displacement \ensuremath⟨\ensuremathΔ\ensuremathΦ2(t)\ensuremath⟩\ensuremath∼tc\ensuremath{\langle}\mathrm{\ensuremath{\Delta}}{\mathrm{\ensuremath{\Phi}}}^{2}(t)\ensuremath{\rangle}\ensuremath{\sim}{t}^{c}\ensuremath⟨\ensuremathΔ\ensuremathΦ2(t)\ensuremath⟩\ensuremath∼tc as c=\ensuremathγ/(\ensuremathνzc)c=\ensuremath{\gamma}/(\ensuremath{\nu}{z}_{c})c=\ensuremathγ/(\ensuremathνzc​), and from the numerically obtained value c\ensuremath≈0.80c\ensuremath{\approx}0.80c\ensuremath≈0.80, we calculate zc{z}_{c}zc​. For different values of the coupling constant \ensuremathλ\ensuremath{\lambda}\ensuremathλ, we report that zc=2.17\ifmmode±\else\textpm\fi0.03{z}_{c}=2.17\ifmmode\pm\else\textpm\fi{}0.03zc​=2.17\ifmmode±\else\textpm\fi0.03 and zc=2.19\ifmmode±\else\textpm\fi0.03{z}_{c}=2.19\ifmmode\pm\else\textpm\fi{}0.03zc​=2.19\ifmmode±\else\textpm\fi0.03 for the two methods, respectively. Our results indicate that zc{z}_{c}zc​ is independent of \ensuremathλ\ensuremath{\lambda}\ensuremathλ, and is likely identical to that for the 2D Ising model. Additionally, we demonstrate that the generalized Langevin equation formulation with a memory kernel, identical to those applicable for the Ising model and polymeric systems, consistently captures the observed anomalous diffusion behavior.
We investigate the dynamical critical behavior of the two- and three-dimensional Ising models with Glauber dynamics in equilibrium. In contrast to the usual standing, we focus on the mean-squared deviation of the magnetization M,MSDMM, {\mathrm{MSD}}_{M}M,MSDM​, as a function of time, as well as on the autocorrelation function of MMM. These two functions are distinct but closely related. We find that MSDM{\mathrm{MSD}}_{M}MSDM​ features a first crossover at time ${\ensuremath{\tau}}_{1}\ensuremath{\sim}{L}^{{z}_{1}}$, from ordinary diffusion with ${\mathrm{MSD}}_{M}\ensuremath{\sim}t$, to anomalous diffusion with ${\mathrm{MSD}}_{M}\ensuremath{\sim}{t}^{\ensuremath{\alpha}}$. Purely on numerical grounds, we obtain the values z1=0.45(5){z}_{1}=0.45(5)z1​=0.45(5) and $\ensuremath{\alpha}=0.752(5)$ for the two-dimensional Ising ferromagnet. Related to this, the magnetization autocorrelation function crosses over from an exponential decay to a stretched-exponential decay. At later times, we find a second crossover at time ${\ensuremath{\tau}}_{2}\ensuremath{\sim}{L}^{{z}_{2}}$. Here, MSDM{\mathrm{MSD}}_{M}MSDM​ saturates to its late-time value $\ensuremath{\sim}{L}^{2+\ensuremath{\gamma}/\ensuremath{\nu}}$, while the autocorrelation function crosses over from stretched-exponential decay to simple exponential one. We also confirm numerically the value z2=2.1665(12){z}_{2}=2.1665(12)z2​=2.1665(12), earlier reported as the single dynamic exponent. Continuity of MSDM{\mathrm{MSD}}_{M}MSDM​ requires that $\ensuremath{\alpha}({z}_{2}\ensuremath{-}{z}_{1})=\ensuremath{\gamma}/\ensuremath{\nu}\ensuremath{-}{z}_{1}$. We speculate that z1=1/2{z}_{1}=1/2z1​=1/2 and $\ensuremath{\alpha}=3/4$, values that indeed lead to the expected z2=13/6{z}_{2}=13/6z2​=13/6 result. A complementary analysis for the three-dimensional Ising model provides the estimates ${z}_{1}=1.35(2), \ensuremath{\alpha}=0.90(2)$, and z2=2.032(3){z}_{2}=2.032(3)z2​=2.032(3). While z2{z}_{2}z2​ has attracted significant attention in the literature, we argue that for all practical purposes z1{z}_{1}z1​ is more important, as it determines the number of statistically independent measurements during a long simulation.
This dataset and code repository accompanies the article of the same title (version 5, May 2026) and provides all simulation data, source code, and raw terminal output necessary to reproduce and verify the results. Study summary We investigate the critical dynamics of the two-dimensional Ising model at the exact critical point (Tc ≈ 2.269185) using the local Metropolis Monte Carlo algorithm. The central focus is the characterization and comparison of two widely used correlation time estimators: the exponential decay criterion τ₁/e and the integrated autocorrelation time τ_int, following the convention of Sokal (1997). Simulations were performed for seven system sizes L ∈ {16, 24, 32, 48, 64, 96, 128} with up to 50 independent replicas per system size, using both Python (NumPy/PCG64) and C++ (std::mt19937, compiled with -O2) implementations. The two implementations were developed independently, enabling cross-validation of results across languages. Main findings The coefficient of variation (CV = std/mean) of τ_int converges to a physical floor of approximately 0.44–0.52 regardless of the number of replicas (8 or 50) or programming language, confirming that this floor is determined by the physics of the estimator under local Metropolis dynamics at criticality — not by statistical sampling limitations. The distribution of τ₁/e is genuinely bimodal at the critical point: individual replicas alternate between low (~200–500 MC steps) and high (~3000–5000 MC steps) correlation times for the same system size, with mean-to-median ratios of up to 2.6× and bimodal ratios (mean of upper half / mean of lower half) of up to 23.5×. This bimodality does not disappear with more replicas, confirming it is a physical property of the estimator. No detectable power-law scaling τ ~ L^z is found for L ≤ 128 in any of the six independent runs (R² < 0.14 in all log-log fits using all 7 system sizes). The accepted asymptotic value z ≈ 2.17 (Li & Sokal, 1989) implies an expected τ_int of approximately 73,600 MC steps at L = 128 — 76× larger than the observed value of 972. This quantifies the dominance of finite-size corrections in the range studied, and justifies treating the effective dynamic exponent z_eff as a continuously varying quantity rather than a fixed constant. The C++ implementation achieves speedups of 98× to 782× relative to Python depending on system size, enabling the full 50-replica run across all seven system sizes to complete in approximately 33 minutes. Cross-validation caveat Cross-validation between Python and C++ is fully valid for τ₁/e. For τ_int, the Python scripts used a fixed equilibration of n_equil = 3000 steps for all L — documentably insufficient for L ≥ 24 where τ_int > 600 (n_equil/τ < 3–5τ). This generates systematic underestimation of up to 86% at L = 64. The valid cross-validation for τ_int is between the C++-8 and C++-50 runs, where CV agreement (~0.52 in both) confirms the physical floor is robust. Connection to the EIE/ZIE framework This study provides the empirical foundation for the EIE/ZIE adaptive meta-control framework (Patent BR 10 2025 024459 4), justifying three architectural decisions: (1) τ_int as the preferred estimator of the correlation time τ_corr; (2) the effective dynamic exponent z_eff treated as a continuously estimated dynamic variable rather than a fixed constant; and (3) the dynamic freezing time τ_freeze introduced as an observable independent of τ_corr, motivated by the bimodal behavior of τ₁/e which reveals that the system alternates between regimes of high and low correlation persistence — a phenomenon invisible to conventional autocorrelation-based diagnostics. Repository contents Cpp/ — C++ source code (3 scripts) and simulation data (6 CSV files + raw terminal output) Python/ — Python source code (2 scripts) and simulation data (4 CSV files + raw screen output) Paper/ — Article in English (PDF) and Portuguese (PDF) README.md — Full repository documentation including file descriptions, column definitions, reproduction instructions, and physical parameters LICENSE.txt — CC BY 4.0 for data; MIT for code; patent notice Joemerson da Silva Lima — Independent Researcher, Cuiabá, Mato Grosso, Brazil joemerson93lima@gmail.com Patent BR 10 2025 024459 4
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}.
Values of dynamic critical exponents are numerically estimated for various models with the nonequilibrium relaxation method to test the dynamic universality hypothesis. The dynamics used here are single-spin update with Metropolis-type transition probabities. The estimated values of nonequilibrium relaxation exponent of magnetization λ m (=β/ z ν) of Ising models on bcc and fcc lattices are estimated to be 0.251(3) and 0.252(3), respectively, which are consistent with the value of the model on simple-cubic lattice, 0.250(2). The dynamic critical exponents of three-states Potts models on square, honeycomb and triangular lattices are also estimated to be 2.193(5), 2.198(4), and 2.199(3), respectively. They are consistent within the error bars. It is also confirmed that Ising models with regularly modulated coupling constants on square lattice have the same dynamic critical exponents with the uniformly ferromagnetic Ising model.
Via extensive Monte Carlo simulations along with systematic analyses of corrections to scaling, we estimate the order parameter critical exponent β\betaβ of the directed Ising universality class. The value of β\betaβ was conjectured to be 1314\frac{13}{14}1413​ and Monte Carlo simulation studies in the literature have repeatedly reproduced the consistent values to the conjecture. In this paper, however, we find that the conjectured value of β\betaβ is not correct. Rather, we obtain β\betaβ = 1.020(5) from different models belonging to the directed Ising class.