We perform a comprehensive numerical study of how variations in microscopic coupling strength affect critical behavior in the two-dimensional Ising model. Using extensive Monte Carlo simulations based on the Wolff cluster algorithm, we investigate the clean Ising model on a square lattice for interaction strengths in the range 0.80 ≤ J ≤ 1.20. While the exact solution at J = 1 has long established the model’s critical properties, a systematic numerical assessment of universality under continuous coupling rescaling has not been performed. Here, we address this gap by analyzing systems of linear size L = 20 − 500, allowing for high-precision finite-size scaling of thermodynamic observables. For each coupling, the critical temperature T_C(J) is determined via finite-size extrapolation of the susceptibility maxima, yielding results in quantitative agreement with Onsager’s exact prediction. At the extrapolated critical points, we extract the critical exponents ν, β, γ, and α through detailed scaling analyses. These exponents remain consistent with the exact Ising benchmarks across the full range of interaction strengths, confirming that the universal aspects of the phase transition are preserved. Our findings demonstrate that modifying the microscopic coupling strength rescales the critical temperature without altering the asymptotic critical behavior, thereby providing a clear and quantitative confirmation of the robustness of the two-dimensional Ising universality class.
We perform a comprehensive numerical study of how variations in microscopic coupling strength affect critical behavior in the two-dimensional Ising model. Using extensive Monte Carlo simulations based on the Wolff cluster algorithm, we investigate the clean Ising model on a square lattice for interaction strengths in the range 0.80 ≤ J ≤ 1.20. While the exact solution at J = 1 has long established the model’s critical properties, a systematic numerical assessment of universality under continuous coupling rescaling has not been performed. Here, we address this gap by analyzing systems of linear size L = 20 − 500, allowing for high-precision finite-size scaling of thermodynamic observables. For each coupling, the critical temperature T_C(J) is determined via finite-size extrapolation of the susceptibility maxima, yielding results in quantitative agreement with Onsager’s exact prediction. At the extrapolated critical points, we extract the critical exponents ν, β, γ, and α through detailed scaling analyses. These exponents remain consistent with the exact Ising benchmarks across the full range of interaction strengths, confirming that the universal aspects of the phase transition are preserved. Our findings demonstrate that modifying the microscopic coupling strength rescales the critical temperature without altering the asymptotic critical behavior, thereby providing a clear and quantitative confirmation of the robustness of the two-dimensional Ising universality class.
On Critical Temperature and Finite Size Scaling of Continuous Spin 2d Ising Model
In this paper, we have studied the critical temperature Tc of continuous spin 2d square-lattice Ising model using Monte-Carlo simulation. We have considered spins s in a bounded interval, where s∈[−1,+1] in square-lattice configuration with periodic boundary condition. We have observed that the critical temperature Tc is approximately 0.925, showing a clear second order phase transition. Considering finite size scaling, we have also obtained the critical exponents associated with susceptibility, specific heat, magnetization and we find that these values are in good agreement with the corresponding values obtained for the standard 2d Ising universality class.
Abstract We investigated the Ising model on a square lattice with ferro and antiferromagnetic interactions modulated by the quasiperiodic Octonacci sequence in both directions of the lattice. We have applied the replica exchange Monte Carlo (parallel tempering) technique to calculate the thermodynamic quantities of the system. We obtained the order parameter, the associated magnetic susceptibility ( χ ) and the specific heat ( c ) in order to characterize the universality class of the phase transition. Also, we use the finite size scaling method to obtain the critical temperature of the system and the critical exponents β , γ and ν . In the low temperature limit we have obtained a continuous transition with critical temperature around T c ≈ 1.413 . The system obeys the Ising universality class with logarithmic corrections. We found estimatives for the correction exponents β ^ , γ ^ and λ ^ by using the finite size scaling technique.
We address the study of quasiperiodic interactions on a square lattice by using an Ising model with ferromagnetic and antiferromagnetic exchange interactions following a quasiperiodic Fibonacci sequence in both directions of a square lattice. We applied the Monte Carlo method, together with the Metropolis algorithm, to calculate the thermodynamic quantities of the system. We obtained the Edwards-Anderson order parameter q_{EA}, the magnetic susceptibility χ, and the specific heat c in order to characterize the universality class of the phase transition. We also use the finite size scaling method to obtain the critical temperature of the system and the critical exponents β,γ, and ν. In the low-temperature limit we obtained a spin-glass phase with critical temperature around T_{c}≈2.274, and the critical exponents β,γ, and ν, indicating that the quasiperiodic order induces a change in the universality class of the system. Also, we discovered a spin-glass ordering in a two-dimensional system which is rare and, as far as we know, the unique example is an under-frustrated Ising model.
In this work, we simulated a quenched random mixed-spin Ising model on the square lattice. The model system consists of two different particles with spins σ = 1/2 (states ±1/2) and S = 1 (states ±1, 0). These particles are randomly distributed on the lattice, and we considered only nearest-neighbor interactions. This model can represent a random magnetic binary alloy AxB1-x, obtained from the high-temperature quenching of a liquid mixture. We performed Monte Carlo simulations for several lattice sizes and temperatures, and we found its critical temperature through the reduced fourth-order cumulant. We also determined the magnetization, the susceptibility, and the specific heat as a function of temperature. We used finite-size scaling arguments to estimate the critical exponents β, γ, and ν of the model. We showed that the quenched model is in the same universality class of the two-dimensional pure Ising model. We also investigated the sample to sample fluctuations that occur in the values of the thermodynamic quantities in order to describe the self-averaging properties at the critical point.
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.
We use a new updated algorithm scheme to investigate the critical behaviour of the two-dimensional ferromagnetic Ising model on a triangular lattice with the nearest neighbour interactions. The transition is examined by generating accurate data for lattices with L = 8, 10, 12, 15, 20, 25, 30, 40 and 50. The updated spin algorithm we employ has the advantages of both a Metropolis algorithm and a single-update method. Our study indicates that the transition is continuous at T c = 3.6403(2). A convincing finite-size scaling analysis of the model yields ν = 0.9995(21), β/ν = 0.12400(17), γ/ν = 1.75223(22), γ'/ν = 1.7555(22), α/ν = 0.00077(420) (scaling) and α/ν = 0.0010(42) (hyperscaling). The present scheme yields more accurate estimates for all the critical exponents than the Monte Carlo method, and our estimates are shown to be in excellent agreement with their predicted values.