Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Critical Temperature and Finite Size Scaling of Continuous Spin $2d$ Ising Model

Swapna Mahapatra, Rudra Majhi, Jahangir Mohammed, Subhashree Mohanty, Priyanka Priyadarshini Pruseth, Masoom Singh

TL;DR

This study investigates a bounded continuous spin $2d$ Ising model with spins in $[-1, +1]$, no external field, and no on-site potential, using Metropolis Monte Carlo on square lattices with periodic boundaries. It identifies a finite-temperature second-order phase transition at $T_c \approx 0.925$ and employs finite-size scaling to extract critical exponents, finding values $\nu=1$, $\gamma \approx 1.73$, and $\beta \approx 0.126$ with $C_v$ scaling as $\log L$, all consistent with the $2d$ Ising universality class. The work demonstrates that bounded continuous spins without an explicit confining potential can exhibit Ising-like criticality, and it highlights the role of finite-size effects in shaping observed critical behavior. These findings provide a baseline for studying continuous-spin systems and motivate future explorations of boundary conditions, on-site potentials, external fields, and alternative computational frameworks such as cellular automata.

Abstract

In this paper, we have studied the critical temperature $T_c$ of continuous spin $2d$ square-lattice Ising model using Monte-Carlo simulation. We have considered spins $s$ in a bounded interval, where $s \in [-1,+1]$ in square-lattice configuration with periodic boundary condition. We have observed that the critical temperature $T_c$ 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.

On Critical Temperature and Finite Size Scaling of Continuous Spin $2d$ Ising Model

TL;DR

This study investigates a bounded continuous spin Ising model with spins in , no external field, and no on-site potential, using Metropolis Monte Carlo on square lattices with periodic boundaries. It identifies a finite-temperature second-order phase transition at and employs finite-size scaling to extract critical exponents, finding values , , and with scaling as , all consistent with the Ising universality class. The work demonstrates that bounded continuous spins without an explicit confining potential can exhibit Ising-like criticality, and it highlights the role of finite-size effects in shaping observed critical behavior. These findings provide a baseline for studying continuous-spin systems and motivate future explorations of boundary conditions, on-site potentials, external fields, and alternative computational frameworks such as cellular automata.

Abstract

In this paper, we have studied the critical temperature of continuous spin square-lattice Ising model using Monte-Carlo simulation. We have considered spins in a bounded interval, where in square-lattice configuration with periodic boundary condition. We have observed that the critical temperature is approximately , 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 Ising universality class.
Paper Structure (7 sections, 4 equations, 3 figures, 1 table)

This paper contains 7 sections, 4 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Phase Transition and Critical Temperature
  3. Numerical Simulation
  4. Results and Discussion
  5. Critical Temperature
  6. Finite-size Scaling
  7. Summary and perspective

Figures (3)

  • Figure 1: Temperature evolution of susceptibility $\chi$, specific heat $C_v$, magnetization $M$, and energy per site $E$ for a $64 \times 64$ lattice, showing the characteristic signatures of the ferromagnetic-paramagnetic phase transition.
  • Figure 2: (a): $|M|$ vs $T$ , (b): $C_v$ vs $T$ , and (c): $\chi$ vs $T$ for lattice sizes $L = 8, 16, 32, 64, 128$.
  • Figure 3: Critical exponents characterizing the ferromagnetic phase transition under periodic boundary condition. Values of $\gamma$ (susceptibility exponent), $\beta$ (order parameter exponent), and $C_0$ (specific heat exponent) are obtained from finite-size scaling analysis.