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Investigation of Phase Transition in 2D Ising Model

Subeksha Bhattarai

Published Aug 22, 2025• 0 citations
Publisher

Abstract

2D Ising model is simple yet powerful model to describe ferromagnetism. In this project, the phase transition and critical behaviour of the 2D Ising model using metropolis algorithm was investigated. The magnetisation vs temperature graph for different lattice size L = 5, L= 10, L = 15, L=20, L=30, was analyzed and compared. The analysis demonstrated a trend where the critical point becomes more sharply defined on increasing lattice size. The obtained value of critical temperature from graph closely approximates to the theoretical value. Hence the simulation successfully demonstrated the phase transition of magnetic material from ordered to disordered state. The significance of Ising model further extends far beyond its application in understanding magnetism. This study provides a deeper understanding of phase transition and critical behaviour.

DOI
10.20944/preprints202508.1548.v2

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2D Ising model is simple yet powerful model to describe ferromagnetism. In this project, the phase transition and critical behaviour of the 2D Ising model using metropolis algorithm was investigated. The magnetisation vs temperature graph for different lattice size L = 5, L= 10, L = 15, L=20, L=30, was analyzed and compared. The analysis demonstrated a trend where the critical point becomes more sharply defined on increasing lattice size. The obtained value of critical temperature from graph closely approximates to the theoretical value. Hence the simulation successfully demonstrated the phase transition of magnetic material from ordered to disordered state. The significance of Ising model further extends far beyond its application in understanding magnetism. This study provides a deeper understanding of phase transition and critical behaviour.
2D Ising model is simple yet powerful model to describe ferromagnetism. In this project, the phase transition and critical behaviour of the 2D Ising model using metropolis algorithm was investigated. The magnetisation vs temperature graph for different lattice size L = 5, L= 10, L = 15, L=20, L=30, was analyzed and compared. The analysis demonstrated a trend where the critical point becomes more sharply defined on increasing lattice size. The obtained value of critical temperature from graph closely approximates to the theoretical value. Hence the simulation successfully demonstrated the phase transition of magnetic material from ordered to disordered state. The significance of Ising model further extends far beyond its application in understanding magnetism. This study though simple but it provides a deeper understanding of phase transition and critical behaviour.
The 2D Ising model is a simple yet powerful model to describe ferromag- netism. In this work, the phase transition and critical behavior of the 2D Ising model using the metropolis algorithm were investigated. Magnetization was an- alyzed as a function of temperature for lattice sizes L = 5, 10, 15, 20, 30. The analysis demonstrated a trend where the critical point becomes more sharply de- fined on increasing the lattice size. The value of the critical temperature obtained from the graph closely approximates the theoretical value. Hence, the simulation successfully demonstrated the phase transition of magnetic material from ordered to disordered state. The significance of Ising model further extends far beyond its application in understanding magnetism. This study provides a deeper under- standing of phase transition and critical behavior.
Many problems in statistical physics resist exact solution once the number of interacting particles becomes large, and the two-dimensional Ising model is a classic case where numerical simulation fills that gap. This paper reports a Monte Carlo study of the two-dimensional Ising model using the Metropolis algorithm, undertaken to examine how magnetization, internal energy, specific heat, and magnetic susceptibility change with temperature. Spin configurations were generated on a square lattice under periodic boundary conditions, with each proposed spin flip accepted or rejected according to the Boltzmann weight of the resulting energy change. The simulation reproduces the expected transition from an ordered ferromagnetic state at low temperature to a disordered paramagnetic state at high temperature, with magnetization falling sharply and both specific heat and susceptibility peaking near a critical temperature of Tc ≈ 2.3–2.4 (in units of J/kB), consistent with the exact value obtained analytically for this system. These results confirm that the Metropolis algorithm, despite its simplicity, remains a dependable tool for locating second-order phase transitions and extracting thermodynamic quantities in lattice spin systems, and the same computational framework can be adapted to more elaborate models used in condensed matter and materials research.
To demonstrate the application of computational physics in modern statistical physics, this paper investigates the critical behavior of a finite-size ferromagnetic system based on the two-dimensional square-lattice Ising model.A nearest-neighbor Ising Hamiltonian is constructed, and periodic boundary conditions are adopted to reduce finite-boundary effects.The Metropolis Monte Carlo algorithm is used to sample spin configurations at different temperatures, while energy, magnetization, specific heat, magnetic susceptibility and Binder cumulant are calculated to describe the transition from an ordered state to a disordered state.The simulation results show that the system exhibits clear long-range order at low temperatures, where the order parameter remains high.As temperature increases, thermal fluctuations become stronger, magnetization decreases rapidly, and the system gradually evolves into a disordered state.The specific heat curve shows a pronounced peak at approximately (T = 2.269), while the susceptibility peak shifts slightly toward the high-temperature side due to finite-size effects and the discrete temperature grid.Overall, the simulated critical region is consistent with the theoretical critical temperature of the two-dimensional square-lattice Ising model.Spin-configuration analysis further reveals enhanced fluctuations and domain reconstruction near the critical region.In addition, a thermalization criterion and an extended sampling test for L = 48 near the critical region are added to examine the influence of finite sampling and equilibration.These results indicate that Monte Carlo simulation of the two-dimensional Ising model can effectively reproduce continuous phase transition, critical fluctuation and finite-size effects, providing a basic computational case for understanding critical phenomena in complex systems and data-driven physical analysis.
We simulate the Ising Model with the Monte Carlo method and use the algorithms of Metropolis to update the distribution of spins. We found that, in the specific case of the three-dimensional Ising Model, methods of Metropolis are efficient. Studying the system near the point of phase transition, we observe that the magnetization goes to zero. In our simulations we analyzed the behavior of the magnetization and magnetic susceptibility to verify the phase transition in a paramagnetic to ferromagnetic material. The behavior of the magnetization and of the magnetic susceptibility as a function of the temperature suggest a phase transition around KT/J ≈ 4.5 and was evidenced the problem of finite size of the lattice to work with large lattice.
Phase transitions are widespread physical phenomena in nature and the phase transition of ferromagnetic substances which can be described by Ising model is an important part of physics. This paper discusses the phase transition of the low-dimensional Ising model. For one-dimensional Ising model, the exact solution is given and no phase transition is found when there is no external magnetic field. For two-dimensional Ising model, Monte Carlo method along with importance sampling, careful balance conditions and the Metropolis algorithm is introduced to numerically simulate it. The trends of order parameters like magnetization, specific heat and magnetic susceptibility are analyzed and the mutation of some order parameters is found around critical temperature. The influences of external magnetic field and the scale of system are discussed and critical temperature of phase transition is explored by several methods, which is eventually found around 2.269 ( kBT/J). Lastly, compared with theoretical values, the disadvantages are pointed out and optimization suggestions are raised for improvement.