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Journal of Physics C Solid State Physics

A computer simulation on the dilute Ising ferromagnet

Takeshi Takahashi, M Shimizu

Published Apr 20, 1981• 3 citations
Publisher

Abstract

A computer simulation on the dilute Ising ferromagnet, where the magnetic atoms are distributed randomly in a two-dimensional matrix of non-magnetic atoms, is carried out. The interaction between the magnetic atoms is assumed to be long range. The numerical results for the temperature dependence of the magnetic specific heat show broad and round peaks. This rounding is too large to be attributed to the small size of the system. It is concluded that most of the rounding can be attributed to the spatial distribution of the local field due to the random distribution of the magnetic atoms. On the other hand, the results for the magnetic susceptibility show rather sharp peaks.

DOI
10.1088/0022-3719/14/11/025

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A computer simulation on the dilute Ising ferromagnet, where the magnetic atoms are distributed randomly in a two-dimensional matrix of non-magnetic atoms, is carried out. The interaction between the magnetic atoms is assumed to be long range. The numerical results for the temperature dependence of the magnetic specific heat show broad and round peaks. This rounding is too large to be attributed to the small size of the system. It is concluded that most of the rounding can be attributed to the spatial distribution of the local field due to the random distribution of the magnetic atoms. On the other hand, the results for the magnetic susceptibility show rather sharp peaks.
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.
Using the Monte Carlo technique the authors have studied a two-dimensional Ising model with quenched non-magnetic impurities. In particular they calculated the temperature dependence of the order parameter, of the zero-field susceptibility and of the specific heat. These results indicate that a second-order phase transition is preserved for p>or=0.9 where 1-p denotes the concentration of randomly distributed non-magnetic sites. Moreover, they yield beta approximately=1/8 and gamma approximately=7/4, indicating that these exponents equal the corresponding ones for the pure Ising model.
PACS. 05.50 – Lattice theory and statistics; Ising problems PACS. 75.10 – General theory and models of magnetic ordering We describe results of the cluster algorithm Special Purpose Processor simulations of the 2D Ising model with impurity bonds. Use of large lattices, with the number of spins up to 10 6, permitted to define critical region of temperatures, where both finite size corrections and corrections to scaling are small. High accuracy data unambiguously show increase of magnetization and magnetic susceptibility effective exponents β and γ, caused by impurities. The M and χ singularities became more sharp, while the specific heat singularity is smoothed. The specific heat is found to be in a good agreement with Dotsenko-Dotsenko theoretical predictions in the The problem of inhomogenities influence on phase transitions has a long history. The first model with some kind of impurities to be studied, is, due to its simplicity, the Ising model. Effect of randomness on the critical behaviour of different Ising
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.
The two-dimensional random bond Ising model with two distinct ferromagnetic nearest-neighbour couplings, J and J ', distributed randomly over the lattice with concentration 1/2 is studied for various ratios J '/ J at criticality. The critical temperature is exactly known from self-duality. The size-dependent specific heat shows a crossover from the perfect Ising to the randomness-dominated critical behaviour of doubly logarithmic form. In the latter region, our estimates for the critical exponents of the size-dependent magnetization and susceptibility are in very good agreement with analytic results suggesting β/ν = 1/8 and γ/ν = 7/4.
We have performed Monte Carlo calculations of the thermodynamic properties of the bond-dilute Ising model on the square lattice. The results are obtained for the magnetization, internal energy, specific heat, and susceptibility for several concentrations of missing bonds. The dependence of the Curie temperature on dilution is in good agreement with the results obtained by other methods. The phase transition is found to remain sharp with the same value of the susceptibility exponent as for the pure Ising model.