The Ising Model: Brief Introduction and Its Application (original) (raw)
Related papers
The magnetization of the 3D Ising model
Journal of Physics A: Mathematical and General, 1996
We present highly accurate Monte Carlo results for simple cubic Ising lattices containing up to 256 3 spins. These results were obtained by means of the Cluster Processor, a newly built special-purpose computer for the Wolff cluster simulation of the 3D Ising model. We find that the magnetization M(t) is perfectly described by M(t) = (a 0 − a 1 t θ − a 2 t)t β , where t = (T c − T )/T c , in a wide temperature range 0.0005 < t < 0.26. If there exist corrections to scaling with higher powers of t, they are very small. The magnetization exponent is determined as β = 0.3269 (6). An analysis of the magnetization distribution near criticality yields a new determination of the critical point: K c = J/k B T c = 0.2216544, with a standard deviation of 3·10 −7 .
Physica A: Statistical Mechanics and its Applications, 2005
Magnetic properties of a mixed spin-1/2 and spin-3/2 Ising model on honeycomb lattice are investigated within the framework of an exact star-triangle mapping transformation. The particular attention is focused on the effect of uniaxial and biaxial crystal-field potentials that basically influence a magnetic behavior of the spin-3/2 atoms. Our results for basic thermodynamic quantities as well as dynamical time-dependent autocorrelation function indicate a spin tunneling between |± 3 2 and |∓ 1 2 states in two different magnetically ordered phases OP 1 and OP 2 , respectively.
Theory and Simulation of the Ising Model
2021
We have provided a concise introduction to the Ising model as one of the most important models in statistical mechanics and in studying the phenomenon of phase transition. The required theoretical background and derivation of the Hamiltonian of the model have also been presented. We finally have discussed the computational method and details to numerically solve the twoand threedimensional Ising problems using Monte Carlo simulations. The related computer codes in both Python and Fortran, as well as a simulation trick to visualize the spin lattice, have also been provided.
Kinetic phase transition in the mixed-spin Ising model
Brazilian Journal of Physics, 2004
In this work we studied a ferromagnetic mixed-spin Ising model including a single ion crystal-field term. The model system consists of two interpenetrating sublattices with spins σ = 1/2 and S = 1. The spins σ = 1/2 occupy the sites of one sublattice, their nearest-neighbours are spins S on the other sublattice, and vice versa. The system is in contact with a heat bath, the spins flipping according to the Metropolis transition rate and, at the same time, subject to an external flow of energy, which is simulated by a two-spin flip process. The model is studied via the dynamical pair approximation and through Monte Carlo simulations. We have determined the phase diagram of the model in the plane crystal-field D versus competition parameter p. The parameter p accounts for the competition between the one-and two-spin flip processes. In the pair approximation, the phase diagram, at high temperatures, present three phases separated by two transition lines: a continuous transition line between the ferromagnetic and paramagnetic phases, and a first-order transition line between the paramagnetic and antiferromagnetic phases. However, Monte Carlo simulations predict the same topology for the phase diagram as the pair approximation, but all the transition lines are continuous for any value of the temperature.
Critical behavior of the classical spin-1 Ising model for magnetic systems
AIP Advances, 2022
In this work, the critical properties of the classical spin-1 Ising Hamiltonian applied to magnetic systems characterized by the first-neighbors biquadratic exchange, the anisotropy and the external magnetic field contributions are theoretically investigated. The first-neighbors bilinear exchange interaction is set equal to zero. For magnetic systems the bicubic exchange interaction must be set equal to zero as it would break the time-reversal invariance of the exchange Hamiltonian. To determine the critical behavior, the spin-1 Ising Hamiltonian is mapped onto the spin-1/2 Ising Hamiltonian by using the Griffith’s variable transformation. The critical surface of a 2D square magnetic lattice is determined in the parameter space as a function of the magnetic parameters and the phase transition occurring across it is quantitatively discussed by calculating, for each spin, the free energy and the magnetization. The free energy of the 2D square magnetic lattice, described via the three-...
A simulation of the ising model
The Ising Model provides an entirely new understanding of how phase transitions in various systems take place and gives us a bet- ter idea of the magnetic behavior/properties of certain systems. This project aims at analyzing phase transitions and magnetic properties of some systems through Monte Carlo simulations based on the 2D Ising Model.
Quantum phase transitions in the transverse 1-D Ising model with four-spin interactions
In this work we investigate phase transitions in the transverse Ising model with four-spin interactions, induced by quantum fluctuations. The model is relevant to the physics of poly(vinylidenefluoride-trifluoroethylene)[P(VDF-TrFE)]. We calculate the ground state and the first excited state energies of the system using Lanczos method. Our calculations are performed using rings up to 20 spins. Finite size scaling is applied to the energy gap to obtain the boundary region where a ferromagnetic to paramagnetic transition takes place, as well as the corresponding critical exponents. A new degenerate phase region is found. The first-order transition boundary between this new phase and the paramagnetic phase is determined by analyzing the behavior of the transverse spin magnetization as the system moves from one region to the other.
Influence of the Pair Correlations on the Phase Transition in an Ising Lattice
Physical Review
We propose a method to incorporate pair correlations of an Ising lattice system in molecular field theory to higher orders. The theory is applied to ferromagnetism, where a system of equations is obtained for the response function (here the susceptibility). We obtain important corrections on the Weiss model and the spherical model. This is illustrated by an explicit calculation of the critical temperature and the spurious phase, appearing for T > T., near the critical point in the case of a simple cubic lattice.
Recent Results of Multimagnetical Simulations of the Ising Model
International Journal of Modern Physics C, 1992
To investigate order-order interfaces, we perform multimagnetical Monte Carlo simulations of the 2D and 3D Ising model. Stringent tests of the numerical methods are performed by reproducing with high precision exact 2D results. In the physically more interesting 3D case we estimate the amplitude [Formula: see text] of the critical interfacial tension.
Physica A-statistical Mechanics and Its Applications - PHYSICA A, 2004
The transverse spin-1 Ising model with a longitudinal crystal field presents a rich variety of critical phenomena. Using the effective field theory with a probability distribution technique that accounts for the self-spin correlation functions, the phase diagrams and the tricritical points are investigated for the square lattice (N=4). The results show that the tricritical points exist for certain values of the strength of the transverse field and the strength of the longitudinal crystal field. The longitudinal and the transverse magnetizations as well as the longitudinal and the transverse quadrupolar moments are also examined. These quantities as functions of the temperature, the strength of the longitudinal crystal field and the strength of the applied transverse field are calculated numerically and some interesting results are obtained.