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Journal of Applied Physics

First-Order Phase Transition in a Rigorously Solvable Magnetic System

Martin Blume, Richard E. Watson

Published Mar 1, 1967• 14 citations
Publisher

Abstract

We have considered the paramagnetic—ferromagnetic transition for the Ising system with ``infinite range'' interactions (i.e., each spin interacts equally with all other spins) and in the presence of a single-ion anisotropy term: H=−2jΣ lim i>i SizSjz+ΔΣ lim i Siz2−gμHΣ lim iSiz. Kittel and Shore1 have shown that this system (in the absence of the term in Δ) is exactly soluble, and leads rigourously, in the limit of an infinite number of particles, to the equations of the molecular-field approximation, and to a second-order phase transition. The single-ion term can cause the transition to become first order2,3 in that there is a discontinuity in the magnetization as a function of temperature. We have solved for the relevant thermodynamic parameters as a function of temperature for the case of ions with spin one. The development of the first-order transition as a function of the number of particles will be discussed. This work will be submitted for publication in the Physical Review.

DOI
10.1063/1.1709712

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The effects of locally random magnetic fields are considered in a nonequilibrium Ising model defined on a square lattice with nearest-neighbors interactions. In order to generate the random magnetic fields, we have considered random variables {h} that change randomly with time according to a double-gaussian probability distribution, which consists of two single gaussian distributions, centered at +ho and −ho, with the same width σ. This distribution is very general, and can recover in appropriate limits the bimodal distribution (σ → 0) and the single gaussian one (ho = 0). We performed Monte Carlo simulations in lattices with linear sizes in the range L = 32 − 512. The system exhibits ferromagnetic and paramagnetic steady states. Our results suggest the occurence of first-order phase transitions between the above-mentioned phases at low temperatures and large random-field intensities ho, for some small values of the width σ. By means of finite size scaling, we estimate the critical exponents in the low-field region, where we have continuous phase transitions. In addition, we show a sketch of the phase diagram of the model for some values of σ. PACS numbers: 05.10.Ln, 05.50.+q, 64.60.De, 75.10Hk, 75.40.MgFirst-order phase transition in a 2D random-field Ising model with conflicting dynamics2 1.

•Nov 20, 2012•10 citations
We have considered the paramagnetic—ferromagnetic transition for the Ising system with ``infinite range'' interactions (i.e., each spin interacts equally with all other spins) and in the presence of a single-ion anisotropy term: H=−2jΣ lim i>i SizSjz+ΔΣ lim i Siz2−gμHΣ lim iSiz. Kittel and Shore1 have shown that this system (in the absence of the term in Δ) is exactly soluble, and leads rigourously, in the limit of an infinite number of particles, to the equations of the molecular-field approximation, and to a second-order phase transition. The single-ion term can cause the transition to become first order2,3 in that there is a discontinuity in the magnetization as a function of temperature. We have solved for the relevant thermodynamic parameters as a function of temperature for the case of ions with spin one. The development of the first-order transition as a function of the number of particles will be discussed. This work will be submitted for publication in the Physical Review.
In Chapter 3, we examined two examples of phase transitions, the paramagnetic–ferromagnetic transition (Section 3.1.4) and the liquid–vapour transition (Section 3.5.4). In the latter case, thermodynamic functions such as entropy or specific volume are discontinuous at the transition. When, for example, one varies the temperature T at constant pressure, the transition takes place at a temperature T c where the Gibbs potentials of the two phases are equal, but the two phases coexist at T c and are either stable or metastable in the vicinity of T c . Each of the phases carries with it its own entropy, specific volume, etc., which are in general different at T = T c , hence the discontinuities. Such a phase transition is called a first order phase transition . The picture is quite different in the paramagnetic–ferromagnetic phase transition: the thermodynamic quantities are continuous at the transition, the transition is not linked to the crossing of two thermodynamic potentials and one never observes metastability. Such a transition is called a second order, or continuous, phase transition . The transition temperature is called the critical temperature and is denoted by T c . The new and remarkable feature which we shall encounter in continuous phase transitions is the existence of cooperative phenomena. To be specific, think of a spin system, for example an Ising or a Heisenberg model, where the interactions between spins are limited to nearest neighbours. More generally we shall consider short range interactions , which decrease rapidly as a function of the distance between spins, and we exclude from our study all long range interactions.
The complete thermodynamics of a model hamiltonian with general-spin infinite range Ising 2-point, 4-point,... interaction terms is found exactly in two different statistical mechanical ensembles which prove to be essentially equivalent. The coefficients of the above terms may be chosen to give critical behaviour which is largely consistent with the idea of scaling and which has delta equal to any odd positive integer and gamma =1; alternatively a first order phase transition may be obtained, a finite spontaneous magnetization appearing as the temperature is decreased through that of the transition. The Heisenberg case is also considered and found to be closely similar to the Ising case.
The ferromagnetic transition is considered in detail for the Heisenberg Hamiltonian with all spins coupled equally. For this special model the statistical problem is solved exactly. The transition develops very slowly as the number NNN of particles in the system is increased; the spin order at the nominal Curie temperature is proportional to ${N}^{\ensuremath{-}\frac{1}{4}}$. The order in the most probable state differs from the mean value of the order, except for $N\ensuremath{\rightarrow}\ensuremath{\infty}$. The connections are studied with the molecular-field, spin-wave, and self-consistent approximations. The failure of the self-consistent approximation is relatively severe as compared with the nearest-neighbor problem. The antiferromagnetic ground state is very close in energy to the N\'eel state.
The infinite range Ising model is usually investigated in the weak coupling limit. Here the model is solved with ferromagnetic coupling at fixed and finite strength. Exact analytical expressions are found for the thermodynamic potentials as functions of enthalpy and external field. These results differ from the potentials  for the weak coupling limit. The model shows a temperature driven first order phase transition from a paramagnetic phase at high temperatures into a low temperature phase from which thermal fluctuations are absent.
The effects of locally random magnetic fields are considered in a nonequilibrium Ising model defined on a square lattice with nearest-neighbor interactions. In order to generate the random magnetic fields, we have considered random variables { h } that change randomly with time according to a double-Gaussian probability distribution, which consists of two single-Gaussian distributions, centered at + h o and − h o , with the same width σ. This distribution is very general and can recover in appropriate limits the bimodal distribution ( ) and the single-Gaussian one ( h o = 0). We performed Monte Carlo simulations in lattices with linear sizes in the range L = 32–512. The system exhibits ferromagnetic and paramagnetic steady states. Our results suggest the occurrence of first-order phase transitions between the above-mentioned phases at low temperatures and large random-field intensities h o for some small values of the width σ. By means of finite size scaling, we estimate the critical exponents in the low-field region, where we have continuous phase transitions. In addition, we show a sketch of the phase diagram of the model for some values of σ.
The effects of locally random magnetic fields are considered in a nonequilibrium Ising model defined on a square lattice with nearest-neighbors interactions. In order to generate the random magnetic fields, we have considered random variables {h} that change randomly with time according to a double-gaussian probability distribution, which consists of two single gaussian distributions, centered at +ho and −ho, with the same width σ. This distribution is very general, and can recover in appropriate limits the bimodal distribution (σ → 0) and the single gaussian one (ho = 0). We performed Monte Carlo simulations in lattices with linear sizes in the range L = 32 − 512. The system exhibits ferromagnetic and paramagnetic steady states. Our results suggest the occurence of first-order phase transitions between the above-mentioned phases at low temperatures and large random-field intensities ho, for some small values of the width σ. By means of finite size scaling, we estimate the critical exponents in the low-field region, where we have continuous phase transitions. In addition, we show a sketch of the phase diagram of the model for some values of σ. PACS numbers: 05.10.Ln, 05.50.+q, 64.60.De, 75.10Hk, 75.40.MgFirst-order phase transition in a 2D random-field Ising model with conflicting dynamics2 1.