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arXiv:0705.3890 · Physical Review B

Phase transition in compressible Ising systems at fixed volume

Akira Ōnuki, Akihiko Minami

Published Nov 15, 2007• 5 citations
arXivHTMLPDF

Abstract

Using a Ginzburg-Landau model, we study the phase transition behavior of compressible Ising systems at constant volume by varying the temperature $T$ and the applied magnetic field $h$. We show that two phases can coexist macroscopically in equilibrium within a closed region in the $T\text{\ensuremath{-}}h$ plane. Its occurrence is favored near tricriticality. We find a field-induced critical point, where the correlation length diverges, the difference of the coexisting two phases and the surface tension vanish, but the isothermal magnetic susceptibility does not diverge in the mean field theory. We also investigate phase ordering numerically.

DOI
10.1103/physrevb.76.174427

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Using a Ginzburg-Landau model, we study the phase transition behavior of compressible Ising systems at constant volume by varying the temperature TTT and the applied magnetic field hhh. We show that two phases can coexist macroscopically in equilibrium within a closed region in the T\ensuremath-hT\text{\ensuremath{-}}hT\ensuremath-h plane. Its occurrence is favored near tricriticality. We find a field-induced critical point, where the correlation length diverges, the difference of the coexisting two phases and the surface tension vanish, but the isothermal magnetic susceptibility does not diverge in the mean field theory. We also investigate phase ordering numerically.
I consider the dynamics of a soft-spin Ising magnet, subject to the time-varying external magnetic field h(t)=h cos\ensuremath{\Omega}t. The system is modeled with a time-dependent Ginzburg-Landau equation and is studied at a mean-field level. The time-averaged magnetization (scrM) acts as the order parameter and divides the temperature-h plane into two phases. In contrast to a previous result that predicted a dynamical tricritical point separating a line of continuous and discontinuous transitions in scrM, I find that the transition is always continuous. The previous work utilized Glauber dynamics which, it is here argued, becomes inapplicable near the phase boundary line due to a critical slowing down.
Based on the foundations of thermodynamics and the equilibrium conditions for the coexistence of two phases in a magnetic Ising-like system, we show, first, that there is a critical point where the isothermal susceptibility diverges and the specific heat may remain finite, and second, that near the critical point the entropy of the system, and therefore all free energies, do obey scaling. Although we limit ourselves to such a system, we elaborate about the possibilities of finding universality, as well as the precise values of the critical exponents using thermodynamics only.
Within the droplet model for the Ising spin system it is shown that the magnetic specific heat Cp{C}_{p}Cp​ and the susceptibility ${\ensuremath{\chi}}_{p}$ for constant vanishing pressure and magnetic field remain finite in a compressible lattice as the critical point Tc{T}_{c}Tc​ is approached from below. This regularization of Cp{C}_{p}Cp​ and ${\ensuremath{\chi}}_{p}$ is caused by the long-range indirect spin-spin interaction, which is a consequence of the dependence of the exchange integral on the lattice spacing.
The interplay between disorder and compressibility in Ising magnets is studied. Contrary to pure systems in which a weak compressibility drives the transition first order, we find from a renormalization group analysis that it has no effect on disordered systems which keep undergoing continuous transition with rigid random-bond Ising model critical exponents. The mean field calculation exhibits a dilution-dependent tricritical point beyond which, at stronger compressibility the transition is first order. The different behavior of XY and Heisenberg magnets is discussed.
Phase transitions in Ising models with tree surfaces are studied from various points of view, including a phenomenological Landau theory, high-temperature series expansions, and a scaling theory for thermodynamic quantities and correlation functions. In the presence of a surface a number of new critical exponents must be defined. These arise because of the existence of "surface" terms in the thermodynamic functions, and because of the anisotropy of space and lack of translational symmetry introduced by the surface. The need for these new critical exponents already appears in the phenomenological theory, which is discussed in detail and related to the microscopic mean-field approximation. The essential new parameter appearing in this theory is an extrapolation length $\ensuremath{\lambda}$ which enters the boundary condition on the magnetization at the surface. For magnetic systems this length is of the order of the interaction range, in contrast to superconductors, where it is usually much larger. In order to go beyond the mean-field theory, high-temperature series expansions are carried out for the Ising half-space, to tenth order in two dimensions and to eighth order in three dimensions. A scaling theory is developed both for thermodynamic functions and for spin correlations near the surface, and relations are found among the exponents of the half-space. Both the scaling theory and the numerical calculations are compared with the exact solution of the Ising half-plane (two dimensions) by McCoy and Wu, and agreement is found wherever the theory is applicable. In analogy to the bulk situation, the scaling theory is found to agree with mean-field theory in four dimensions. The prediction of the present work which is most easily accessible to experiment is the temperature dependence of the magnetization at the surface, with critical exponent estimated to be ${\ensuremath{\beta}}_{1}=\frac{2}{3}$. The mean-field result, ${\ensuremath{\beta}}_{1}=1$, seems to agree more closely with presently available experiment, and more work is needed to clarify the situation.
The dynamic phase transition has been studied in the three-dimensional kinetic Ising model in the presence of a time-varying(sinusoidal)magnetic field by Monte Carlo simulation.The nature of the transition is characterized by studying the distribution of the order parameter.The system is in contact with an isothermal heat bath at temperature T.We varied the magnitude of the magnetic field,keeping the frequency of the magnetic field unchanged.For the lower values of the environment temperature the transition is discontinuous and it is continuous for higher values of the environment temperature,indicating the existence of a tricritical point (TCP) on the phase boundary.The TCP shifts towards higher temperature region with the decrease of frequency.i.e:TTCP=1.33×e(-ω/30.7)