The variety of singularities in models of first order phase transitions

Koo-Chul Lee

Alain H. Billoire

W. Klein, D. J. Wallace, Royce K. P. Zia

Renormalization group ideas are applied to the problem of first-order phase transitions in Ising-like models. They support the existence of essential singularities as the coexistence region is approached, at all temperatures below the critical temperature.

Susan R. McKay, A. Nihat Berker

The three-dimensional random-field Ising model has been approximately solved by pursuing the global renormalization-group trajectories of the full, coupled probability distributions of local fields and bonds. This study, which in effect is the global analysis of the renormalization-group flows of 220 physical quantities (based on a generalization of the Migdal–Kadanoff approximation), yields several results that were previously unsuspected. The boundary between the ferromagnetic and paramagnetic phases is hybrid order, in the sense that the magnetization is discontinuous (as in a first-order transition) and the specific heat has a power-law singularity (as in a second-order transition). The magnetic properties of the d=3 random-field Ising model are presented in the form of equimagnetization lines. The study has been repeated for a variety of dimensions d. The lower critical dimension dl=2 is correctly obtained. Moreover, this study indicates the existence of an intermediate-critical dimension dt (between the lower- and upper-critical dimensions) where a singularity occurs in the critical properties as a function of dimension.

Y Achiam, Y. Imry

The renormalization-group method is applied to the analysis of phase transitions in systems where the order parameter $s$ is coupled to a nonordering additional variable $y$. A variety of critical and tricritical behaviors or first-order transitions is found as a function of the physical variables and possible macroscopic constraints imposed on the system. For ${s}^{2}y$ coupling, the correlation function of $y$ was found to be governed by a correlation length which is proportional to that of the order parameter, and by a critical index ${\ensuremath{\eta}}_{y}\ifmmode\cdot\else\textperiodcentered\fi{}{\ensuremath{\eta}}_{y}=2\ensuremath{-}2\ensuremath{\alpha}$; $\ensuremath{\alpha}$ here is the specific-heat exponent of the appropriate unconstrained system. The singular part of the susceptibility, ${\ensuremath{\chi}}_{y}$, has a critical exponent equal to $\ensuremath{\alpha}$, the true specific-heat exponent. When the coupling is ${s}^{2}{y}^{2}$, a weaker singularity of ${\ensuremath{\chi}}_{y}$ appears. The crossover between this behavior and the one typical to ${s}^{2}y$ coupling is calculated. $〈y〉$ has a singular part with an exponent $1\ensuremath{-}\ensuremath{\alpha}$, in the unconstrained case. The breakdown of the scaling law related to the correlation function of $y$ in the constrained case is discussed.

Andrzej Radosz