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Journal of Experimental and Theoretical Physics

Equation of state and universal combinations of thermodynamic critical amplitudes in the Ising impurity model

N. A. Shpot

Published Nov 1, 1990• 0 citations
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The critical behavior of a d -dimensional random Ising ferromagnet described in the critical region by the Ginzburg-Landau model with a random temperature has been considered. The critical temperature dependences of thermodynamic quantities, the equation of state, and the spin correlation function at the Curie point have been calculated by the renormalization group method. The most profound questions of the theory of impurity ferromagnets are the problem of universality and the role of the Griffiths singularities.
The behavior of the magnetic susceptibility of a two-dimensional Ising model with nonmagnetic impurities is investigated numerically. A new method for determining the critical amplitudes and critical temperature is developed. The results of a numerical investigation of the ratio of the critical amplitudes of the magnetic susceptibility are presented. It is shown that the ratio of the critical amplitudes is universal right up to impurity concentrations q ≤ 0.25 (the percolation point of a square lattice is q c = 0.407254). The behavior of the effective critical exponent γ( q ) of the magnetic susceptibility is discussed. Apparently, a transition from Ising-type universal behavior to percolation behavior should occur in a quite narrow concentration range near the percolation point of the lattice.
We introduce a new way of reconstructing the equation of state of a thermodynamic system near a second-order critical point from a finite set of Taylor coefficients computed away from the critical point. We focus on the Ising universality class (Z_{2} symmetry) and show that, in the crossover region of the phase diagram, it is possible to efficiently extract the location of the nearest thermodynamic singularity, the Lee-Yang edge singularity, from which one can (i) determine the location of the critical point, (ii) constrain the nonuniversal parameters that maps the equation of state to that of the Ising model in the scaling regime, and (iii) numerically evaluate the equation of state in the vicinity of the critical point. This is done by using a combination of Padé resummation and conformal maps. We explicitly demonstrate these ideas in the celebrated Gross-Neveu model.