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Physical review. B, Solid state

Equations of state and renormalization-group recursion relations

Joseph Rudnick, David R. Nelson

Published Mar 1, 1976• 264 citations
Publisher

Abstract

Renormalization-group recursion relations are used to construct equations of state to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ for continuous spin models of critical behavior. The recursion relations map Hamiltonians out of the critical regime into regions of small correlation length, where Landau theory with fluctuation corrections is employed. This technique is used to discuss critical behavior along a $\ensuremath{\lambda}$ line of critical points, together with the effect of imposing an ordering field. An application of the method to tricritical phenomena is described briefly.

DOI
10.1103/physrevb.13.2208

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Renormalization-group recursion relations are used to construct equations of state to first order in \ensuremathϵ=4\ensuremath−d\ensuremath{\epsilon}=4\ensuremath{-}d\ensuremathϵ=4\ensuremath−d for continuous spin models of critical behavior. The recursion relations map Hamiltonians out of the critical regime into regions of small correlation length, where Landau theory with fluctuation corrections is employed. This technique is used to discuss critical behavior along a \ensuremathλ\ensuremath{\lambda}\ensuremathλ line of critical points, together with the effect of imposing an ordering field. An application of the method to tricritical phenomena is described briefly.
Recent theoretical work has led to closed form expressions for the thermodynamic functions which characterize complicated multricritcal phenomeana. Renormalization group recursion relations are used to map multicritical systems out of the critical regime, at which point Landau theory with fluctuation corrections can be employed. We review here applications of this approach to tricritical points, spin flop transitions, dipolar systems, and Heisenberg ferromagnets below Tc. Although the calculations have presently been carried out only to first order ε=4−d, they nevertheless give a good qualtiative understanding of multicritical behavior. In particular, they provide a detailed picture of the crossover between competing kinds of critical behavior, and allow treatment of ordered phases with spontaneously broken symmetries.
Renormalization-group recursion relations are used to calculate to order $\ensuremath{\epsilon}=4\ensuremath{-}d$ the free-energy, ordering-susceptibility, and magnetization-crossover scaling functions for tricritical points in isotropic nnn-vector models. The equations of state describe critical and tricritical behavior in both ordered and disordered phases.
The renormalization-group method for studying critical phenomena is generalized to a class of dynamical systems---the time-dependent Ginzburg-Landau models. The effects of conservation laws on the critical dynamics are investigated through the study of models with different conservation properties for the energy and the space integral of the order parameter. Dynamic critical exponents near four dimensions ($d\ensuremath{\approx}4$) are obtained from recursion relations, analogous to those of Wilson and Fisher. The physical significance of the time-dependent Ginzburg-Landau models is explored and the applicability of the results to experiments on the NMR linewidth of FeF2{\mathrm{F}}_{2}F2​ is discussed.
Thermodynamic functions of an nnn-component magnetic system coupled to an isotropic elastic medium in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions are calculated to leading order in $\ensuremath{\epsilon}$. Solutions to renormalization-group recursion relations correct to $O(\ensuremath{\epsilon})$ along with trajectory-integral methods are employed in these calculations. For n> 2n>~2n> 2 the finite compressibility modifies the critical-point behavior producing new corrections to scaling in the thermodynamic functions which are given. In the n=1n=1n=1 case renormalization-group predictions of a first-order transition are verified. Thermodynamic functions are given and analyzed in this case. In particular the transition temperature and discontinuity in order parameter are determined and the phase diagram is constructed.
By expressing the free energy as a line integral along a renormalization-group trajectory, a technique for calculating the crossover scaling functions which describe tricritical systems, anisotropic spin systems, etc., is developed. This formalism is applied to the model recursion relations of Riedel and Wegner, which simulate crossover behavior. A simple mechanism for the breakdown of dimensionality dependent hyperscaling relationships emerges from the analysis. The specific-heat crossover scaling function describing crossovers from Gaussian to Heisenberg critical behavior is constructed to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$.
The conventional renormalization-group analysis of the tricritical behavior of a metamagnet is completed by a description of the first-order transition below the tricritical temperature, and the conditions for the discontinuity in the magnetization are explicitly verified in an Ising spin model. The singularity structure of the crossover scaling function is derived from relations between the scaling fields for the critical and tricritical fixed points. Numerical calculations illustrating various aspects of the theory are given for a square-lattice Ising model in a four-cell cluster approximation.