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Physical review. B, Solid state

Renormalization-group theory and calculations of tricritical behavior

Bernard Nienhuis, Michael Nauenberg

Published Mar 1, 1976• 49 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.13.2021

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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.
The form of the field distribution, in a random-field Ising model, is known to play an important role in determining the order of the phase transition and the existence of a tricritical point. We have studied a random-field Ising model with a trimodal distribution (sum of three \ensuremath{\delta} functions), which simulates a Gaussian distribution for a certain range of its parameters, within the mean-field renormalization-group scheme. We have considered clusters of one, two, and four spins, and the recursion relations for the determination of the fixed points are numerically analyzed. Using this analysis we have constructed the phase diagram of the system in the space of coupling (K), field (h), and zero-field probability (p). It is shown that the line of tricritical points ends for values of p near (2/3, very different from that obtained in the usual mean-field approximation. Mean-field renormalization-group (p\ensuremath{\simeq}(2/3) results are found to be consistent with the equivalence of the trimodal and Gaussian (absence of tricritical points) distributions until the fourth-order moments for p=(2/3.
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.
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.
The expectations of phenomenological crossover scaling theory are worked out for two-point correlation functions in zero field above the primary critical point. Renormalization-group recursion relations are then used to construct the two-point correlation function for the changeover from tricritical to critical-line behavior, above the tricritical point, to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$, for $3\ensuremath{\le}d<4$. An explicit expression is obtained for the normalized double-scaling function ${\stackrel{^}{\ensuremath{\Gamma}}}_{N}({q}^{2}{\ensuremath{\xi}}^{2}, \frac{g}{{t}^{\ensuremath{\phi}}})$, for the general scaling variable ${q}^{2}{\ensuremath{\xi}}^{2}$, and the crossover variable $w=\frac{g}{{t}^{\ensuremath{\phi}}}$, $q\ensuremath{\ll}{a}^{\ensuremath{-}1}$ being the wave vector; $\ensuremath{\xi}\ensuremath{\gg}a$ is a second-moment correlation length and aaa the lattice spacing. In the tricritical regime, $w\ensuremath{\simeq}0$, ${\stackrel{^}{\ensuremath{\Gamma}}}_{N}$ (x2,w{x}^{2}, wx2,w) has the Ornstein-Zernicke form and the changeover to the critical regime $w\ensuremath{\simeq}\ensuremath{\infty}$ is explicitly discussed. The limitation of the results, within our calculations to first order in $\ensuremath{\epsilon}$, are pointed out.
The description of first-order phase transitions using the linear renormalization group is illustrated with the two-dimensional Ising model. Different linear renormalization groups should be used to describe the system in the neighborhood of criticality and away from it. In the latter case, the renormalization-group trajectories are lines of constant spin expectation value, which map a system below critical temperature in a small magnetic field to a system of free spins at a finite magnetic field. A finite-lattice approximation is also presented. The equation of state and several thermodynamic quantities are obtained; a smooth matching of the behavior near and away from criticality occurs naturally.