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Physical review. B, Solid state

First-order phase transitions and linear renormalization group

Krishnappa Subbarao

Published May 1, 1976• 12 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.13.3939

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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.
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 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.
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Accurate calculations of the thermodynamic properties of a system near its critical point are now possible using finite-lattice approximations to renormalization groups. In order to investigate some features of such approximations, we introduce a linear-renormalization-group transformation appropriate to a system of continuous spins like the Gaussian model of Berlin and Kac. We solve the renormalization-group equations for the Gaussian model exactly, and then study in detail a finite-lattice approximation to the renormalization group for the three-dimensional case. The renormalization-group transformation contains a parameter bbb. The exact transformation has fixed points only for $b={2}^{\ensuremath{-}\frac{5}{2}}$. The approximate transformation has fixed points for a range of values of bbb around ${2}^{\ensuremath{-}\frac{5}{2}}$. Eigenvalues of the transformation, which determine critical exponents, depend on the value of bbb used in the calculation. We study the problem of identifying the correct value of bbb.
The spin-5/2 Blume–Capel model was studied using the mean-field approximation and the Migdal–Kadanoff renormalization group method for two- and three-dimensional systems. We determined the phase diagrams in the (crystal field, temperature) plane where the system exhibited first- and second-order phase transitions as well as isolated critical, bicritical and triple points. In order to show first-order transitions at low temperature, we presented the total magnetization per site and the derivative of the free energy as a function of the crystal field. Moreover, the critical exponents of the system were calculated by linearizing the renormalization transformation at the vicinity of the second-order fixed points.
These lectures describe the physics of phase transitions and critical phenomena. After introducing the thermodynamics of phase transitions, these general considerations are made explicit by presenting statistical mechanical calculations on a lattice gas model in the mean-field approximation. Then the Landau theory of phase transitions is presented. The form of free energy in the Landau theory is motivated by an explicit mean field calculation for the Ising model. This is followed by a discussion of spatial correlations and the breakdown of the mean field theory due to increase of spatial fluctuations near the critical point. This calculation also brings out the role of dimension of the system for the critical behaviour. The basic ideas of Renormalization Group (RG) are then introduced and illustrated by a simple RG calculation for the one-dimensional Ising model. Real-space RG calculations are then applied to the two-dimensional Ising model on the triangular lattice, which, in fact, brings forth several key features of the RG technique. A discussion of more general aspects of the RG formalism, such as scaling for spatial correlation functions, relevant and irrelevant couplings, universality, upper critical dimension, etc. are introduced.