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Physical review. B, Condensed matter

Renormalization group for first-order phase transitions: Equation of state of the compressible Ising magnet

J. Bruno, Josef Sak

Published Oct 1, 1980• 42 citations
Publisher

Abstract

Thermodynamic functions of an $n$-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 $n>~2$ the finite compressibility modifies the critical-point behavior producing new corrections to scaling in the thermodynamic functions which are given. In the $n=1$ 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.

DOI
10.1103/physrevb.22.3302

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Physical review. B, Solid state•Jun 1, 1976•48 citations
Thermodynamic functions of an nnn-component magnetic system coupled to an isotropic elastic medium in d=4\ensuremath−\ensuremathϵd=4\ensuremath{-}\ensuremath{\epsilon}d=4\ensuremath−\ensuremathϵ dimensions are calculated to leading order in \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ. Solutions to renormalization-group recursion relations correct to O(\ensuremathϵ)O(\ensuremath{\epsilon})O(\ensuremathϵ) 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 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.
A renormalization-group calculation has been carried out in $d=6\ensuremath{-}\ensuremath{\epsilon}$ dimensions to lowest order in $\ensuremath{\epsilon}$, using the continuum generalization of the Potts model. The equation of state and the various thermodynamic functions have been calculated explicitly by integrating up the recursion relations in both the ordered and disordered phases. The transition is found to be first order for p>2p>2p>2. For 2<p<1032<p<\frac{10}{3}2<p<310​ the transition is first order in spite of the fact that there is an accessible stable fixed point for all p<103p<\frac{10}{3}p<310​. At the first-order transition, instead of the usual expansion in temperature, the various thermodynamic functions are studied as functions of ($p\ensuremath{-}2$). Finally it is shown that the free energy of the Potts model has an essential singularity at the first-order transition temperature Tc{T}_{c}Tc​, due to instantons, which is of the same form as that of the Ising model at H=0H=0H=0.
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.
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.
We apply an approximate form of the exact one-particle-irreducible renormalization-group generator to the calculation of the equation of state. Several approaches are explored. (i) Global nonlinear trajectories for the Hamiltonian parameters are exploited to give nonlinear crossover equations of state. Logarithms of the reduced temperature $t\ensuremath{\equiv}\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$ are automatically exponentiated to give power-law behavior. (ii) The form and asymptotic properties of the equation of state are described for those systems whose complete nonlinear trajectories cannot be explicitly obtained. (iii) Nonlinear trajectories for the irreducible Green's functions are solved to give a fully exponentiated equation of state containing no logarithmic terms. (iv) Simple critical (noncrossover) equations of state are obtained by iteration of the generator with linear and quasilinear trajectories. (v) Operator nonlinear equations are solved to give the crossover equation of state for an arbitrary order $\ensuremath{\theta}$ critical system, both in an $\ensuremath{\epsilon}$ expansion and at the borderline dimension. (vi) Combination of the Green's functions are formed into Green's eigenfunctions or operators and their nonlinear trajectories used to calculate fully exponentiated equations of state for order $\ensuremath{\theta}$ Ising systems. Systems considered include the usual Wilson-Fisher Hamiltonian [methods (i)-(iv)], the Sak-model compressible ferromagnet [(i)-(ii)], the nm\mathrm{nm}nm-hypercubical model (of which the dilute quenched random ferromagnet forms the $n\ensuremath{\rightarrow}0$ limit) [(ii)], and the isotropic nnn-component order $\ensuremath{\theta}$ ferromagnet [(iv)-(vi)]. The tricritical ($\ensuremath{\theta}=3$) case is given explicitly as an example of the general formalism of methods (iv)-(vi). Throughout, we use a bare critical propagator which is an arbitrary generalized homogeneous function of the components of the wave vector. This allows us to simultaneously describe ordinary critical systems and anisotropic Lifschitz points as well as certain structural and spin-reorientation phase transitions.
We study a model equation of motion describing the dynamics of isotropic antiferromagnets near the critical point. We perform a renormalization-group analysis of this equation of motion correct to $O(\ensuremath{\epsilon})$, where $\ensuremath{\epsilon}=4\ensuremath{-}d$, finding recursion relations and analyzing their fixed points and their stability. For the one stable fixed point we obtain the dynamical index $z=\frac{d}{2}=2\ensuremath{-}\frac{\ensuremath{\epsilon}}{2}$. We also calculate the response and correlation functions in the scaling region using perturbation theory correct to $O(\ensuremath{\epsilon})$. Our calculation gives analytical forms for the correlation functions for the staggered magnetization and the magnetization. These correlation functions can be written in the dynamical scaling form. The shape function that characterizes the frequency spectrum for the staggered magnetization is Lorentzian in the hydrodynamical regime, but shows fluctuation induced peaks at and near the N\ifmmode\dagger\else\textdagger\fi{}eel temperature. The shape function for the magnetization is essentially Lorentzian for all $x=q\ensuremath{\xi}$ (where qqq is the wave number and $\ensuremath{\xi}$ is the correlation length) and shows a narrow lydrodynamical width as $x\ensuremath{\rightarrow}0$.