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Physical Review Letters

Renormalized Critical Behavior or First-Order Phase Transitions?

Leon Gunther, David J. Bergman, Y. Imry

Published Aug 30, 1971• 58 citations
Publisher

Abstract

The exactly solvable Baker-Essam model for a compressible Ising lattice is re-examined. We find that, depending on the type of constraint imposed upon the system, the second-order phase transition either gets renormalized or is changed into a first-order transition. We resolve the apparent disagreement of this result with Fisher's renormalization theory by showing that, in fact, our result is typical of the correct state of affairs to be expected on the basis of the general theory of critical points in systems with hidden variables.

DOI
10.1103/physrevlett.27.558

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The exactly solvable Baker-Essam model for a compressible Ising lattice is re-examined. We find that, depending on the type of constraint imposed upon the system, the second-order phase transition either gets renormalized or is changed into a first-order transition. We resolve the apparent disagreement of this result with Fisher's renormalization theory by showing that, in fact, our result is typical of the correct state of affairs to be expected on the basis of the general theory of critical points in systems with hidden variables.
The critical behavior of a compressible Ising system of the Baker-Essam type is examined for positive pressures. In addition to satisfying the assumptions of Fisher's theory of renormalization of critical exponents, the model, allows one to calculate the temperature region around Tc{T}_{c}Tc​ where the renormalized exponents should show up. The region is $\ensuremath{\Delta}{T}_{\ifmmode\times\else\texttimes\fi{}}\ensuremath{\sim}{(P{J}_{1}^{2})}^{\frac{1}{\ensuremath{\alpha}}}$ for small pressures, where J1{J}_{1}J1​ is the spin-lattice coupling and $\ensuremath{\alpha}$ is the specific-heat exponent of the corresponding rigid Ising system.
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.
Renormalization group methods are applied to study the critical behaviour of a compressible n-component Ising model with short range interactions at d = 4 and a one component Ising model with dipolar interactions at d = 3. The recursion equations are exactly solved in the case of an elastic system of spherical symmetry (d = 4) or cylindrical symmetry (d = 3); new types of logarithmic corrections, corresponding to a Fisher renormalization at marginal dimensions, are found. It is shown that the system exhibits a first-order transition for constant pressure external conditions or when anisotropy is taken into account. The relevance of the calculations to the critical behaviour of uniaxial ferroelectrics is discussed.
In papers devoted to superfluidity, the generally accepted statement that the dynamics of the corresponding phase transition is described by the stochastic model F or E can be frequently found. Nevertheless, the dynamical critical index has not been found even in the leading order of the perturbation theory. It is also unknown which model, E or F, in fact corresponds to this system. We use two different approaches to study this problem. First, we study the dynamics of the critical behavior in a neighborhood of the λ point using the renormalization group method based on a quantum microscopic model in the formalism of the time-dependent Green’s functions at a finite temperature. Second, we study the stochastic model F to find whether it is stable under compressibility effects. Both approaches lead to the same very unexpected result: the dynamics of the phase transition to the superfluid state are described by the stochastic model A with a known dynamical critical index.
The interplay between disorder and compressibility in Ising magnets is studied. Contrary to pure systems in which a weak compressibility drives the transition first order, we find from a renormalization group analysis that it has no effect on disordered systems which keep undergoing continuous transition with rigid random-bond Ising model critical exponents. The mean field calculation exhibits a dilution-dependent tricritical point beyond which, at stronger compressibility the transition is first order. The different behavior of XY and Heisenberg magnets is discussed.
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.