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Physical review. B, Condensed matter

Critical and crossover behavior in the double-Gaussian model on a lattice

George Allen Baker Jr., A. R. Bishop, Klaus Fesser, Paul D. Beale, JAMES A. KRUMHANSL

Published Sep 1, 1982• 17 citations
Publisher

Abstract

The double-Gaussian model, as recently introduced by Baker and Bishop, is studied in the context of a lattice-dynamics Hamiltonian belonging to the familiar ${\ensuremath{\varphi}}^{4}$ class. Advantage is taken of the partition-function factorability (into Ising and Gaussian components) to place bounds on the Ising-class critical temperature for various lattice dimensions and all degrees of displaciveness in the bare Hamiltonian. Further, a simple criterion for a noncritical and nonuniversal crossover from order-disorder to Gaussian behavior is evaluated in numerical detail. In one and two dimensions these critical and crossover properties are compared with predictions based on real-space decimation renormalization-group flows, as previously exploited in the ${\ensuremath{\varphi}}^{4}$ model by Beale et al. The double-Gaussian model again introduces some unique analytical advantages.

DOI
10.1103/physrevb.26.2596

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The double-Gaussian model, as recently introduced by Baker and Bishop, is studied in the context of a lattice-dynamics Hamiltonian belonging to the familiar \ensuremathφ4{\ensuremath{\varphi}}^{4}\ensuremathφ4 class. Advantage is taken of the partition-function factorability (into Ising and Gaussian components) to place bounds on the Ising-class critical temperature for various lattice dimensions and all degrees of displaciveness in the bare Hamiltonian. Further, a simple criterion for a noncritical and nonuniversal crossover from order-disorder to Gaussian behavior is evaluated in numerical detail. In one and two dimensions these critical and crossover properties are compared with predictions based on real-space decimation renormalization-group flows, as previously exploited in the \ensuremathφ4{\ensuremath{\varphi}}^{4}\ensuremathφ4 model by Beale et al. The double-Gaussian model again introduces some unique analytical advantages.
A real-space renormalization-group transformation has been carried out on the classical ${\ensuremath{\varphi}}^{4}$ model on a lattice in two dimensions. The critical line has been found for $0<\ensuremath{\theta}<\ensuremath{\infty}$ where $\ensuremath{\theta}$ is the ratio of the site well depth to the nearest-neighbor harmonic coupling energy. For $\ensuremath{\theta}\ensuremath{\ll}1$ (near the displacive limit) the critical temperature qualitatively agrees with a rigorously derived function for ${T}_{c}(\ensuremath{\theta})$ in that limit. In contrast, a previously published Kadanoff-Migdal transformation on the same model predicts that ${T}_{c}(\ensuremath{\theta})$ is linear in $\ensuremath{\theta}$ for $\ensuremath{\theta}\ensuremath{\ll}1$. A crossover to noncritical Gaussian-like behavior is found at temperature ${T}_{0}(\ensuremath{\theta})>{T}_{c}(\ensuremath{\theta})$ for sufficiently small $\ensuremath{\theta}$.
The critical behavior of a two-component lattice-gas model (the 1-2 model) in two and three dimensions is investigated by means of standard and nonstandard methods of series analysis on exact series expansions for the thermodynamic properties of the model. The model exhibits a critical line that passes from the Ising critical point at finite temperature to a critical point at zero temperature. In two dimensions the Ising values of the critical exponents found along the critical line and at zero temperature support the expectation that the model lies in the Ising universality class. In three dimensions Ising values of the critical exponents are found away from zero temperature, but near and at zero temperature the series analysis suggests a crossover to non-Ising values of the critical exponents. This would imply that the model belongs to the Ising universality class at nonzero temperatures and to a different universality class at zero temperature---a quite unexpected result. Various possible explanations for this unusual behavior of the series results are discussed.
We study the crossover between classical and nonclassical critical behaviors. The critical crossover limit is driven by the Ginzburg number G. The corresponding scaling functions are universal with respect to any possible microscopic mechanism which can vary G, such as changing the range or the strength of the interactions. The critical crossover describes the unique flow from the unstable Gaussian to the stable nonclassical fixed point. The scaling functions are related to the continuum renormalization-group functions. We show these features explicitly in the large-NNN limit of the $\mathrm{O}(N) {\ensuremath{\varphi}}^{4}$ model. We also show that the effective susceptibility exponent is nonmonotonic in the low-temperature phase of the three-dimensional Ising model.
We revisit the two-dimensional quantum Ising model by computing renormalization group flows close to its quantum critical point. The low but finite temperature regime in the vicinity of the quantum critical point is squashed between two distinct non-Gaussian fixed points: the classical fixed point dominated by thermal fluctuations and the quantum critical fixed point dominated by zero-point quantum fluctuations. Truncating an exact flow equation for the effective action, we derive a set of renormalization group equations and analyze how the interplay of quantum and thermal fluctuations, both non-Gaussian in nature, influences the shape of the phase boundary and the region in the phase diagram where critical fluctuations occur. The solution of the flow equations makes this interplay transparent: we detect finite temperature crossovers by computing critical exponents and we confirm that the power law describing the finite temperature phase boundary as a function of control parameter is given by the correlation length exponent at zero temperature as predicted in an ϵϵϵ-expansion with ϵ=1ϵ=1ϵ=1 by Sachdev [Phys. Rev. B 55, 142 (1997)].
A cutoff-dependent formulation of renormalized perturbation theory is used to study the Gaussian-to-Heisenberg crossover, describing the changeover from tricritical to critical-line behavior, for the specific heat, to second order in $\ensuremath{\epsilon}=4\ensuremath{-}d$, in extension of earlier work by Bruce and Wallace for the susceptibility. The role of the nonhomogeneous part of the renormalization-group equation for the bare vertex function ${\ensuremath{\Gamma}}^{(2,0)}$ - proportional to the specific heat - with two ${\ensuremath{\varphi}}^{2}$ insertions is discussed in detail in an isotropic ${\ensuremath{\varphi}}^{4}$ field model. An explicit crossover scaling function for the specific heat is calculated and the effective exponent ${\ensuremath{\alpha}}_{\mathrm{eff}}$ is found to differ considerably from an earlier first-order calculation.
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.