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Physical review. B, Condensed matter

Tricritical behavior in two dimensions. II. Universal quantities from the ε expansion

A. L. Lewis, F. W. Adams

Published Nov 1, 1978• 37 citations
Publisher

Abstract

The tricirical exponents ${\ensuremath{\eta}}_{t}$ and ${\ensuremath{\varphi}}_{t}$ are calculated through order ${\ensuremath{\epsilon}}^{3}$ and ${\ensuremath{\epsilon}}^{2}$, respectively, in the isotropic $n$-component model, where $\ensuremath{\epsilon}=3\ensuremath{-}d$. The estimate for ${\ensuremath{\eta}}_{t}$, in two dimensions for the Ising case, is 0.027; the series for ${\ensuremath{\varphi}}_{t}$ is quite ill behaved, producing a negative estimate at this order. Beginning at this order in $\ensuremath{\epsilon}$, the spherical-model limit fails to exist. A scaling function for the spin-spin correlation function, appropriate for nonexceptional paths of approach to the tricritical point in the disordered phase, is calculated through ${\ensuremath{\epsilon}}^{2}$; its large momentum expansion is shown to involve the crossover exponent. These results are generalized to points where $\ensuremath{\theta}$ coexisting phases become critical.

DOI
10.1103/physrevb.18.5099

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The tricirical exponents \ensuremathηt{\ensuremath{\eta}}_{t}\ensuremathηt​ and \ensuremathφt{\ensuremath{\varphi}}_{t}\ensuremathφt​ are calculated through order \ensuremathϵ3{\ensuremath{\epsilon}}^{3}\ensuremathϵ3 and \ensuremathϵ2{\ensuremath{\epsilon}}^{2}\ensuremathϵ2, respectively, in the isotropic nnn-component model, where \ensuremathϵ=3\ensuremath−d\ensuremath{\epsilon}=3\ensuremath{-}d\ensuremathϵ=3\ensuremath−d. The estimate for \ensuremathηt{\ensuremath{\eta}}_{t}\ensuremathηt​, in two dimensions for the Ising case, is 0.027; the series for \ensuremathφt{\ensuremath{\varphi}}_{t}\ensuremathφt​ is quite ill behaved, producing a negative estimate at this order. Beginning at this order in \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ, the spherical-model limit fails to exist. A scaling function for the spin-spin correlation function, appropriate for nonexceptional paths of approach to the tricritical point in the disordered phase, is calculated through \ensuremathϵ2{\ensuremath{\epsilon}}^{2}\ensuremathϵ2; its large momentum expansion is shown to involve the crossover exponent. These results are generalized to points where \ensuremathθ\ensuremath{\theta}\ensuremathθ coexisting phases become critical.
In Part I the global phase diagram of the general spin-1, nearest-neighbor, ferromagnetic Ising model was studied for d=2d=2d=2 dimensions using an approximate renormalization-group approach of the Migdal-Kadanoff type. Here general ddd is considered with emphasis on $\ensuremath{\epsilon}=d\ensuremath{-}1$ expansions and on the variation of the tricritical exponents with d< 4d<~4d< 4. A dilution field is introduced which can be chosen to yield tricritical exponents for d=2d=2d=2, in better agreement with more reliable estimates and conjectured exact results. The three-phase monohedra, which describe the densities of coexisting phases near a tricritical point, are calculated quantitatively for d=2d=2d=2, and their shape contrasted with that predicted by classical theory: they are found, in particular, to be significantly flattened.
We analyze exactly in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$, the nnn-component continuous-spin model with a cubic field term ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}}_{3}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{s}}{|\stackrel{\ensuremath{\rightarrow}}{\mathrm{s}}|}^{2}$. Full "four-field" tricritical behavior is exhibited for dimensionalities $d\ensuremath{\ge}3$. However, orthodox tricritical scaling is shown to be impossible for 3<d<43<d<43<d<4: to obtain the correct nonclassical spherical-model exponents on the $\ensuremath{\lambda}$ line (H=H3=0,T>TtH={H}_{3}=0, T>{T}_{t}H=H3​=0,T>Tt​) it is essential to allow for a dangerous irrelevant scaling variable $p\ensuremath{\propto}\frac{1}{{R}_{0}^{d}}$, where R0{R}_{0}R0​ is the range of the pair interactions. The appropriate crossover exponent is ${\ensuremath{\varphi}}_{p}=3\ensuremath{-}d$ so that ppp is marginal for d=3d=3d=3: orthodox scaling is then possible but the scaling functions are nonuniversal. On the disordered symmetry plane (H=H3=0H={H}_{3}=0H=H3​=0) only the corrections to scaling survive and describe crossover to Gaussian tricritical behavior. For T>TtT>{T}_{t}T>Tt​ bicritical crossover from spherical to classical critical behavior occurs when H3{H}_{3}H3​ varies but scaling is fully obeyed. Some inferences for systems with finite nnn are drawn.
Renormalisation of a field theory model for tricritical behaviour in which the operator phi 6 is essential for thermodynamic stability is discussed in d=4- epsilon dimensions. The crossover scaling form of the equation of state is obtained explicitly to first order in epsilon for both positive and negative values of the four-spin coupling constant. Orthodox scaling, involving Gaussian tricritical exponents is obtained, but is shown to be physically inappropriate in the ordered region of the symmetry plane. A reformulation of scaling, using classical tricritical exponents is possible, but involves an additional parameter p with its own scaling exponent phi p =- 1 / 2 (1- epsilon ). This parameter, introduced by Sarbach and Fisher for the many-component, spherical model limit, is related to the coefficient of (nabla phi ) 2 in the field-theoretic Hamiltonian.
The critical behavior of a system of nnn-component classical "spins"with anisotropic pair interactions is discussed using renormalization-group techniques for dimension $d=4\ensuremath{-}\ensuremath{\epsilon}$. To first order in $\ensuremath{\epsilon}$ the correlation and susceptibility exponents in the isotropic limit are $2\ensuremath{\nu}=\ensuremath{\gamma}=1+[\frac{(n+2)}{2(n+8)}]\ensuremath{\epsilon}$, while the anisotropy or crossover exponent is $\ensuremath{\varphi}=1+[\frac{n}{2(n+8)}]\ensuremath{\epsilon}$. When $n\ensuremath{\rightarrow}\ensuremath{\infty}$ these expansions agree with exact spherical-model results. For n=3n=3n=3 and d=3d=3d=3 series expansions indicate $\ensuremath{\varphi}\ensuremath{\simeq}1.2$ (compared with $\ensuremath{\gamma}\ensuremath{\simeq}1.38$).
In a recent work [M. Castellana and G. Parisi, Phys. Rev. E 82, 040105(R) (2010)], the large-scale behavior of the simplest non-mean-field spin-glass system has been analyzed, and the critical exponent related to the divergence of the correlation length has been computed at two loops within the ε-expansion technique by two independent methods. By performing the explicit calculation of the critical exponents at two loops, one obtains that the two methods yield the same result. This shows that the underlying renormalization group ideas apply consistently in this disordered model, in such a way that an ε-expansion can be set up. The question of the extension to high orders of this ε-expansion is particularly interesting from the physical point of view. Indeed, once high orders of the series in ε for the critical exponents are known, one could check the convergence properties of the series, and find out if the ordinary series resummation techniques, yielding very accurate predictions for the Ising model, work also for this model. If this is the case, a consistent and predictive non-mean-field theory for such a disordered system could be established. In that regard, in this work we expose the underlying techniques of such a two-loop computation. We show with an explicit example that such a computation could be quite easily automatized, i.e., performed by a computer program, in order to compute high orders of the ε-expansion, and so eventually make this theory physically predictive. Moreover, all the underlying renormalization group ideas implemented in such a computation are widely discussed and exposed.
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.