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Journal of Physics A Mathematical and General

Critical behaviour of the long-range ( ϕ2)2model in the short-range limit

Juha Honkonen

Published Mar 7, 1990• 33 citations
Publisher

Abstract

The relation between scaling regimes described by ( ϕ 2 ) 2 field theories with long-range and short-range exchange is analysed. It is shown that the crossover from the long-range regime to the short-range regime takes place due to the singular behaviour of three-loop and higher-order contributions to the beta function of the renormalization group, which controls the scaling behaviour of the long-range model. The critical exponents eta and nu are shown to be continuous functions of the parameter alpha , which characterises the power-like fall-off of the exchange interaction.

DOI
10.1088/0305-4470/23/5/024

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The relation between scaling regimes described by ( ϕ 2 ) 2 field theories with long-range and short-range exchange is analysed. It is shown that the crossover from the long-range regime to the short-range regime takes place due to the singular behaviour of three-loop and higher-order contributions to the beta function of the renormalization group, which controls the scaling behaviour of the long-range model. The critical exponents eta and nu are shown to be continuous functions of the parameter alpha , which characterises the power-like fall-off of the exchange interaction.
The relation between scalar field theories with short-range and long-range exchange or correlations is studied. It is shown to all orders in perturbation theory that the critical exponents of fields and composite operators are continuous functions of the parameters alpha ( xi ) characterising the decay rate of the long-range exchange (correlations) with power-like falloff 1/rd+2-2 alpha (1/rd-2 xi ). The scaling law (d-2 xi )v=2, where v is the correlation length exponent, as well as the Harris criteria (1970) are shown to be exact to all orders in perturbation theory. A discrepancy between two widely used approaches to the crossover problem is pointed out.
The critical exponents for two ${\ensuremath{\varphi}}^{3}$-field theories with long-range (LR) interactions decaying as 1/${R}^{d+\ensuremath{\sigma}}$, \ensuremath{\sigma}>0, are calculated to two-loop order in renormalized perturbation theory in d=3\ensuremath{\sigma}-\ensuremath{\epsilon}' dimensions. One is the continuum version of the p-state Potts model and the other is the scalar field theory with imaginary coupling that describes the Yang-Lee edge-singularity problem. The two crossover exponents for quadratic symmetry breaking discussed by Wallace and Young and in recent work by the present authors are also calculated in the first case. By means of renormalization-group recursion relations to one-loop order, it is shown that the LR fixed point is stable for all \ensuremath{\sigma}2 whenever ${\ensuremath{\eta}}_{\mathrm{SR}}$, the critical correlation-function exponent for short-range (SR) interactions, is negative, implying a discontinuity of critical exponents at \ensuremath{\sigma}=2. This is the case for the (p2)-state Potts model and for the Yang-Lee edge-singularity problem, and is in agreement with recent results by Chang and Sak for the Ising spin-glass problem. For the (p>2)-state Potts model there is an indication of a continuous crossover to SR behavior at \ensuremath{\sigma}=2-${\ensuremath{\eta}}_{\mathrm{SR}}$, with ${\ensuremath{\eta}}_{\mathrm{SR}}$>0. It is pointed out that a number of exact results [\ensuremath{\beta}=(1-\ensuremath{\epsilon}'/2\ensuremath{\sigma})\ensuremath{\sigma}\ensuremath{\nu}, \ensuremath{\sigma}^=(d-\ensuremath{\sigma})/(d+\ensuremath{\sigma}), in which \ensuremath{\sigma}^ is the Yang-Lee edge-singularity exponent, and ${\ensuremath{\nu}}^{\mathrm{\ensuremath{-}}1}$=(d-\ensuremath{\sigma})/2 for a scalar theory] may apply within the LR expansion depending on \ensuremath{\eta}=2-\ensuremath{\sigma} (shown here to hold at least to two-loop order) being exact, to all orders.
Tricriticdal behavior, in particular, critical exponents (η, γ, φ, ν, α), of classical N-component spin systems with a short-range or long-range interaction are studied in the dimension d (≪ 3 σ/2; σ: potential range parameter, 1 ≪ σ≤2) by means of the renormalization theory. The obtained expressions for all these critical exponents in the long-range interaction system and for the crossover exponent φ in the short-range one are new. The corrections of these tricritical exponents to the classical ones are shown to be smaller than those in the corresponding systems undergoing the second-order phase transition.
Random-field models are considered with long-range exchange, J(r) approximately 1/r d+ sigma , and/or long-range correlated random fields, (h(r)h(r')) approximately 1/ mod r-r' mod d- rho . A general scaling analysis is presented, in which the existence of three independent critical exponents is emphasized. Four different universality classes are identified, according to whether the fixed point controlling the critical behaviour is of short-range or long-range character for the exchange and/or the random-field correlations. The lines in the rho - sigma plane at which the long-range couplings first become relevant are expressed in terms of the exponents eta and eta of the corresponding short-range problem: long-range exchange is relevant for sigma <2- eta , long-range field correlations for rho >2 eta - eta . The scaling results are confirmed and amplified by explicit calculations of three independent exponents of the n-component spin model, correct to order 1/n. The results reveal errors in published expansions around the upper critical dimension.
The influence of long-range interactions decaying in d dimensions as 1/R^{d+σ} on the critical behavior of systems with Fisher's correlation-function exponent for short-range interactions η_{SR}<0, is re-examined. Such systems, typically described by Φ^{3}-field theories, are e.g. the Potts-model in the percolation-limit, the Edwards-Anderson spin-glass, and the Yang-Lee edge singularity. In contrast to preceding studies, it is shown by means of Wilson's momentum-shell renormalization-group recursion relations that the long-range interactions dominate as long as σ<2-η_{SR}. Exponents change continuously to their short-range values at the boundary of this region.
We explicitly compute the critical exponents associated with logarithmic corrections (the so-called hatted exponents) starting from the renormalization group equations and the mean field behavior for a wide class of models at the upper critical behavior (for short and long range φ n -theories) and below it.This allows us to check the scaling relations among these critical exponents obtained by analysing the complex singularities (Lee-Yang and Fisher zeroes) of these models.Moreover, we have obtained an explicit method to compute the ϙ exponent [defined by ξ ∼ L(log L) ϙ] and, finally, we have found a new derivation of the scaling law associated with it.