The critical behavior of an $n$-component system in $d$ dimensions with long-range (LR) interactions decaying as $\frac{1}{{|\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}|}^{d+\ensuremath{\sigma}}}$, for $\ensuremath{\sigma}>0$, is reexamined near the crossover to short-range (SR) behavior, where $\ensuremath{\sigma}=2$, by means of renormalized perturbation theory in ${\ensuremath{\epsilon}}^{\ensuremath{'}}=2\ensuremath{\sigma}\ensuremath{-}d$. It is pointed out that $2\ensuremath{-}\ensuremath{\sigma}$ should not be viewed as an expansion parameter, and consequently the critical exponents turn out to be discontinuous at $\ensuremath{\sigma}=2$ without the intermediate region of weakly LR interactions found earlier by Sak. For any $\ensuremath{\sigma}<2$ it is shown that (i) the LR expansion is stable to weak SR perturbations and (ii) the SR expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$ breaks down under a weak LR perturbation.
The critical behavior of an n-component system in d dimensions with long-range (LR) interactions decaying as ∣r\ensuremath→∣d+\ensuremathσ1, for \ensuremathσ>0, is reexamined near the crossover to short-range (SR) behavior, where \ensuremathσ=2, by means of renormalized perturbation theory in \ensuremathϵ\ensuremath′=2\ensuremathσ\ensuremath−d. It is pointed out that 2\ensuremath−\ensuremathσ should not be viewed as an expansion parameter, and consequently the critical exponents turn out to be discontinuous at \ensuremathσ=2 without the intermediate region of weakly LR interactions found earlier by Sak. For any \ensuremathσ<2 it is shown that (i) the LR expansion is stable to weak SR perturbations and (ii) the SR expansion in \ensuremathϵ=4\ensuremath−d breaks down under a weak LR perturbation.
The establishment of the Wilson-Fisher fixed point (WFP) for O(n) spin models in d=4−ε dimensions stands as a cornerstone of the renormalization group (RG) theory for critical phenomena. However, when long-range (LR) interactions, algebraically decaying as ∝1/rd+σ, are introduced, the fate of the short-range WFP (SR-WFP) has remained a subject of intense debate since the 1970s. We employ two complementary techniques -- the standard field-theoretic RG and a perturbative bootstrap scheme, and perform the ε-expansion calculations up to the two-loop level. We show that, as long as σ<2, the SR-WFP becomes unstable and a stable LR-WFP emerges, and, in the non-classical regime with d/2<σ<2, the critical exponents, including the anomalous dimension, are functions of ε, δ=2−σ and n, which reduce to the exact results in the limiting cases ε→0, δ→0 or n→∞. Our (4−ε)-expansion calculations support the scenario that the threshold between the LR- and SR-WFP occurs strictly at σ∗=2, well consistent with the recent high-precision numerical study while different from the widely accepted Sak's criterion.
The influence of long-range interactions decaying in d dimensions as ${1/R}^{d+\ensuremath{\sigma}}$ on the critical behavior of systems with negativeFisher's correlation-function exponent for short-range interactions, ${\ensuremath{\eta}}_{\mathrm{SR}}<0,$ is reexamined. Such systems, typically described by ${\ensuremath{\varphi}}^{3}$-field theories, are, e.g., the Potts model in the percolation limit, the Edwards-Anderson spin-glass model, 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 $\ensuremath{\sigma}<2\ensuremath{-}{\ensuremath{\eta}}_{\mathrm{SR}}.$ Exponents change continuously to their short-range values at the boundary of this region.
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.
The critical exponents r and a are derived in powers of 1/n to first order for the n- vector model (or n-component system) with long-range interactions.The results show that the exponents have discontinuity with respect to the potential-range " at <T=2 for d>2 (d: dimensionality).It is also shown that they depend upon d and "• rather independently.The connection of the present results with the ones derived in E-expansion is briefly discussed.The 1/n-expansion of other critical exponents such as {J, iJ, "' and 71 are also discussed.§ 1. Introduction and summary Critical exponents characteristic of critical phenomena depend upon the dimensionality d, symmetry (or the degrees of freedom, n) and potential-range s (or effective potential-range rJ).Among these three parameters, the d-and n- dependences of critical exponents have been investigated 1 l-4 l mainly by using an expansion method combined with renormalization group techniques. 5> However, the potential-range s (or rJ) is also a very important parameter for critical phenomena, in particular for the purpose of discussing critical behavior in systems of restricted dimensionality, in which there would occur no phase transition otherwise.6>One of our main purposes in this paper is to investigate how the critical exponents depend upon the three parameters d, n and (J, particularly to clarify the mutual relationship between dimensionality d and potential-range s (or rJ).For this purpose, the critical exponents for general d and s (or rJ) are calculated, (I) by applying 1/n-expansion 7 > to n-vector model with potential-range s, and (II) by extending s-expansion 1 >• 2 > to a generalized effective Hamiltonian of Wilson-type 8 >• 6 l (essentially equivalent to the n-vector model with long-range interactions).Following Wilson 2 l and Fisher,ll for any real value d(>1), we make use of the formulae (2n)-a S d"-kf(k~) =Ka f"ka-~(k~)dk,Downloaded from https://academic.oup.com/ptp/article/49/2/424/1856924 by guest on 12 August 2023Critical Exponents for Long-Range Interactions.I with Ka = 2-<"-t>n-"12 [T(d/2) J-t.The effective potential-range (J is defined by rJ=s for O<s<2 and rJ=2 for s>2 through the interaction
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.
The 6−ε Expansion for Long-Range Lee--Yang and Percolation Criticality
The crossover from long-range (LR) to short-range (SR) criticality in percolation has remained unsettled because previous renormalization-group (RG) analysis within the ε′=3σ−d expansion fixes the anomalous dimension at η=2−σ, whereas SR percolation has ηSR<0 near d=6. Sak's matching condition then places the crossover above σ=2, outside the regime in which the LR interaction dominates. In spatial dimension d=6−ε, we formulate a perturbative expansion for the LR ϕ3 field theory and perform a one-loop RG analysis throughout the perturbatively accessible nonclassical regime 0<δ<ε/3, where δ=2−σ. We derive the one-loop corrections to the critical exponents η and ν, which acquire nontrivial dependence on ε and δ. They reduce to their mean-field values at the LR upper critical line and continuously recover the SR 6−ε results as σ→2. These results support a crossover threshold σ∗=2 and remove the apparent discontinuity of η between the LR and SR values within this framework. The same approach also yields the anomalous and edge exponents of the LR Lee--Yang universality class and q-state Potts universality classes with q<2.