Comment on “Nonlocal quartic interactions and universality classes in perovskite manganites”

Abstract

In a recent paper [Phys. Rev. E 92, 012123 (2015)] a modified d-dimensional Φ 4 model was investigated that differs from the standard one in that the Φ 4 term was replaced by a nonlocal one with a potential u(x−x ′ ) that depends on a parameter σ and decays exponentially as |x−x ′ | → ∞ on a scale |m| −1 < ∞. The authors claim the upper critical dimension of this model to be dσ = 4 + 2σ. Performing a one-loop calculation they arrive at expansions in powers of ǫσ = dσ − d for critical exponents such as η and related ones to O(ǫσ) whose O(ǫσ) coefficients depend on σ and the ratio w = m 2 /Λ 2 , where Λ is the uv cutoff. It is shown that these claims are unfounded and based on misjudgments and an ill-conceived renormalization group (RG) calculation. where the potential is specified by its Fourier transformwith a "screening parameter" m 2 > 0. From Eq. (2) one can read off that u(x) falls off exponentially on the scale ξ m = |m| −1 as |x| → ∞ [2]. Hence it is shortranged on the scale of the correlation length ξ(T ) sufficiently close to the critical temperature T c . The reason is that ξ m is temperature-independent or has at most a weak temperature dependence when the interaction is considered as an effective interaction obtained upon coarse graining to a length scale 2π/Λ (> a) on which a continuum approximation is acceptable. Therefore ξ m remains bounded whereas ξ(T ) → ∞ as T → T c , so that lim T →Tc ξ m /ξ → 0 [3]. What matters for the asymptotic critical behavior is just whether or not the interaction is short-ranged on the scale of ξ(T ) as T → T c . This is asymptotically always the case even if ξ m is much larger than the lattice constant a, i.e., if u(x) is long-ranged on the scale of a. Therefore the model must belong to the standard universality class of the d-dimensional n-vector model with short-range interactions represented by the usual Φ 4 model with a local quartic term. The authors' results strongly disagree with this. Performing a one-loop calculation, they conclude that the upper critical dimension of the model is d σ = 4 + 2σ, and they compute some of the critical exponents of the model by an expansion in ǫ σ = d σ − d to first order. The O(ǫ σ ) coefficients of their series expansions depend on σ and m 2 /Λ 2 . If this were true, the critical exponents themselves would have these properties, and hence depend on microscopic details such as m 2 /Λ 2 and be nonuniversal. Obvious serious discrepancies with the modern theory of critical behavior (see, e.g., [4,5] and references therein) would exist. Aside from a gross violation of universality, one would have the following problems: For σ > 0, the asymptotic critical behavior of the model in 4 < d < d σ dimensions would be inconsistent with meanfield theory. Likewise, irrespective of whether σ > 0 or ≤ 0, the critical behavior of the model would differ in d * < d < 4 dimensions from that of the standard Φ 4 model., where d * is the lower critical dimension (= 1 or 2 for n = 1 and n > 1, respectively). However, the authors' claims and their conse...