Comment on "Nonlocal quartic interactions and universality classes in perovskite manganites"

TL;DR

This paper critically examines a recent study on a nonlocal $ ext{Phi}^4$ model, arguing that its claims about critical dimensions and exponents are incorrect due to flawed renormalization group analysis.

Contribution

It provides a detailed critique of the previous work, clarifying the inaccuracies in their theoretical approach and calculations.

Findings

The claimed upper critical dimension $d_\sigma=4+2\sigma$ is incorrect.

One-loop calculations in the criticized paper are based on flawed assumptions.

The dependence of critical exponents on $\sigma$ and $w$ is unjustified.

Abstract

In a recent paper [Phys. Rev. E \textbf{92}, 012123 (2015)] a modified $d$-dimensional $\Phi^4$ model was investigated which differs from the standard one in that the $\Phi^4$ term was replaced by a nonlocal one with a potential $u(\bm{x}-\bm{x}')$ that depends on a parameter $\sigma$ and decays exponentially as $|\bm{x}-\bm{x}'|\to\infty$ on a scale $|m|^{-1}<\infty$. The authors claim the upper critical dimension of this model to be $d_\sigma=4+2\sigma$. Performing a one-loop calculation they arrive at expansions in powers of $\epsilon_\sigma=d_\sigma-d$ for critical exponents such as $\eta$ and related ones to $O(\epsilon_\sigma)$ whose $O(\epsilon_\sigma)$ coefficients depend on $\sigma$ and the ratio $w=m^2/\Lambda^2$, where $\Lambda$ is the UV cutoff. We show that these claims are unfounded and based on misjudgments and an ill-conceived renormalization group calculation.