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 ${\mathrm{\ensuremath{\Phi}}}^{4}$ model was investigated that differs from the standard one in that the ${\mathrm{\ensuremath{\Phi}}}^{4}$ term was replaced by a nonlocal one with a potential $u(\mathbit{x}\ensuremath{-}{\mathbit{x}}^{\ensuremath{'}})$ that depends on a parameter $\ensuremath{\sigma}$ and decays exponentially as $|\mathbit{x}\ensuremath{-}{\mathbit{x}}^{\ensuremath{'}}|\ensuremath{\rightarrow}\ensuremath{\infty}$ on a scale ${|m|}^{\ensuremath{-}1}<\ensuremath{\infty}$. The authors claim the upper critical dimension of this model to be ${d}_{\ensuremath{\sigma}}=4+2\ensuremath{\sigma}$. Performing a one-loop calculation, they arrive at expansions in powers of ${\ensuremath{\epsilon}}_{\ensuremath{\sigma}}={d}_{\ensuremath{\sigma}}\ensuremath{-}d$ for critical exponents such as $\ensuremath{\eta}$ and related ones to $O({\ensuremath{\epsilon}}_{\ensuremath{\sigma}})$ whose $O({\ensuremath{\epsilon}}_{\ensuremath{\sigma}})$ coefficients depend on $\ensuremath{\sigma}$ and the ratio $w={m}^{2}/{\mathrm{\ensuremath{\Lambda}}}^{2}$, where $\mathrm{\ensuremath{\Lambda}}$ is the uv cutoff. It is shown that these claims are unfounded and based on misjudgments and an ill-conceived renormalization-group calculation.