Comment on `Fluctuation-dominated phase ordering at a mixed order transition'

TL;DR

This paper critically reviews a recent study on a one-dimensional spin system with long-range interactions, emphasizing the importance of accounting for non-stationarity in renewal processes to correctly determine correlation functions.

Contribution

It clarifies the impact of non-stationarity in renewal processes on the analysis of long-range spin systems, correcting previous misconceptions.

Findings

Non-stationary renewal processes affect correlation function calculations.

Discarding non-stationarity leads to incorrect critical behavior analysis.

Properly accounting for non-stationarity yields accurate correlation functions.

Abstract

Renewal processes generated by a power-law distribution of intervals with tail index less than unity are genuinely non-stationary. This issue is illustrated by a critical review of the recent paper by Barma, Majumdar and Mukamel 2019 \textit{J.~Phys.~A} {\bf 52} 254001, devoted to the investigation of the properties of a specific one-dimensional equilibrium spin system with long-range interactions. We explain why discarding the non-stationarity of the process underlying the model leads to an incorrect expression of the critical spin-spin correlation function, even when the system, subjected to periodic boundary conditions, is translation invariant.