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Journal of Physics A Mathematical and Theoretical

Reply to “Comment on ‘Fluctuation-dominated phase ordering at a mixed order transition’”

Mustansir Barma, Satya N. Majumdar, David Mukamel

Published Dec 24, 2020• 0 citations
Publisher

Abstract

Abstract Godrèche, in Comment on ‘Fluctuation dominated phase ordering at a mixed order transition’ [2021 J. Phys. A: Math. Theor. 54 038001], has commented on our recent paper Fluctuation dominated phase ordering at a mixed order transition (2019 J. Phys. A: Math. Theor. 52 254001). This comment concerns the prefactor of the cusp-like small-argument singularity of the scaled spin-spin correlation function at criticality. We remark that the approach used in our paper is adequate for computing the cusp exponent, which is what is really needed to establish fluctuation-dominated phase ordering. Computing the precise value of the prefactor of the cusp singularity is irrelevant for this purpose, or for the physics behind the relation between the mixed order phase transition and the fluctuation-dominated phase ordering—the understanding of which was the main purpose of our paper.

DOI
10.1088/1751-8121/abd34c

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Fluctuation-dominated phase ordering at a mixed order transition

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Abstract Godrèche, in Comment on ‘Fluctuation dominated phase ordering at a mixed order transition’ [2021 J. Phys. A: Math. Theor. 54 038001], has commented on our recent paper Fluctuation dominated phase ordering at a mixed order transition (2019 J. Phys. A: Math. Theor. 52 254001). This comment concerns the prefactor of the cusp-like small-argument singularity of the scaled spin-spin correlation function at criticality. We remark that the approach used in our paper is adequate for computing the cusp exponent, which is what is really needed to establish fluctuation-dominated phase ordering. Computing the precise value of the prefactor of the cusp singularity is irrelevant for this purpose, or for the physics behind the relation between the mixed order phase transition and the fluctuation-dominated phase ordering—the understanding of which was the main purpose of our paper.
Abstract Mixed order transitions are those which show a discontinuity of the order parameter as well as a divergent correlation length at the transition (critical) point. We show that the behavior of the order parameter correlation function along the transition line of mixed order transitions can change from normal critical behavior with power law decay, to fluctuation-dominated phase ordering as a parameter is varied. The defining features of fluctuation-dominated order are anomalous fluctuations which remain large in the thermodynamic limit, and correlation functions which approach a finite value through a cusp singularity as the separation scaled by the system size approaches zero. We demonstrate that fluctuation-dominated order sets in along a portion of the transition line of an Ising model with truncated long-range interactions which was earlier shown to exhibit mixed order transitions. We also argue that this connection may hold more generally.
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 J. Phys. A 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.
We have recently shown that in nonequilibrium spin systems at criticality the limit ${X}^{\ensuremath{\infty}}$ of the fluctuation-dissipation ratio X(t,tw)X(t,{t}_{w})X(t,tw​) for t⪢tw⪢1t⪢{t}_{w}⪢1t⪢tw​⪢1 can be measured using observables such as magnetization or energy [Phys. Rev. E 68, 016116 (2003)]. Pleimling argues in a Comment [preceding paper, Phys. Rev. 70, 018101 (2004)] on our paper that for such observables correlation and response functions are dominated by one-time quantities dependent only on ttt, and are therefore not suitable for a determination of ${X}^{\ensuremath{\infty}}$. Using standard scaling forms of correlation and response functions, as used by Pleimling, we show that our data do have a genuine two-time dependence and allow X(t,tw)X(t,{t}_{w})X(t,tw​) and ${X}^{\ensuremath{\infty}}$ to be measured, so that Pleimling's criticisms are easily refuted. We also compare with predictions from renormalization-group calculations, which are consistent with our numerical observation of a fluctuation-dissipation plot for the magnetization that is very close to a straight line. A key point remains that coherent observables make measurements of ${X}^{\ensuremath{\infty}}$ easier than the traditionally used incoherent ones, producing fluctuation-dissipation plots whose slope is close to ${X}^{\ensuremath{\infty}}$ over a much larger range.
Abstract In their comment, Angelini et al object to the conclusion of (2019 J. Phys. A: Math. Theor. 52 445002), where we show that in (2013 Phys. Rev. B 87 134201) the exponent ν has been obtained by applying a mathematical relation in a regime where this relation is not valid. We observe that the criticism above on the mathematical validity of such relation has not been addressed in the comment. Our criticism thus remains valid, and disproves the conclusions of the comment. This constitutes the main point of this reply. In addition, we provide a point-by-point response and discussion of Angelini et al ’s claims. First, Angelini et al claim that the prediction 2 1/ ν = 1 of (2019 J. Phys. A: Math. Theor . 52 445002) is incorrect, because it results from the relation λ max = 2 1/ ν between the largest eigenvalue of the linearized renormalization-group (RG) transformation and ν , which cannot be applied to the ensemble renormalization group (ERG) method, given that for the ERG λ max = 1. However, the feature λ max = 1 is specific to the ERG transformation, and it does not give any grounds for questioning the validity of the general relation λ max = 2 1/ ν specifically for the ERG transformation. Second, Angelini et al claim that ν should be extracted from an early RG regime (A), as opposed to the asymptotic regime (B) used to estimate ν in (2019 J. Phys. A: Math. Theor . 52 445002), and that (B) is dominated by finite-size effects. However, (A) is a small-wavelength, non-critical regime, which cannot characterize the critical exponent ν related to the divergence of the correlation length. Also, the fact that (B) involves finite-size effects is a feature specific to the ERG, and gives no rationale for extracting ν from (A). Finally, we refute the remaining claims made by Angelini et al As a result of our analysis, we stand by our assertion that the ERG method yields a prediction given by 2 1/ ν = 1.
We reply to the comment on our published paper `` Universal Fluctuations in Correlated Systems'',Phys. Rev. Lett. Vol; 84, p3744 (2000), by B. Zheng and S. Trimper, cond-mat/0109003. We argue that their results confirm our conjecture, that the probability distribution for order parameter fluctuations in the 2D and 3D Ising models at a temperature T∗(L)T^{\ast}(L)T∗(L) slightly below TCT_CTC​ for the infinite, system approximates the universal functional form of the 2D-XY model in its low temperature phase. We discuss the relevance of the temperature interval TC−T∗T_C-T^{\ast}TC​−T∗, considered to be large by Zheng and Trimper and explain why the observed phenomena is a critical phenomena.
We reply to a recent comment by Diehl and Shpot ( 2001 J. Phys. A: Math. Gen. 34 9101 ) criticizing our paper (Albuquerque L C and Leite M M 2001 J. Phys. A: Math. Gen. 34 L327 ). We show that the approximation we use for evaluating higher loop integrals is consistent with homogeneity. A new renormalization group approach is presented in order to compare the two methods with high-precision numerical data concerning the uniaxial case. We stress that isotropic behaviour cannot be obtained from anisotropic behaviour.