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arXiv:cond-mat/0109107

Reply to Comment on "Universal Fluctuations in Correlated Systems", by B. Zheng and S. Trimper, cond-mat/0109003

S. T. Bramwell, Kim Christensen, Jean-Yves Fortin, P. C. W. Holdsworth, Henrik Jeldtoft Jensen, Stefano Lise, Juan M. López, Mario Nicodemi, Jean-François Pinton, Mauro Sellitto

Published Sep 6, 2001• 0 citations
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Abstract

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 $T^{\ast}(L)$ slightly below $T_C$ 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 $T_C-T^{\ast}$, considered to be large by Zheng and Trimper and explain why the observed phenomena is a critical phenomena.

DOI
10.48550/arxiv.cond-mat/0109107

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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 propose an interpretation of previous experiments and numerical experiments showing that, for a large class of systems, distributions of global quantities are similar to a distribution originally obtained for the magnetization in the 2D-XY model (Bramwell S. T. et al. , Nature , 396 (1998) 512). This approach, developed for the Ising model, is based on previous numerical observations (Clusel M. et al. , Phys. Rev. E , 70 (2004) 046112). We obtain an effective action using a perturbative method, which successfully describes the order parameter fluctuations near the phase transition. This leads to a direct link between the D-dimensional Ising model and the XY model in the same dimension, which appears to be a generic feature of many equilibrium critical systems and which is at the heart of the above observations.
This Comment is dedicated to the investigation of many-body localization in a quantum Ising model with long-range power-law interactions ${r}^{\ensuremath{-}\ensuremath{\alpha}}$, relevant for a variety of systems ranging from electrons in Anderson insulators to spin excitations in chains of cold atoms. It has earlier been argued [arXiv:cond-mat/0611387 (2005); Phys. Rev. B 91, 094202 (2015)] that this model obeys the dimensional constraint suggesting the delocalization of all finite-temperature states in the thermodynamic limit for $\ensuremath{\alpha}\ensuremath{\le}2d$ in a ddd-dimensional system. This expectation conflicts with the recent numerical studies of the specific interacting spin model of Li et al. [Phys. Rev. A 94, 063625 (2016)]. To resolve this controversy we reexamine the model of Li et al. [Phys. Rev. A 94, 063625 (2016)] and demonstrate that the infinite-temperature states there obey the dimensional constraint. The earlier developed scaling theory for the critical system size required for delocalization is extended to small exponents $0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}d$. The disagreements between the two works are explained by the nonstandard selection of investigated states in the ordered phase in the work of Li et al. [Phys. Rev. A 94, 063625 (2016)].
Reply to Comment on 'A new method to calculate the spin-glass order parameter of the two-dimensional ±J\pm J±J Ising model'
In recent works [S. T. Bramwell, P. C. W. Holdsworth, and J.-F. Pinton, Nature (London) 396, 552 (1998); S. T. Bramwell et al., Phys. Rev. Lett. 84, 3744 (2000)], a generalized universality has been proposed, linking phenomena as dissimilar as two-dimensional (2D) magnetism and turbulence. To test these ideas, we performed Monte Carlo simulations of the 2D XY model. We found that the shape of the probability distribution function for the magnetization M is non-Gaussian and independent of the system size-in the range of the lattice sizes studied-below the Kosterlitz-Thoules temperature. However, our results suggest that in the full 2D XY model the shape of these distributions has a slight dependence on temperature-for finite volume-below the lattice-shifted critical temperature T*(L). This behavior can be explained by using renormalization group arguments and an extended finite-size scaling analysis, and by the existence of bounds for M.
Order parameter fluctuations for the two-dimensional Ising model in the region of the critical temperature are presented. A locus of temperatures T(*) (L) and a locus of magnetic fields B(*) (L) are identified, for which the probability density function is similar to that for the two-dimensional XY model in the spin wave approximation. The characteristics of the fluctuations along these points are largely independent of universality class. We show that the largest range of fluctuations relative to the variance of the distribution occurs along these loci of points, rather than at the critical temperature itself and we discuss this observation in terms of intermittency. Our motivation is the identification of a generic form for fluctuations in correlated systems in accordance with recent experimental and numerical observations. We conclude that a universality-class-dependent form for the fluctuations is a particularity of critical phenomena related to the change in symmetry at a phase transition.