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Physical review. B, Condensed matter

Comment on “Effect of long-range interactions on the critical behavior of the continuous Ising model”

Silvano Romano

Published Jul 1, 2000• 1 citation
Publisher

Abstract

Available mathematical results entail the absence of a proper ordering transition taking place at finite temperature in the case $d=1,$ $\ensuremath{\sigma}=1.1,$ where the authors of the article commented on find evidence of such a transition. Moreover, in contrast to the findings of the authors of the article, the case $d=2,$ $\ensuremath{\sigma}=0$ entails a ground-state energy per particle diverging logarithmically with increasing sample size, hence an infinite transition temperature.

DOI
10.1103/physrevb.62.1464

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Available mathematical results entail the absence of a proper ordering transition taking place at finite temperature in the case d=1,d=1,d=1, \ensuremathσ=1.1,\ensuremath{\sigma}=1.1,\ensuremathσ=1.1, where the authors of the article commented on find evidence of such a transition. Moreover, in contrast to the findings of the authors of the article, the case d=2,d=2,d=2, \ensuremathσ=0\ensuremath{\sigma}=0\ensuremathσ=0 entails a ground-state energy per particle diverging logarithmically with increasing sample size, hence an infinite transition temperature.
Critical behavior of the one- and two-dimensional general continuous Ising models with long-range ferromagnetic interactions decaying as ${1/r}^{d+\ensuremath{\sigma}}$ is studied using a histogram Monte Carlo technique. A continuous Ising model means that a spin can take any value between $\ensuremath{-}1$ and 1. It is found that the system exhibits a second-order phase transition with nonstandard critical exponents which depend on $\ensuremath{\sigma}.$ Results for various values of $\ensuremath{\sigma}$ will be shown and compared to predictions from renormalization-group theory. Though there is an agreement with the overall tendency predicted, there are several fundamental differences. Discussion is given.
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)].
Recently, it has been found that an effective long-range interaction is realized among local bistable variables (spins) in systems where the elastic interaction causes ordering of the spins. In such systems, we generally expect both long-range and short-range interactions to exist. In the short-range Ising model, the correlation length diverges at the critical point. In contrast, in the long-range interacting model the spin configuration is always uniform and the correlation length is zero. As long as a system has nonzero long-range interactions, it shows criticality in the mean-field universality class, and the spin configuration is uniform beyond a certain scale. Here we study the crossover from the pure short-range interacting model to the long-range interacting model. We investigate the infinite-range model (Husimi-Temperley model) as a prototype of this competition, and we study how the critical temperature changes as a function of the strength of the long-range interaction. This model can also be interpreted as an approximation for the Ising model on a small-world network. We derive a formula for the critical temperature as a function of the strength of the long-range interaction. We also propose a finite-size scaling form for the spin correlation length at the critical point, which is finite as long as the long-range interaction is included, though it diverges in the limit of the pure short-range model. These properties are confirmed by extensive Monte Carlo simulations.
The statement that any phase transition is related to the appearance or disappearance of long-range spatial correlations and precludes a finite transition temperature in one-dimensional (1D) systems is rooted in condensed matter theory. In this paper, we demonstrate that the 1D Ising model with short-range exchange interactions exhibits a second-order phase transition at a finite temperature relying on the proper choice of the order parameter. We combined analytical calculations and high-precision entropic sampling simulations and chose a slightly different order parameter, namely, the modulus of the magnetization. Notably, we detected a phase transition with a corresponding critical temperature around 15 K, which is in excellent agreement with experimental results. Our study indicates that the incorrect choice of the order parameter may mask phase transitions in 1D systems.
We have considered classical spin systems, consisting of n-component unit vectors (n=2,3), associated with a one-dimensional lattice {uk, k ∈ Z}, and interacting via translationally and rotationally invariant pair potentials, of the long-range and ferromagnetic form [Formula: see text] here ε is a positive constant setting energy and temperature scales (i.e. T*=k B T/ε). Available theorems entail the existence of an ordering transition at finite temperature when 0<σ<1, and its absence for σ≥1. When σ=1, spin-wave arguments and previous simulation results suggest the existence of a transition to a low-temperature phase with slow decay of correlations and infinite susceptibility, i.e. a Berezhinskiǐ–Kosterlitz–Thouless-like transition. We have reexamined this case by a more detailed simulation study, and especially addressed the sample-size dependence of calculated observables: simulation results suggest the existence of such a transition, and the proposed estimates for the transition temperatures are Θ2=0.775±0.025 and Θ3=0.500±0.025, when n=2 and n=3, respectively.
The ordering of charges on half-filled hypercubic lattices is investigated numerically, where electroneutrality is ensured by background charges. This system is equivalent to the s=1/2s=1/2s=1/2 Ising lattice model with antiferromagnetic 1/r1/r1/r interaction. The temperature dependences of specific heat, mean staggered occupation, and of a generalized susceptibility indicate continuous order-disorder phase transitions at finite temperatures in two- and three-dimensional systems. In contrast, the susceptibility of the one-dimensional system exhibits singular behavior at vanishing temperature. For the two- and three-dimensional cases, the critical exponents are obtained by means of a finite-size scaling analysis. Their values are consistent with those of the Ising model with short-range interaction, and they imply that the studied model cannot belong to any other known universality class. Samples of up to 1400, 1122{112}^{2}1122, and 223{22}^{3}223 sites are considered for dimensions 1 to 3, respectively.