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arXiv:cond-mat/0112472 · Physical Review Letters

Boundary between Long-Range and Short-Range Critical Behavior in Systems with Algebraic Interactions

Erik Luijten, Henk W. J. Blöte

Published Jun 20, 2002• 116 citations
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Abstract

We investigate phase transitions of two-dimensional Ising models with power-law interactions, using an efficient Monte Carlo algorithm. For slow decay, the transition is of the mean-field type; for fast decay, it belongs to the short-range Ising universality class. We focus on the intermediate range, where the critical exponents depend continuously on the power law. We find that the boundary with short-range critical behavior occurs for interactions depending on distance r as r(-15/4). This answers a long-standing controversy between mutually conflicting renormalization-group analyses.

DOI
10.1103/physrevlett.89.025703

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We investigate phase transitions of two-dimensional Ising models with power-law interactions, using an efficient Monte Carlo algorithm. For slow decay, the transition is of the mean-field type; for fast decay, it belongs to the short-range Ising universality class. We focus on the intermediate range, where the critical exponents depend continuously on the power law. We find that the boundary with short-range critical behavior occurs for interactions depending on distance r as r(-15/4). This answers a long-standing controversy between mutually conflicting renormalization-group analyses.
We investigate the universality class of the finite-temperature phase transition of the two-dimensional Ising model with the algebraically decaying ferromagnetic long-range interaction, J_{ij}=|r[over ⃗]_{i}-r[over ⃗]_{j}|^{-(d+σ)}, where d (=2) is the dimension of the system and σ is the decay exponent, by means of the order-N cluster-algorithm Monte Carlo method. In particular, we focus on the upper and lower critical decay exponents, the boundaries between the mean-field-universality, intermediate, and short-range-universality regimes. At the critical decay exponents, it is found that the standard Binder ratio of magnetization at the critical temperature exhibits extremely slow convergence as a function of the system size. We propose more effective physical quantities, namely the combined Binder ratio and the self-combined Binder ratio, both of which cancel the leading finite-size corrections of the conventional Binder ratio. Utilizing these techniques, we clearly demonstrate that in two dimensions, the lower and upper critical decay exponents are σ=1 and 7/4, respectively, contrary to the recent Monte Carlo and renormalization-group studies [M. Picco, arXiv:1207.1018; T. Blanchard et al., Europhys. Lett. 101, 56003 (2013)EULEEJ0295-507510.1209/0295-5075/101/56003].
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.
We perform a numerical study of the long-range (LR) ferromagnetic Ising model with power law decaying interactions (J∝r{-d-σ}) on both a one-dimensional chain (d=1) and a square lattice (d=2). We use advanced cluster algorithms to avoid the critical slowing down. We first check the validity of the relation connecting the critical behavior of the LR model with parameters (d,σ) to that of a short-range (SR) model in an equivalent dimension D. We then study the critical behavior of the d=2 LR model close to the lower critical σ, uncovering that the spatial correlation function decays with two different power laws: The effect of the subdominant power law is much stronger than finite-size effects and actually makes the estimate of critical exponents very subtle. By including this subdominant power law, the numerical data are consistent with the standard renormalization group (RG) prediction by Sak [Phys. Rev. B 8, 281 (1973)], thus making not necessary (and unlikely, according to Occam's razor) the recent proposal by Picco [arXiv:1207.1018] of having a new set of RG fixed points in addition to the mean-field one and the SR one.
Extensive Monte Carlo simulations are employed in order to study the dynamic critical behavior of the one-dimensional Ising magnet, with algebraically decaying long-range interactions of the form 1rd+σ\frac{1}{r^{d+σ}}rd+σ1​, with σ=0.75σ=0.75σ=0.75. The critical temperature, as well as the critical exponents, is evaluated from the power-law behavior of suitable physical observables when the system is quenched from uncorrelated states, corresponding to infinite temperature, to the critical point. These results are compared with those obtained from the dynamic evolution of the system when it is suddenly annealed at the critical point from the ordered state. Also, the critical temperature in the infinite interaction limit is obtained by means of a finite-range scaling analysis of data measured with different cutoffs of the interaction range. All the estimated static critical exponents (γ/νγ/νγ/ν, β/νβ/νβ/ν, and 1/ν1/ν1/ν) are in good agreement with Renormalization Group (RG) predictions and previously reported numerical data obtained under equilibrium conditions. It is found that the dynamic exponent zzz is different for quenching and annealing experiments, most likely due to the influence of the Kosterlitz-Thouless transition occurring at relatively similar algebraic decay of the interactions with σ=1σ=1σ=1. However, for annealing experiments the measured exponent zzz is close to the RG predictions. On the other hand, the relevant exponents of the dynamic behavior (zzz and θθθ) are slightly different than the RG predictions, most likely due to the fact that they may depend on the especific dynamics used (Metropolis in the present paper).
The d-dimensional long-range Ising model, defined by spin-spin interactions decaying with the distance as the power 1/r^{d+s}, admits a second-order phase transition with continuously varying critical exponents. At s=s_{*}, the phase transition crosses over to the usual short-range universality class. The standard field-theoretic description of this family of models is strongly coupled at the crossover. We find a new description, which is instead weakly coupled near the crossover, and use it to compute critical exponents. The existence of two complementary UV descriptions of the same long-range fixed point provides a novel example of infrared duality.
Determining the threshold value σ∗σ_*σ∗​ that separates the short-range (SR) and long-range (LR) universality classes in phase transitions remains a controversial issue. While Sak's criterion, σ∗=2−ηSRσ_* = 2 - η_{\mathrm{SR}}σ∗​=2−ηSR​, has been widely accepted, recent studies of two-dimensional (2D) models with long-range interactions have challenged it. In this work, we focus on the crossover between LR and SR criticality in several classical 2D statistical models, including the XY, Heisenberg, percolation, and Ising models, whose interactions decay as 1/r2+σ1/r^{2+σ}1/r2+σ. Our previous simulations for the XY, Heisenberg, and percolation models consistently indicate a universal boundary at σ∗=2σ_* = 2σ∗​=2. Here, we complete the picture by performing large-scale Monte Carlo simulations of the 2D LR-Ising model, reaching lattice sizes up to L=8192L = 8192L=8192. By analyzing the Fortuin-Kasteleyn critical polynomial RpR_pRp​, the Binder ratio QmQ_mQm​, and the anomalous dimension ηηη, we obtain convergent and self-consistent evidence that the universality class already changes sharply at σ=2σ= 2σ=2. Taken together, these results establish a unified scenario for LR interacting systems: across all studied models, the crossover from LR to SR universality occurs at σ∗=2σ_* = 2σ∗​=2.