Finite-size scaling above the upper critical dimension in Ising models with long-range interactions

Abstract

The correlation length plays a pivotal role in finite-size scaling and hyperscaling at continuous phase transitions. Below the upper critical dimension, where the correlation length is proportional to the system length, both finite-size scaling and hyperscaling take conventional forms. Above the upper critical dimension these forms break down and a new scaling scenario appears. Here we investigate this scaling behaviour by simulating one-dimensional Ising ferromagnets with long-range interactions. We show that the correlation length scales as a non-trivial power of the linear system size and investigate the scaling forms. For interactions of sufficiently long range, the disparity between the correlation length and the system length can be made arbitrarily large, while maintaining the new scaling scenarios. We also investigate the behavior of the correlation function above the upper critical dimension and the modifications imposed by the new scaling scenario onto the associated Fisher relation.

Cluster Monte Carlo and dynamical scaling for long-range interactions

“…Naive approaches exhibit unfavorable O(N 2 ) scaling with the number of spins N , but it is possible to formulate cluster-update algorithms that combine O(N ln N ) or even O(N ) scaling of the run-time per sweep with the additional benefit of a reduced critical slowing down of systems close to continuous phase transitions. The scaling of autocorrelation times in the mean-field regime 0 ≤ σ ≤ 1/2 of the 1D power-law Ising model is explained in terms of the modified QFSS approach to finite-size scaling [39,40]. We introduced a single-cluster algorithm based on the generalized Fortuin-Kasteleyn representation (9) that is the only known algorithm with strictly linear scaling of run times.…”

“…The corresponding Ewald summation is discussed, e.g., in Refs. [40,45]. Another aspect is the problem of measuring the energy for the long-range interactions, a task that in itself has O(N 2 ) scaling in the straightforward approach.…”

“…We hence expect a scaling of |C| ∼ ξ γ/ν and thus |C| /N ∼ L ϙγ/ν−d . For the mean-field cases σ = 0.1 and σ = 0.2 the values γ/ν = σ [40] and ϙ = 1/2σ imply ϙγ/ν − d = −1/2. What is more, the long-range model has the pecularity that the value of γ/ν = 2 − η = σ does not acquire any corrections beyond mean-field even for 1/2 ≤ σ ≤ 1, such that |C| /N scales as L σ−1 there.…”

“…By analogy to the mean-field limit, we conjecture that z SW int = σ/2 for multi-cluster and z Wolff int = 0 for single-cluster updates in the same regime. Due to the particular scaling of the correlation length with system size according to ξ ∼ L ϙ above the upper critical dimension [39], where ϙ = d/d c and d c = 4 for short-range models and ϙ = d/2σ for the interactions (3) [40], we expect the following finite-size scaling of the autocorrelation times,…”

“…Our simulations were performed for systems with periodic boundary conditions and for the interaction-range exponents σ = 0.1 and σ = 0.2 in the mean-field range σ < 1/2 as well as a number of values σ ≥ 0.5 in the non-trivial long-range regime. We used an Ewald summed form of the interaction (3) [40] and performed simulations at the previously determined critical temperatures T c = 21.0013 for σ = 0.1, T c = 10.8419 for σ = 0.2, T c = 4.36395 for σ = 0.5, T c = 3.54886 for σ = 0.6, T c = 2.93061 for σ = 0.7, T c = 2.43267 for σ = 0.8, and T c = 2.00144 for σ = 0.9 [41]. To provide autocorrelation times on a common time scale for all algorithms it is convenient to consider updates such that, on average, each spin is touched once (one sweep).…”