#paperOfTheDay "Relations between short-range and long-range Ising models" from 2014.
The basic version of the Ising model in
#statistical #physics is a lattice where every site contains a binary variable, a "spin" that can point up or down. There is a nearest-neighbour interaction which energetically prefers neighbouring spins to point in the same direction. Then, there are "long range" versions, where the interaction also takes into account spins at larger distance, with a weighting factor that decays with some power law with exponent sigma (where sigma=2 reproduces the conventional short-range model). In this class of models, there are thus two parameters: The dimension d, and the parameter sigma.
The present paper is mostly a numerical Monte Carlo study of various such systems at different d and sigma. The guiding question is whether instead of two, there is actually only one parameter. Phrased differently: Given some d and sigma, can I find some other D such that the short-range (sigma=2) model at this D is equivalent to the long-range one at d? It is intuitively plausible that this works close to the interface between long-range and short-range models, making it slightly long-range is mostly the same as slightly changing dimension. However, the authors demonstrate that such relations are in fact only an approximation, and fail for generic values of sigma and d.
A second finding concerns the structure of the correlation functions of certain long-range models, which decay according to a power law (as expected from the
#renormalization group), but where the leading correction is another power law (and not exponentially small).
https://journals.aps.org/pre/abstract/10.1103/PhysRevE.89.062120 https://arxiv.org/abs/1401.6805 
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