Relations between Short Range and Long Range Ising models

Maria Chiara Angelini

IPhT

Mon, Apr. 28th 2014, 14:00-15:00

Salle Claude Itzykson, Bât. 774, Orme des Merisiers

Long Range (LR) models in d dimensions with interactions decaying with a given power law exponent sigma, can have properties similar to those of short range (SR) models in D dimensions (with d different from D). Thus the effective dimension D can be changed just varying the parameter sigma, a very useful property for numerical simulations. For this reason often LR models (both ferromagnetic and disordered ones) have been used to extract properties of the analogous SR effective models, however it is not clear whether this operation is really justified. par We perform a numerical study of the LR ferromagnetic Ising model both on a one-dimensional chain and on a square lattice. We first check the validity of the relation connecting the critical behavior of the LR model with a given sigma in d dimensions to that of a SR model in an equivalent dimension D, emphasizing its limitations. We then study the critical behaviour of the d=2 LR model close to the lower critical exponent for which the LR model should become equivalent to the SR model in D=d dimensions. We uncover 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. That is the reason why there is not a consensus on the behaviour of the model in this region, after more than 40 years of works.

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MISGUICH