Critical Exponents for Long-Range Interactions

Abstract

Long-range components of the interaction in statistical mechanical systems may affect the critical behavior, raising the system's 'effective dimension'. Presented here are explicit implications to this effect of a collection of rigorous results on the critical exponents in ferromagnetic models with one-component lsing (and more generally Griffiths-Simon class) spin variables. In particular, it is established that even in dimensions d < 4 if a ferromagnetic Ising spin model has a reflection-positive pair interaction with a sufficiently slow decay, e.g. as Jx = 1/Ixl a+~ with 0 < a~< d/2, then the exponents ~, 6, ? and A 4 exist and take their mean-field values. This proves rigorously an early renormalization-group prediction of Fisher, Ma and Nickel. In the converse direction: when the decay is by a similar power law with try> 2, then the long-range part of the interaction has no effect on the existent critical exponent bounds, which coincide then with those obtained for short-range models.

Coherent-incoherent transition in the sub-Ohmic spin-boson model

“…Yet the comparison with the results known for the Ising model with 1/r 1+s interactions indicates that higher-order corrections to Silbey-Harris ansatz should be necessary to describe the close proximity of the transition, since for s > 0 the transition in the Ising model is of 2nd-order. 21 The numerical renormalisation group analysis by Bulla, Tong and Vojta 9 found that the system is localized at s = 0, and their perturbative RG results suggest that for s > 0, the transition is continuous as a function of α. As our method is based on a variational ansatz, we cannot make any strong statement about the exact nature of the transition.…”

Phase transitions in one-dimensional nonequilibrium systems

“…Indeed, the Ising model with long-range interactions decaying with distance as J r r ,1, has been well studied 14 and it has been demonstrated that the one-dimensional system orders at low temperatures for 1 15 . Also, of course we require non-zero temperature so that entropy comes into play, otherwise the two fully ordered states ground states would dominate the partition sum and the system would be frozen into them.…”

Coherent phenomena in mesoscopic systems

“…We see that the interaction (3) is a long-range interaction. This indicates in particular that thermodynamic fluctuations of the current will be strongly supressed 6 .…”