Non-perturbative corrections to mean-field critical behavior: the spherical model on a spider-web graph

作者：Balram Ajit C^*; Dhar Deepak

来源：Journal of Physics A-Mathematical and Theoretical, 2012, 45(12): 125006.

DOI：10.1088/1751-8113/45/12/125006

We consider the spherical model on a spider-web graph. This graph is effectively infinite dimensional, similar to the Bethe lattice, but has loops. We show that these lead to non-trivial corrections to the simple mean-field behavior. We first determine all normal modes of the coupled springs problem on this graph, using its large symmetry group. In the thermodynamic limit, the spectrum is a set of delta-functions, and all the modes are localized. The fractional number of modes with frequency less than omega varies as exp(-C/omega) for omega tending to zero, where C is a constant. For an unbiased random walk on the vertices of this graph, this implies that the probability of return to the origin at time t varies as exp(-C%26apos;t(1/3)), for large t, where C%26apos; is a constant. For the spherical model, we show that while the critical exponents take the values expected from the mean-field theory, the free energy per site at temperature T, near and above the critical temperature T-c, also has an essential singularity of the type exp[-K(T - T-c)(-1/2)].

• 出版日期2012-3-30