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

TL;DR

This paper investigates the spherical model on a spider-web graph, revealing non-trivial corrections to mean-field behavior, including localized modes, unusual spectral properties, and essential singularities in free energy near criticality.

Contribution

It provides a detailed analysis of the spherical model on a complex graph, showing how loops induce non-mean-field corrections and characterizing the spectral and thermodynamic properties.

Findings

Modes are localized with a spectrum of delta functions.

Return probability decays as exp(-C' t^{1/3}) for large t.

Free energy exhibits an essential singularity near T_c.

Abstract

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 $\exp(- C' t^{1/3})$, for large $t$, where $C'$ 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 $\exp[ -K {(T - T_c)}^{-1/2}]$.