Solving the Frustrated Spherical Model with q-Polynomials

TL;DR

This paper analytically solves the frustrated spherical model using q-Hermite polynomials, revealing its q-periodic behavior and contrasting it with related models that may exhibit metastability phenomena.

Contribution

The paper introduces an exact solution for the frustrated spherical model via q-oscillator algebra representation and explores its relation to a matrix model with oscillating potential.

Findings

The model's low-temperature phase is smooth and non-glassy across q-values.

The solution employs q-Hermite polynomials to represent q-oscillator algebra.

The related matrix model shows potential metastability due to oscillating interactions.

Abstract

We analyse the Spherical Model with frustration induced by an external gauge field. In infinite dimensions, this has been recently mapped onto a problem of q-deformed oscillators, whose real parameter q measures the frustration. We find the analytic solution of this model by suitably representing the q-oscillator algebra with q-Hermite polynomials. We also present a related Matrix Model which possesses the same diagrammatic expansion in the planar approximation. Its interaction potential is oscillating at infinity with period log(q), and may lead to interesting metastability phenomena beyond the planar approximation. The Spherical Model is similarly q-periodic, but does not exhibit such phenomena: actually its low-temperature phase is not glassy and depends smoothly on q.