Ground states for a class of deterministic spin models with glassy behaviour

TL;DR

This paper investigates the ground states of a deterministic spin model with glassy behavior, revealing how their degeneracy depends on the number-theoretic properties of system size and connecting the model's matrix to symplectic transformations.

Contribution

It provides an explicit construction of eigenvectors for the interaction matrix and analyzes the asymptotic degeneracy of ground states based on number theory.

Findings

Ground state degeneracy varies with system size and prime factorization.

Explicit eigenvector construction using symplectic matrix properties.

Degeneracy disappears for even system sizes.

Abstract

We consider the deterministic model with glassy behaviour, recently introduced by Marinari, Parisi and Ritort, with \ha\ $H=\sum_{i,j=1}^N J_{i,j}\sigma_i\sigma_j$, where $J$ is the discrete sine Fourier transform. The ground state found by these authors for $N$ odd and $2N+1$ prime is shown to become asymptotically dege\-ne\-ra\-te when $2N+1$ is a product of odd primes, and to disappear for $N$ even. This last result is based on the explicit construction of a set of eigenvectors for $J$, obtained through its formal identity with the imaginary part of the propagator of the quantized unit symplectic matrix over the $2$-torus.