Non glassy ground-state in a long-range antiferromagnetic frustrated model in the hypercubic cell

TL;DR

This paper investigates a long-range antiferromagnetic model on a hypercube, revealing conditions for stable antiferromagnetic phases and identifying a non-glassy, infinitely degenerate ground state depending on the hypercube dimension.

Contribution

It introduces a novel analysis of a hypercube-based antiferromagnetic model linked to Boolean functions, highlighting phase behavior and ground state degeneracy.

Findings

Low temperature antiferromagnetic phase exists for certain dimensions

Ground state is infinitely degenerated and non-glassy in some regimes

Results also extend to ferromagnetic version of the model

Abstract

We analize the statistical mechanics of a long-range antiferromagnetic model defined on a D-dimensional hypercube, both at zero and finite temperatures. The associated Hamiltonian is derived from a recently proposed complexity measure of Boolean functions, in the context of neural networks learning processes. We show that, depending of the value of D, the system either presents a low temperature antiferromagnetic stable phase or the global antiferromagnetic order disappears at any temperature. In the last case the ground state is an infinitely degenerated non-glassy one, composed by two equal size anti-aligned antiferromagnetic domains. We also present some results for the ferromagnetic version of the model.