Next nearest neighbour Ising models on random graphs

TL;DR

This paper investigates the phase behavior of the next nearest neighbour Ising model on random graphs, revealing paramagnetic, spin glass, and dynamical transition phenomena through advanced statistical physics methods.

Contribution

It introduces a comprehensive analysis of the next nearest neighbour Ising model on various random graph ensembles, including phase diagrams and transition mechanisms, using the cavity method.

Findings

Periodic free energy variations in graph ensembles.

Paramagnetic phases at zero temperature for integer coupling ratios.

Spin glass phases at low temperatures for anti-ferromagnetic couplings.

Abstract

This paper develops results for the next nearest neighbour Ising model on random graphs. Besides being an essential ingredient in classic models for frustrated systems, second neighbour interactions interactions arise naturally in several applications such as the colour diversity problem and graphical games. We demonstrate ensembles of random graphs, including regular connectivity graphs, that have a periodic variation of free energy, with either the ratio of nearest to next nearest couplings, or the mean number of nearest neighbours. When the coupling ratio is integer paramagnetic phases can be found at zero temperature. This is shown to be related to the locked or unlocked nature of the interactions. For anti-ferromagnetic couplings, spin glass phases are demonstrated at low temperature. The interaction structure is formulated as a factor graph, the solution on a tree is developed. The replica symmetric and energetic one-step replica symmetry breaking solution is developed using the cavity method. We calculate within these frameworks the phase diagram and demonstrate the existence of dynamical transitions at zero temperature for cases of anti-ferromagnetic coupling on regular and inhomogeneous random graphs.