# Ground states for the SOS model with an external field on the Cayley tree

## TL;DR

This paper characterizes ground states of a multi-spin SOS model with external fields on Cayley trees, including translation-invariant and periodic configurations, expanding understanding of such models' low-energy states.

## Contribution

It provides a detailed description of ground states for the SOS model with external fields on Cayley trees, including translation-invariant and periodic cases.

## Key findings

- Characterization of translation-invariant ground states for specific parameters.
- Description of periodic ground states under periodic external fields.
- Analysis of ground states for the case k=2, m=2.

## Abstract

We consider a nearest-neighbor solid-on-solid (SOS) model, with several spin values $0,1,2,...,m, m\geq2$ and non zero external field, on a Cayley tree of order $k$. In the case $k=2, m=2$, we describe translation-invariant ground states for the SOS model with a translation-invariant external field. Some periodic ground states for the SOS model with periodic external field are described.