Order by disorder up to arbitrarily high temperature

TL;DR

This paper proves that certain classical lattice models on   exhibit long-range checkerboard order at high temperatures driven purely by entropy, using Pirogov--Sinai theory and Peierls bounds.

Contribution

It establishes high-temperature long-range order in a class of models, including the Han--Huang--Komargodski--Lucas--Popov model, via rigorous mathematical proof.

Findings

Models show long-range checkerboard order at high temperature.

Order mechanism is purely entropic, not energetic.

Proof employs Pirogov--Sinai theory and Peierls bounds.

Abstract

We prove that a class of classical lattice models on $\mathbb{Z}^d$ ($d \geq 2$) with on-site space $\mathbb{N}_0$ and nearest neighbour interaction, exhibits long-range checkerboard order at sufficiently high temperature. The ordering mechanism is purely entropic. The class of models contains the recently introduced model of Han--Huang--Komargodski--Lucas--Popov (arXiv:2503.22789), by which our work is inspired. The proof uses Pirogov--Sinai theory and the key input is a Peierls bound.