Emergence of geometric order from topological constraints in a three-dimensional Coulomb phase

TL;DR

This paper explores how topological constraints in a 3D Coulomb phase lead to emergent geometric order and limit shapes, extending phenomena known in 2D models to three dimensions.

Contribution

It introduces a 3D lattice model with boundary conditions that induce long-range order and demonstrates a 3D generalization of the arctic limit shape phenomenon.

Findings

Partial lifting of ground state degeneracy by boundary conditions

Emergence of long-range magnetic order in the thermodynamic limit

Numerical evidence for a 3D arctic limit shape

Abstract

The emergence of order and geometric limit shapes in a three-dimensional (3D) Coulomb phase subject to domain wall boundary conditions (DWBC) is investigated. While the arctic circle phenomenon -- the spatial segregation of frozen and fluctuating degrees of freedom -- is well-established in the two-dimensional six-vertex model (square ice), its extension to 3D remains largely unexplored. A cubic lattice model with Ising degrees of freedom living on the edges, whose ground state manifold is governed by a divergence-free (3-in/3-out) local constraint, is considered. In the bulk, this model realizes a classical spin liquid characterized by algebraic correlations and pinch-point singularities in reciprocal space. It is demonstrated that applying DWBC partially lifts the extensive ground state degeneracy, inducing long-range magnetic order in the thermodynamic limit. Despite this ordering, it is found that the system retains a fluctuating component that exhibits the signature of a Coulomb phase. Finally, by mapping the local vertex polarization density, compelling numerical support is provided for a 3D generalization of the arctic limit shape, bridging the gap between topological constraints and emergent geometry in higher dimensions.