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

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.

Figures (4)

• Figure 1: The underlying graph of the model. Ising degrees of freedom live on the edges of the graph, such that any configuration maps onto an orientation of the edges. Spheres at the vertices depict the nature of the vertex: gray satisfies the ice rule (3-in/3-out), while blue and red indicate defects (2-in/4-out and 2-out/4-in, respectively). In the constrained ground state manifold, no 6-in or 6-out vertices appear.
• Figure 2: (Up) Magnetic structure factor $S(\mathbf{q})$ of the ground state manifold of the model on a finite graph ($L = 21, N = 3 L^2 (L + 1)$) subject to open boundary conditions. It exhibits the fingerprints of a 3D Coulomb phase: a structured diffuse scattering with characteristic pinch points. These are the reciprocal space signatures of algebraic correlations (cut off in a finite system) related to the local divergence-free constraint Henley2010. The planes forming the Brillouin zone boundaries are visualized and indexed in reciprocal lattice units (r.l.u.). (Bottom) Structure factor of the spin fluctuations, $S_{\text{fluc}}(\mathbf{q})$, under domain wall boundary conditions (DWBC), obtained by subtracting the long range order. Despite the reduced spectral weight due to the frozen boundary, the persistence of clear pinch points confirms that the bulk interior remains in a fluctuating, divergence-free Coulomb phase.
• Figure 3: Evolution of the spatially averaged order parameter $m$ as a function of the inverse system size $1/N$ ($N = 3 L^2 (L + 1)$), for $L = 11, 16, 21, 26, 31$. The solid line is a polynomial fit. The non-vanishing value at the intercept ($1/N \to 0$) indicates that the order persists in the thermodynamic limit, confirming that the boundary conditions select a specific subset of the ground state manifold. The inset schematically represents the global boundary conditions as a single effective vertex. Each of its six edges corresponds to an entire boundary plane of the simulated cubic volume: an inward-pointing arrow indicates that all spins on that specific macroscopic face are fixed to point uniformly into the bulk, defining the global 2-in/4-out configuration. Note that the topological constraint of the DWBC takes here the form of a marginal magnetic charge injection ($N_m = L^2$). This is to be contrasted with the usual DWBC applied in 2d, which are charge neutral.
• Figure 4: Profile of the mean vertex polarization along the diagonal of the cube, for different sizes ($L = 11, 16, 21, 26, 31 - N = 3 L^2 (L+1)$). In the inset, the convex hull of the mean vertex polarization is shown, with a chosen mean vertex value threshold $= 0.1$.