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arXiv:2602.24051

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

B. Canals

Published Feb 27, 2026• 0 citations
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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.

DOI
10.48550/arxiv.2602.24051

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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.
The square ice is a two-dimensional spin liquid hosting a Coulomb phase physics. When constrained under specific boundary conditions, the so-called domain-wall boundary conditions, a phase separation occurs that leads to the formation of a spin liquid confined within a disk surrounded by magnetically ordered regions. Here, we numerically characterize the ground-state properties of this spin liquid, coined the arctic square ice in reference to a phenomenon known in statistical mechanics. Our results reveal that both the vertex distributions and the magnetic correlations are inhomogeneous within the liquid region, and they exhibit a radial dependence. If these properties resemble those of the conventional square ice close to the center of the disk, they evolve continuously as the disk perimeter is approached. There, the spin liquid orders. As a result, pinch points, signaling the presence of algebraic spin correlations, coexist with magnetic Bragg peaks in the magnetic structure factor computed within the disk. The arctic square ice thus appears as an unconventional Coulomb phase sharing common features with a fragmented spin liquid, albeit on a charge-neutral vacuum.
Under certain boundary conditions, the square ice model exhibits a phase separation in which the core of the system is disordered while its outer region remains ordered. This phenomenon, known as the "arctic circle," has been studied theoretically in combinatorial mathematics and statistical mechanics. Here, we realize the physics of the arctic circle experimentally for the first time, using a programmable lattice of superconducting qubits, and investigate its properties under the prism of a highly frustrated magnet. Our work reveals two unexpected properties. First, the disordered spin manifold confined within the arctic curve is a spin liquid whose average spin texture resembles that of an antivortex, i.e., it is a topologically charged Coulomb phase. Second, monopole quasiparticle excitations, which are totally absent in theoretical works, can be isolated in a phase-separated system. Remarkably, a monopole segregation mechanism is observed, in which the monopoles are sorted according to the magnetic charge and magnetic moment they carry, without the application of an external driving force.
We show how the directed-loop Monte Carlo algorithm can be applied to study vertex models. The algorithm is employed to calculate the arrow polarization in the six-vertex model with the domain wall boundary conditions. The model exhibits spatially separated ordered and "disordered" regions. We show how the boundary between these regions depends on parameters of the model. We give some predictions on the behavior of the polarization in the thermodynamic limit and discuss the relation to the Arctic Circle theorem.
We show that quantum square ice-namely, the two-dimensional version of proton or spin ice with tunable quantum tunneling of the electric or magnetic dipole moment-exhibits a quantum spin-liquid phase supporting fractionalized spinons. This phase corresponds to a thermally induced, deconfined quantum Coulomb phase of a two-dimensional lattice gauge theory. It emerges at finite, yet exceedingly low temperatures from the melting of two distinct order-by-disorder phases appearing in the ground state: a plaquette valence-bond solid for low tunneling; and a canted Néel state for stronger tunneling. The latter phases appear via the highly nonlinear effect of quantum fluctuations within the degenerate manifold of ice-rule states, and they can be identified as the two competing ground states of a discrete lattice gauge theory (quantum link model) emerging as the effective Hamiltonian of the system within degenerate perturbation theory.
The thermodynamic limit is foundational to statistical mechanics, underlying our understanding of many-body phases. It assumes that, as the system size grows infinitely at fixed density of particles, unambiguous macroscopic phases emerge that are independent of the system’s boundary shape. We present explicit quantum spin and dimer Hamiltonians whose ground states violate this principle. Our construction relies on the previous mathematical work on classical dimers on the Aztec diamond and the square-octagon fortress, where geometry-dependent phase behaviors are observed in the infinite-size limit. We reverse engineer quantum spin Hamiltonians on the square and the square-octagon lattices whose ground states at the Rokhsar-Kivelson points are described by classical dimer coverings. On diamond-shaped domains, we find macroscopic boundary regions exhibiting distinct quantum phases from those on square-shaped domains. We study the nature of these phases by calculating the dimer-dimer and vison correlators and adapt Kasteleyn matrix based analytical and numerical methods for computing the vison correlator, which are significantly more efficient than standard Monte Carlo techniques. Our results show that the square-octagon lattice supports a single gapped short-range entangled phase, with exponentially decaying dimer correlators and a constant vison correlator. When the same model is considered on a diamond-shaped domain, two additional macroscopic regions emerge, with one near the corners and exhibiting staggered dimer order, and another exhibiting critical correlations.
Time-reversal invariant three-dimensional topological insulators can be defined fundamentally by a topological field theory with a quantized axion angle $\ensuremath{\theta}$ of 0 or $\ensuremath{\pi}$. It was recently shown that fractional quantized values of $\ensuremath{\theta}$ are consistent with time-reversal invariance if deconfined, gapped, fractionally charged bulk excitations appear in the low-energy spectrum due to strong correlation effects, leading to the concept of a fractional topological insulator. These fractionally charged excitations are coupled to emergent gauge fields, which ensure that the microscopic degrees of freedom, the original electrons, are gauge-invariant objects. A first step towards the construction of microscopic models of fractional topological insulators is to understand the nature of these emergent gauge theories and their corresponding phases. In this work, we show that low-energy effective gauge theories of both Abelian or non-Abelian type are consistent with a fractional quantized axion angle if they admit a Coulomb phase or a Higgs phase with gauge group broken down to a discrete subgroup. The Coulomb phases support gapless but electrically neutral bulk excitations while the Higgs phases are fully gapped. The Higgs and non-Abelian Coulomb phases exhibit multiple ground states on boundaryless spatial three-manifolds with nontrivial first homology, while the Abelian Coulomb phase has a unique ground state. The ground-state degeneracy receives an additional contribution on manifolds with boundary due to the induced boundary Chern-Simons term.