N-state Potts ices as generalizations of classical and quantum spin ice

TL;DR

This work develops a comprehensive framework for classical and quantum $N_c$-state Potts ice models, uncovering an emergent abelian gauge theory arising from the Cartan subalgebra of $rak{su}(N_c)$ and rooton-like excitations with root-valued charges. Classical Potts ices exhibit Coulomb-like entropic interactions and pinch-point correlations, while Monte Carlo data corroborate the Coulomb phase and characteristic worm statistics across dimensions and colors. The quantum extension introduces gauge-field dynamics, three-field interactions, and Higgs-like condensation pathways (via gauge mean field theory) that generate multiple photon modes and flux-frustrated vacua, including potential flux-liquid ground states tied to SU$(N_c)$ structure. Flux frustration is further explored with exactly solvable $\,Z_N$ variants and perturbations, suggesting correlated flux liquids coexisting with color liquids and complex topological dynamics. Altogether, the paper links Potts ice physics to non-Abelian gauge theory concepts, offering a tractable platform for simulating SU$(N_c)$-like gauge dynamics in quantum many-body systems with potential experimental realizations in qudit-based platforms.

Abstract

Classical and quantum spin ice models are amongst the most popular settings for the study of spin liquid physics. $N-$state Potts ice models have been constructed that generalize spin ice, hosting multiple emergent $\text{U}(1)$ gauge fields and excitations charged under non-trivial combinations of these fields. We present a general treatment of classical $N-$state Potts ices relating their properties to the $\mathfrak{su}(N)$ Lie algebras, and demonstrate how the properties of charged excitations in the classical model can be related to this symmetry group. We also introduce quantum generalizations of the Potts Ice models, and demonstrate how charge flavor changing interactions unique to $N>2$ models dominate their low energy physics. We further show how symmetries inherited from the $\mathfrak{su}(N)$ can lead to flux vacuum frustration, greatly modifying the dynamical properties of charged excitations.

Figures (12)

• Figure 1: Various realizations of $N_c$-state Antiferromagnetic Potts model ground states on lattices in two and three dimensions: $N_c=3$ model on the edges of the Honeycomb lattice ((a)); $N_c=4$ model on the vertices of the pyrochlore lattice (line-graph of diamond lattice) ((b)); $N_c=6$ model on the edges of the cubic lattice ((c)). In all cases $z=1$, so ground states consists of all edges neighboring each vertex (or each vertex of each cell for the line graph, as in (b)) having a distinct color.
• Figure 2: Worm length statistics for the $N_c=4$ models on (a) the square lattice and (b) the pyrochlore lattice, and (c) the $N_c=6$ model on the cubic lattice. All three exhibit the same behavior observed in the coulomb phases of classical $N_c=2$ spin ice in two and three dimensions respectively PhysRevLett.107.177202. The two dimensional $N_c=4$ model is described by a single power law with a non-universal exponent $n\sim-2.32$, whilst the three dimensional models possess identical statistics: short worm frequency obeys a power law with exponent $n_1=-2.5$, whilst longer worms fall in prevalence with power $n_2=-1.0$.
• Figure 3: Ground state of $N_c=4$ model on the square lattice with worms of different flavors highlighted
• Figure 4: Logarithms of separation frequency divided by the discrete measure $N_r$ displayed as functions of ((a)) $\log(r)$ or ((b) and (c)) $1/r$ for rooton-antirooton dynamics. We find that the effective interaction between particle-antiparticle pairs obeys the form expected of coulomb interactions in two ((a)) and three ((b) and (c)) dimensions, excepting an over representation of states where the excitations sit on neighboring sites.
• Figure 5: Diagrams of the root systems for (a) $\mathfrak{su}(3)$ and (b) $\mathfrak{su}(4)$, with roots labeled as sums of simple positive roots. Each root is associated with the exchange of two Potts state colors. Each edge of the $(N_c-1)$-simplex corresponds to a different matter field flavor. Combinations of fields whose root sum is zero are gauge invariant operators.