Hidden entropic order of emergent charges in a classical Ampère spin liquid

Abstract

Classical spin liquids are characterized by an extensive number of ground-state configurations. Coulomb spin liquids have, in this ground-state manifold, a divergence-free local constraint, which gives dipolar correlations and pinch points in the structure factor. Recently, models with a complementary ground-state constraint, defined by a zero-circulation constraint (Ampère's law), have been proposed. While classical Coulomb spin liquids forbid emergent charges, their Ampère counterparts leave the charges unconstrained. Here we study an Ising Ampère model on the octochlore lattice. The ground states obey a Gauss law for an antisymmetric rank-two field, i.e., three coupled two-dimensional Gauss laws, which are expected to produce pinch points in the structure factor. Nevertheless, maximal charges, vertices with all six arrows in or out, spontaneously accumulate on one of the two sublattices of the cubic lattice, with disordered signs. This order has a purely entropic origin; it washes out the pinch-point singularities at large scales, produces no magnetic Bragg peaks, and coexists with an extensive entropy. The system enters into this sublattice long-range ordered state through a continuous transition in what seems to be the three-dimensional Ising universality class. Moreover, the state has no magnetic order; it is even under time reversal: it is a hidden, entropic order within the Ampère manifold. Thus, a system whose Gauss law would predict an algebraic spin liquid becomes, through a non-magnetic ordering transition, a partially ordered short-range spin liquid. Our study is based on a combination of the constraint-vector theory, exact bounds and enumeration, and Monte Carlo, Wang-Landau, and parallel-tempering simulations.