A Higher-Form Spin Liquid in the Spin Vorticity Model*

Quantum spin ice is perhaps the most important example of a quantum spin liquid in three dimensions, owing to (i) the many pyrochlore compounds which might realize this phase and (ii) the detailed theoretical understanding of the classical nearest-neighbor spin ice (NNSI) model and how quantum perturbations of this model can be mapped to a U(1) lattice gauge theory. In this talk, we discuss a new class of gapless spin liquids in three dimensions: 2-form U(1) quantum spin liquids. To illustrate this phase, we first introduce a minimal classical Ising model on the pyrochlore lattice, analogous to NNSI, that we call the spin "vorticity model", constructed to enforce a local "zero-curl" constraint on every elementary hexagonal loop of the lattice. We give a detailed characterization of the emergent gauge structure of the ground state manifold of this model, which may be described as a condensate of membranes —flipping a collection of spins forming a closed surface costs zero energy. Unlike established spin liquids like NNSI, this model contains no point-like quasiparticle excitations. Instead, the excitations are extended string objects, created by flipping a collection of spins forming an open surface. At finite temperature, the classical model is then described by a gas of string loops in a background of fluctuating membranes. Interestingly, Monte Carlo simulations find that this model exhibits an unusual weak symmetry-breaking transition at finite temperature below which a very small but extensive fraction of the system appears to develop long range order. Nevertheless, we find that the ground state entropy remains extensive, remarkably close to that of NNSI, with correlation functions showing a low-temperature Coulomb-like phase characterized by "inverted" pinch points. By considering a minimal quantum extension of the spin vorticity model, we propose an effective membrane exchange model of the quantum dynamics within the classical ground state manifold which can be mapped to a frustrated 2-form U(1) lattice gauge theory. We briefly discuss the stability of the gapless deconfined phase of this gauge theory, and thus the possibility of realizing a novel phase of quantum matter: a 2-form U(1) quantum spin liquid.