2-Form U(1) Spin Liquids: Classical Model and Quantum Aspects

We introduce a novel geometrically frustrated classical Ising model, dubbed the "spin vorticity model", whose ground state manifold is a novel classical spin liquid, a "2-form Coulomb phase". We study the thermodynamics of this model both analytically and numerically, exposing the presence of algebraically decaying correlations and demonstrating an extensive ground state entropy, and give a comprehensive account of its ground state properties and excitations. Each classical ground state may be decomposed into collections of closed 2-dimensional membranes, supporting fractionalized string excitations attached to the boundaries of open membranes. At finite temperature, the model can then be described as a gas of closed strings in a background of fluctuating membranes. We demonstrate that the emergent gauge structure of the low-temperature phase is naturally captured in the formalism of 2-form electrodynamics, which describes 1-dimensional charged strings coupled to a rank-2 anti-symmetric gauge field. After establishing the classical spin vorticity model, we consider perturbing it with quantum exchange interactions, from which we derive an effective membrane exchange model of the quantum dynamics of these membranes, which maps to a frustrated 2-form U(1) lattice gauge theory. We show the existence of a fine-tuned Rokhsar-Kivelson point where the quantum ground state is an equal weight superposition of all classical ground state configurations. We further demonstrate how to quantize the string excitations, by coupling a 1-form string field to the emergent 2-form U(1) gauge field, thus mapping a quantum spin model to a 2-form gauge-Higgs model. We discuss the stability of the gapless deconfined phase of this gauge theory and the possibility of realizing a novel class of phases of quantum matter: 2-form U(1) quantum spin liquids.