A Tale of Two Quantum Compass Models

We investigate two variants of quantum compass models (QCMs). The first, an orbital-only honeycomb QCM, is shown to exhibit a quantum phase transition (QPT) from a \(XX\)- to \(ZZ\)-ordered phase in the \(3d\)-Ising universality class, in accord with earlier studies. In a fractionalized parton construction, this describes a ``superfluid-Mott insulator'' transition between a higher-order topological superfluid and the toric code, the latter described as a \(p\)-wave resonating valence bond state of the partons. The second variant, the spinless fermion QCM on a square lattice, is of interest in the context of cold-atom lattices with higher-angular momentum states on each atom. We explore finite-temperature orbital order-disorder transitions in the itinerant and localized limits using complementary methods. In the itinerant limit, we uncover an intricate temperature (\(T\))-dependent dimensional crossover from a high-\(T\) quasi-\(1d\) insulator-like state, via an incoherent bad-metal-like state at intermediate \(T\), to a \(2d\) symmetry-broken insulator at low \(T\), well below the ``orbital'' ordering scale. Finally, we discuss how engineering specific, tunable, and realistic perturbations in both these variants can act as a playground for simulating a variety of exotic QPTs between topologically ordered and trivial phases. In the cold-atom context, we propose a novel way to engineer a possible realisation of the exotic exciton Bose liquid phase at a QPT between a Bose superfluid and a charge density wave insulator. We argue that advances in the design of Josephson junction arrays and manipulating cold-atom lattices offer the hope of simulating such novel phases of matter in the foreseeable future.