From Quantum Dimers to the $\pi$-flux Toric Code via Deconfined Multicriticality

Abstract

Two-dimensional Rokhsar-Kivelson (RK) dimer models on bipartite lattices are generally limited to translation-symmetry-broken dimer crystals. We introduce a tensor-product regularisation of the dimer Hilbert space that yields a qubit Hamiltonian interpolating from the RK model to the $\pi$-flux toric code, thereby accessing a deconfined ${\mathbb{Z}}_2$ topological liquid. In this framework, the ${\mathbb{Z}}_2$ liquid descends from a multicritical $U(1)$ spin liquid through condensation of a charge-2 Higgs field, thus avoiding confinement. Using iDMRG together with low-energy field theory, we determine a phase diagram containing two continuous quantum phase transitions -- a $3\mathrm{D}$ XY${}^{\ast }$ transition between the ${\mathbb{Z}}_2$ liquid and the columnar/plaquette-VBS, and a quantum Lifshitz transition between two dimer crystals -- alongside a first-order transition between the staggered crystal and the ${\mathbb{Z}}_2$ liquid. Our field theory suggests a deconfined multicritical point described by an Abelian Higgs model with dynamical critical exponent, $z=2$, where the three transitions meet, highlighting the interplay of fractionalisation and emergent gauge fluctuations.