Kosterlitz--Thouless Criticality in a Dipole-Conserving XY Model
In this Letter, we study finite-temperature phase transitions in a two-dimensional classical statistical model called ``dipole-conserving XY model''. Unlike the conventional XY model, the original phase field $\theta$ has no quasi-long-range order, conventional phase vortices have finite self-energy, and the standard helicity modulus vanishes identically. Analytic vortex energetics and Gaussian continuum theory show that Kosterlitz--Thouless (KT) criticality is instead controlled by phase-gradient (PG) vortices defined in the two compact dipole fields $\chi_\alpha=a\partial_\alpha\theta$, whose self-energies grow logarithmically with system size. We determine the phase diagram using parallel-tempering Metropolis Monte Carlo simulations and three diagnostics tailored to dipole conservation. KT finite-size scaling of dipole-field correlation-ratio crossings locates the critical temperatures; generalized helicity moduli defined through quadratic phase twists measure the stiffness of individual dipole channels; and PG-vortex densities obtained from plaquette winding numbers identify the proliferating vortex species. At isotropic couplings, the two PG-vortex species unbind simultaneously at a single KT transition. Spatial anisotropy separates their unbinding temperatures, yielding two KT transitions and an intermediate phase with quasi-long-range order in only one dipole channel. The transitions merge again when the mixed-derivative channel is removed. Our results establish the finite-temperature phase structure of the dipole-conserving XY model and identify thermal PG-vortex unbinding as a novel route to KT criticality in systems with higher-moment conservation.