Classical origins of Landau-incompatible transitions

Continuous phase transitions where symmetry is spontaneously broken are ubiquitous in physics and often found between `Landau-compatible' phases where residual symmetries of one phase are a subset of the other. However, continuous `deconfined quantum critical' transitions between Landau-incompatible symmetry-breaking phases are known to exist in certain quantum systems, often with anomalous microscopic symmetries. In this Letter, we investigate the need for such special conditions. We show that Landau-incompatible transitions can be found in a family of well-known classical statistical mechanical models with anomaly-free symmetries, introduced by Jos\'{e}, Kadanoff, Kirkpatrick and Nelson (Phys. Rev. B 16, 1217). The models are anisotropic deformations of the classical 2d XY model labelled by a positive integer $Q$. For a range of temperatures, even $Q$ models exhibit two Landau-incompatible partial symmetry-breaking phases and a direct transition between them for $Q \ge 4$. Characteristic features of deconfined quantum criticality, such as enhanced symmetries and melting of charged defects are easily seen in a classical setting. For odd $Q$, and corresponding temperature ranges, two regions of a single partial symmetry-breaking phase appear, split by a stable `unnecessary critical' line. We discuss experimental systems that realize these transitions and present anomaly-free quantum models that also exhibit similar phase diagrams.