  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 paper, 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 on-site microscopic symmetries, introduced by Jos\'{e}, Kadanoff,
Kirkpatick and Nelson (Phys. Rev. B 16, 1217). The models are labeled by a
positive integer $Q$ and constructed by a deformation of the 2d classical XY
model, defined on any lattice, with an on-site potential that preserves a
discrete $Q$-fold spin rotation and reflection symmetry. 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 Landau-incompatible transitions are easily seen,
such as enhanced symmetries and melting of charged defects. For odd $Q$, and
corresponding temperature ranges, two regions of a single partial
symmetry-breaking phase are obtained, split by a stable `unnecessary critical'
line. We present quantum models with anomaly-free symmetries that also exhibit
similar phase diagrams.
