  Superconductivity owes its properties to the phase of the electron pair
condensate that breaks the $U(1)$ symmetry. In the most traditional ground
state, the phase is uniform and rigid. The normal state can be unstable towards
special inhomogeneous superconducting states: the Abrikosov vortex state, and
the Fulde-Ferrell-Larkin-Ovchinnikov state. Here we show that the phase-uniform
superconducting state can go into a fundamentally different and more ordered
non-uniform ground state, that we denote as a phase crystal. The new state
breaks translational invariance through formation of a spatially periodic
modulation of the phase, manifested by unusual superflow patterns and
circulating currents, that also break time-reversal symmetry. We list the
general conditions needed for realization of phase crystals. Using microscopic
theory we then derive an analytic expression for the superfluid density tensor
for the case of a non-uniform environment in a semi-infinite superconductor. We
demonstrate how the surface quasiparticle states enter the superfluid density
and identify phase crystallization as the main player in several previous
numerical observations in unconventional superconductors, and predict existence
of a similar phenomenon in superconductor-ferromagnetic structures. This
analytic approach provides a new unifying aspect for the exploration of
boundary-induced quasiparticles and collective excitations in superconductors.
More generally, we trace the origin of phase crystallization to non-local
properties of the gradient energy, which implies existence of similar
pattern-forming instabilities in many other contexts.
