  The Lieb-Schultz-Mattis (LSM) theorem states that a spin system with
translation and spin rotation symmetry and half-integer spin per unit cell does
not admit a gapped symmetric ground state lacking fractionalized excitations.
That is, the ground state must be gapless, spontaneously break a symmetry, or
be a gapped spin liquid. Thus, such systems are natural spin-liquid candidates
if no ordering is found. In this work, we give a much more general criterion
that determines when an LSM-type theorem holds in a spin system. For example,
we consider quantum magnets with arbitrary space group symmetry and/or
spin-orbit coupling. Our criterion is intimately connected to recent work on
the general classification of topological phases with spatial symmetries and
also allows for the computation of an "anomaly" associated with the existence
of an LSM theorem. Moreover, our framework is also general enough to encompass
recent works on "SPT-LSM" theorems where the system admits a gapped symmetric
ground state without fractionalized excitations, but such a ground state must
still be non-trivial in the sense of symmetry-protected topological (SPT)
phases.
