  The Lieb-Schultz-Mattis (LSM) theorem and its extensions forbid trivial
phases from arising in certain quantum magnets. Constraining infrared behavior
with the ultraviolet data encoded in the microscopic lattice of spins, these
theorems tie the absence of spontaneous symmetry breaking to the emergence of
exotic phases like quantum spin liquids. In this work, we take a new
topological perspective on these theorems, by arguing they originate from an
obstruction to "trivializing" the lattice under smooth, symmetric deformations,
which we call the "lattice homotopy problem." We conjecture that all LSM-like
theorems for quantum magnets (many previously-unknown) can be understood from
lattice homotopy, which automatically incorporates the full spatial symmetry
group of the lattice, including all its point-group symmetries. One consequence
is that any spin-symmetric magnet with a half-integer moment on a site with
even-order rotational symmetry must be a spin liquid. To substantiate the
claim, we prove the conjecture in two dimensions for some physically relevant
settings.
