  Lieb-Schultz-Mattis (LSM) theorems provide powerful constraints on the
emergibility problem, i.e. whether a quantum phase or phase transition can
emerge in a many-body system. We derive the topological partition functions
that characterize the LSM constraints in spin systems with $G_s\times G_{int}$
symmetry, where $G_s$ is an arbitrary space group in one or two spatial
dimensions, and $G_{int}$ is any internal symmetry whose projective
representations are classified by $\mathbb{Z}_2^k$ with $k$ an integer. We then
apply these results to study the emergibility of a class of exotic quantum
critical states, including the well-known deconfined quantum critical point
(DQCP), $U(1)$ Dirac spin liquid (DSL), and the recently proposed
non-Lagrangian Stiefel liquid. These states can emerge as a consequence of the
competition between a magnetic state and a non-magnetic state. We identify all
possible realizations of these states on systems with $SO(3)\times
\mathbb{Z}_2^T$ internal symmetry and either $p6m$ or $p4m$ lattice symmetry.
Many interesting examples are discovered, including a DQCP adjacent to a
ferromagnet, stable DSLs on square and honeycomb lattices, and a class of
quantum critical spin-quadrupolar liquids of which the most relevant spinful
fluctuations carry spin-$2$. In particular, there is a realization of
spin-quadrupolar DSL that is beyond the usual parton construction. We further
use our formalism to analyze the stability of these states under
symmetry-breaking perturbations, such as spin-orbit coupling. As a concrete
example, we find that a DSL can be stable in a recently proposed candidate
material, NaYbO$_2$.
