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$_2$.

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