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Lieb-Schultz-Mattis (LSM) theorems provide powerful constraints on the 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 Gₛ G₈₍ₓ G s × G i n t symmetry, where Gₛ G s is an arbitrary space group in one or two spatial dimensions, and G₈₍ₓ G i n t is any internal symmetry whose projective representations are classified by Z₂ᵏ ℤ 2 k with k 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) 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) Z₂T S O (3) × ℤ 2 T internal symmetry and either p6m p 6 m or p4m p 4 m 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 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 2.
Ye et al. (Mon,) studied this question.