Topological characterization of Lieb-Schultz-Mattis constraints and applications to symmetry-enriched quantum criticality

TL;DR

This paper develops topological tools to understand Lieb-Schultz-Mattis constraints in spin systems with various symmetries and applies them to analyze exotic quantum critical states, revealing new realizations and stability conditions.

Contribution

It introduces topological partition functions characterizing LSM constraints for systems with complex symmetries and applies these to classify and analyze emergent quantum critical states.

Findings

Identifies all realizations of quantum critical states on specific lattices with given symmetries.

Discovers stable Dirac spin liquids on square and honeycomb lattices.

Finds a spin-quadrupolar Dirac spin liquid beyond traditional parton constructions.

Abstract

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$_2$.