Lattice Homotopy Constraints on Phases of Quantum Magnets

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.

Figure 1

Lattice homotopy. (a)–(d) Representations of the rotation group SO(3) fuse following a ${\mathbb{Z}}_2$ rule. Open and filled circles, respectively, denote the representations of integer and half-integer spins. (e) A smooth deformation of a lattice (circles) symmetric under mirror planes (dashed lines) and threefold rotations (about the stars). (f)–(i) There are two inequivalent sites (big and small circles) in each unit cell (shaded) of a mirror-symmetric 1D lattice. Under lattice homotopy, there are four distinct lattice classes. (j) Assuming the symmetries of (e), a honeycomb lattice of half-integer spins is equivalent to a kagome lattice of integer spins, as demonstrated by the depicted smooth deformation.

Figure 2

Flux insertion. (a) A $C_2$ symmetric lattice with a pair of $X$ fluxes (crosses) inserted at $\pm r_{\mathit{X}}$, which leads to “defect regions” (shaded) near the fluxes. Far away from $\pm r_{\mathit{X}}$, flux insertion amounts to choosing a defect line (dash-dot) and twisting the local Hamiltonian by $X$ along the line. (b) As $X^{-1}=X$, the system retains a twisted $C_2^{\prime }$ symmetry, since the transformed defect line can be brought back to the original by applying a gauge transformation on the region $A$.