Lattice Homotopy Constraints on Phases of Quantum Magnets

TL;DR

It is conjecture that all LSM-like theorems for quantum magnets can be understood from lattice homotopy, which automatically incorporates the full spatial symmetry group of the lattice, including all its point-group symmetries.

Abstract

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.

Figures (5)

• Figure 1: Lattice homotopy. (a-d) Representations of the rotation group ${\rm SO}(3)$ 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 three-fold 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 $\pm \bm{r}_{X}$, which leads to "defect regions" (shaded) near the fluxes. Far away from $\pm \bm{r}_X$, 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 $C_2'$ symmetry, since the transformed defect line can be brought back to the original by applying a gauge transformation on the region $A$.
• Figure 3: A $C_4$ symmetric lattice with all the spins living on the sites (orange and red). The rotation center is specified by the red site. $X^a$ fluxes are inserted at 4 locations related to each other by the $C_4$ rotation. The dotted lines are the branch cut associated to the fluxes. The symmetry action associated with each dotted line is given by $X^a$ (e.g. that for the tripped dotted line is $X^{3a}$). The consistency condition requires $X^{4a} = 1$.
• Figure 4: (a) The toric code model. Arrows indicate periodic boundary conditions. (b) A configuration of $C_2$-symmetric $X$-fluxes. The orange diamonds represent plaquette flipped by the $X$ fluxes. (c) Illustration of the ground states after inserting the $X$-fluxes. The blue circle means that the plaquette has $\prod_{i\in p}\hat{Z}_i=-1$. Those on the top (bottom) row has $+1$ ($-1$) eigenvalue of $\hat{Z}=\prod_{i}\hat{Z}_i$.
• Figure 5: New no-go in 3D. The flux-insertion argument for sym-SRE no-go can be generalized to higher dimensions with more complicated geometries. Here we show a compact space obtained by identifying the two marked faces by a $4_2$ screw and the other pairs by lattice translations. A pair of symmetry-related line fluxes (green lines) are inserted, and the system maintains a $C_2$ rotation symmetry about the dashed axis. The defect surface (shaded) and its $C_2$ partner (not shown) together enclose a single site (sphere).