# Lattice Homotopy Constraints on Phases of Quantum Magnets

## TL;DR

This paper introduces a topological perspective on LSM theorems in quantum magnets, framing them as obstructions in lattice homotopy that dictate the emergence of exotic phases like spin liquids.

## Contribution

It proposes that all LSM-like theorems can be understood through lattice homotopy, unifying various constraints under a topological framework that includes full spatial symmetries.

## Key findings

- Lattice homotopy provides a new topological understanding of LSM theorems.
- All spin-symmetric magnets with half-integer moments and certain symmetries are spin liquids.
- The conjecture is proven in two-dimensional cases for relevant physical settings.

## 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.