Lieb-Schultz-Mattis constraints for hyperbolic lattices

TL;DR

This paper extends the Lieb-Schultz-Mattis theorem to hyperbolic lattices, providing constraints on ground states and proposing models for realizing spin liquids in negatively curved spaces.

Contribution

It generalizes LSM constraints to hyperbolic lattices using hyperbolic band theory and flux-threading, offering new insights into hyperbolic quantum matter.

Findings

Derived lower bounds on ground-state degeneracy for hyperbolic lattices.

Identified hyperbolic spin models as potential platforms for spin liquids.

Extended LSM constraints to non-Euclidean translation symmetries.

Abstract

The Lieb-Schultz-Mattis (LSM) theorem and its higher-dimensional extensions forbid the existence of a unique, symmetric, and gapped ground state at fractional fillings in quantum many-body systems with a conserved particle number (or spin angular momentum) and the conventional translation symmetry of Euclidean lattices. In this work, we propose a generalization of the LSM theorem to quantum many-body systems on hyperbolic lattices, i.e., regular tessellations of two-dimensional negatively curved space. By leveraging concepts from hyperbolic band theory in a many-body setting, we adapt Oshikawa's flux-threading argument to periodic hyperbolic lattices with a non-Euclidean (Fuchsian) translation symmetry and compute a lower-bound to the ground-state degeneracy as a function of filling and lattice geometry. We explore the consequences of LSM constraints for gapped phases of hyperbolic quantum matter and suggest frustrated spin models on hyperbolic analogs of the square and triangular lattices as promising platforms for realizing symmetric spin liquids in hyperbolic space.