Lieb-Schultz-Mattis, Luttinger, and 't Hooft -- anomaly matching in lattice systems

TL;DR

This paper investigates 't Hooft anomalies in 1+1d lattice systems with global symmetries, providing a unified framework to analyze their effects, including on translation symmetry and Luttinger constraints, with applications to spin chains and bosonization.

Contribution

It introduces a novel approach to study lattice anomalies via background gauge fields and twisted boundary conditions, unifying known results and deriving new insights into lattice models and their continuum limits.

Findings

Lattice models can realize the $c=1$ boson with full symmetry and anomalies.

Lieb-Schultz-Mattis theorem is interpreted as an 't Hooft anomaly matching.

Filling constraints like Luttinger theorem are related to anomalies in certain cases.

Abstract

We analyze lattice Hamiltonian systems whose global symmetries have 't Hooft anomalies. As is common in the study of anomalies, they are probed by coupling the system to classical background gauge fields. For flat fields (vanishing field strength), the nonzero spatial components of the gauge fields can be thought of as twisted boundary conditions, or equivalently, as topological defects. The symmetries of the twisted Hilbert space and their representations capture the anomalies. We demonstrate this approach with a number of examples. In some of them, the anomalous symmetries are internal symmetries of the lattice system, but they do not act on-site. (We clarify the notion of "on-site action.") In other cases, the anomalous symmetries involve lattice translations. Using this approach we frame many known and new results in a unified fashion. In this work, we limit ourselves to 1+1d systems with a spatial lattice. In particular, we present a lattice system that flows to the $c=1$ compact boson system with any radius (no BKT transition) with the full internal symmetry of the continuum theory, with its anomalies and its T-duality. As another application, we analyze various spin chain models and phrase their Lieb-Shultz-Mattis theorem as an 't Hooft anomaly matching condition. We also show in what sense filling constraints like Luttinger theorem can and cannot be viewed as reflecting an anomaly. As a by-product, our understanding allows us to use information from the continuum theory to derive some exact results in lattice model of interest, such as the lattice momenta of the low-energy states.