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

TL;DR

The paper reframes LSM-type theorems and Luttinger constraints as ’t Hooft anomaly matching conditions in lattice 1+1d systems, using flat background gauge fields and topological defects to probe anomalous symmetry actions. It develops a general lattice framework for twisting in space and time, identifies emanant symmetries arising from lattice translations, and demonstrates how these anomalies constrain low-energy spectra and defect dynamics. By constructing explicit lattice models with anomalous internal symmetries and connecting them to the continuum c=1 compact boson, the authors unify known results and produce new lattice realizations that flow to the same IR theory for all radii, with precise T-duality structures. The work further generalizes to Z_n emanant symmetries from lattice translations, analyzes SU(n) spin chains, and shows how Luttinger-type constraints emerge from anomaly considerations, providing exact correspondences between UV lattice data and IR continuum data, including lattice momenta and defect spectra.

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.

Figures (4)

• Figure 1: Rotating space in the presence of a defect. Space runs horizontally and time runs vertically. We see the effect of the symmetry operator $g$, the $g$-defect, and a pictorial derivation of Eq. \ref{['TOTind']} or its realization \ref{['keytrane']}.
• Figure 2: A momentum defect (in blue) and a winding defect (in red) are shifted to the right in two different orders. These two operations do not commute and differ by a phase, as in equation \ref{['Usdonotcommute']}.
• Figure 3: Spectra of the extended Heisenberg model with even length $L=16$ and $18$. Here the horizontal axis is the lattice momentum and the vertical axis is the energy. We only show states with momentum in $[0,\pi]$, as the $[-\pi,0]$ part are related by parity symmetry. Note that the ground-state momentum is different in these two cases.
• Figure 4: Spectra of the extended Heisenberg model with odd length $L=17$ and $19$. Note that the ground-state momentum is different in these two cases.