[Paper Review] Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries

Sahand Seifnashri|arXiv (Cornell University)|Aug 9, 2023

Atomic and Subatomic Physics ResearchPhysics and Astronomy125 references11 citations

TL;DR

The paper shows that Lieb-Schultz-Mattis (LSM) anomalies on the lattice are obstructions to gauging internal symmetries that may act non-on-site, and it develops a local, defect-based framework to compute the anomaly cocycle and connect lattice anomalies to ’t Hooft anomalies.

ABSTRACT

We study 't Hooft anomalies of global symmetries in 1+1d lattice Hamiltonian systems. We consider anomalies in internal and lattice translation symmetries. We derive a microscopic formula for the "anomaly cocycle" using topological defects implementing twisted boundary conditions. The anomaly takes value in the cohomology group $H^3(G,U(1)) imes H^2(G,U(1))$. The first factor captures the anomaly in the internal symmetry group $G$, and the second factor corresponds to a generalized Lieb-Schultz-Mattis anomaly involving $G$ and lattice translation. We present a systematic procedure to gauge internal symmetries (that may not act on-site) on the lattice. We show that the anomaly cocycle is the obstruction to gauging the internal symmetry while preserving the lattice translation symmetry. As an application, we construct anomaly-free chiral lattice gauge theories. We demonstrate a one-to-one correspondence between (locality-preserving) symmetry operators and topological defects, which is essential for the results we prove. We also discuss the generalization to fermionic theories. Finally, we construct non-invertible lattice translation symmetries by gauging internal symmetries with a Lieb-Schultz-Mattis anomaly.

Motivation & Objective

• Motivate a unified view of ’t Hooft anomalies, bulk-boundary anomalies, and LSM anomalies.
• Introduce symmetry defects as a local, topology-driven description of symmetries on the lattice.
• Provide a local formula for the anomaly cocycle and interpret it as an obstruction to gauging.
• Demonstrate the framework through lattice models that exhibit LSM-type constraints and non-on-site symmetries.

Proposed method

• Represent internal symmetries by topological defects that implement space twists.
• Establish a defect–operator correspondence to relate symmetry actions to defect fusions.
• Define and compute an anomaly cocycle as a U(1)-valued function on the lattice, identified with the obstruction to gauging.
• Show that LSM anomalies correspond to obstructions to gauging while preserving lattice translation symmetry.
• Apply the framework to bosonic systems and discuss fermionic cases with concrete lattice examples.

Experimental results

Research questions

• RQ1 How can lattice symmetries, including non-on-site actions, be gauged consistently?
• RQ2 What is the local structure of the Lieb-Schultz-Mattis anomaly on a 1+1d lattice?
• RQ3 How do symmetry defects encode the fusion and obstruction data of global symmetries on the lattice?
• RQ4 Can LSM-type constraints be reinterpreted as ’t Hooft anomalies via a lattice defect framework and continuum limit?
• RQ5 What are explicit lattice examples where the framework reveals the anomaly and obstruction to gauging?

Key findings

• An anomaly cocycle is identified as a U(1)-valued function capturing the obstruction to gauging internal symmetries on the lattice.
• A one-to-one correspondence is established between locality-preserving global symmetries and topological defects.
• LSM anomalies on the lattice manifest as obstructions to gauging translation-symmetry together with internal symmetry, linking lattice and continuum notions.
• The framework unifies ’t Hooft anomalies, bulk-boundary (SPT) perspectives, and LSM-type constraints within a lattice setting.
• Examples on Heisenberg-like chains, XYZ chains with non-invertible translation, and lattice chiral gauge theories illustrate the computation and interpretation of the anomaly.