Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries

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)) × H^2(G,U(1))H3(G,U(1))×H2(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.

Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space

“…We note that the symmetry operator (11) is constructed as a product of the movement operators U j η that move the defect around the chain. This is a general feature reflecting a one-to-one correspondence between topological defects and symmetry operators; see [53] for a general discussion.…”

“…As we review in Appendix E, there are two kinds of translation defects T + and T − . T + adds a site to our chain and T − removes a site from our chain [44][45][46][47][48][49][50][51][52][53]. Consequently, as emphasized in [53], the width of the defect T −n = (T − ) ⊗n , which removes n sites, is proportional to n. Therefore, for large n (n ∼ L), such defects are nonlocal and hence they cannot be described by a fusion category.…”

“…T + adds a site to our chain and T − removes a site from our chain [44][45][46][47][48][49][50][51][52][53]. Consequently, as emphasized in [53], the width of the defect T −n = (T − ) ⊗n , which removes n sites, is proportional to n. Therefore, for large n (n ∼ L), such defects are nonlocal and hence they cannot be described by a fusion category. Related to that, while we can add an arbitrary number of T + defects, we cannot add an arbitrary number of T − defects.…”

“…Related to that, while we can add an arbitrary number of T + defects, we cannot add an arbitrary number of T − defects. (See more about this point in [52,53].) Finally, in the presence of a defect associated with the symmetry operator R, the group relation corresponding to periodic boundary conditions T L = 1 is modified to T −L = R. Such modifications in the relations are not incorporated in the fusion category.…”

“…As it stands, the system described by (E.1) is manifestly translation invariant. Following [53], we would like to present it in a language similar to the one we used for defects of internal symmetries.…”