Note on Generalized Symmetries, Gapless Excitations, Generalized Symmetry Protected Topological states, and Anomaly

TL;DR

This paper explores quantum many-body systems with generalized symmetries, constructs low-energy states, derives anomalies, and connects boundary states to higher-dimensional SPT phases, revealing new topological and symmetry-protected phenomena.

Contribution

It introduces explicit constructions of low-energy states, derives 't Hooft anomalies for generalized symmetries, and links boundary theories to higher-dimensional SPT states with novel symmetry structures.

Findings

Explicit low-energy excited states for systems with generalized symmetries.

Derivation of 't Hooft anomalies in generalized symmetry Hamiltonians.

Identification of boundary states as lower-dimensional SPT phases or anomalous topological orders.

Abstract

We consider quantum many body systems with generalized symmetries, such as the higher form symmetries introduced recently, and the "tensor symmetry". We consider a general form of lattice Hamiltonians which allow a certain level of nonlocality. Based on the assumption of dual generalized symmetries, we explicitly construct low energy excited states. We also derive the 't Hooft anomaly for the general Hamiltonians after "gauging" the dual generalized symmetries. A 3d system with dual anomalous 1-form symmetries can be viewed as the boundary of a 4d generalized symmetry protected topological (SPT) state with 1-form symmetries. We also present a prototype example of 4d SPT state with mixed 1-form and 0-form symmetry topological response theory as well as its physical construction. The boundary of this SPT state can be a 3d anomalous QED state, or an anomalous 1-form symmetry enriched topological order. Insights are gained by dimensional compatification/reduction. After dimensional compactification, the 3d system with N pairs of dual 1-form symmetries reduces to a 1d system with 2N pairs of dual U(1) global symmetries, which is the boundary of an ordinary 2d SPT state; while the 3d system with the tensor symmetry reduces to a 1d Lifshitz theory, which is protected by the center of mass conservation of the system.