Anomalies on the Lattice, Homotopy of Quantum Cellular Automata, and a Spectrum of Invertible States

TL;DR

This paper develops a topological framework for understanding lattice anomalies and classifies invertible states and quantum cellular automata using $\\Omega$-spectra, advancing the understanding of symmetry protected topological phases.

Contribution

It introduces a rigorous topological theory of lattice anomalies and constructs spectra to classify invertible states and automata up to blend equivalence.

Findings

Classifies anomalies and SPT phases using $\\Omega$-spectra.

Provides a topological obstruction framework for gauging symmetries.

Establishes a classification scheme for invertible states and quantum cellular automata.

Abstract

We develop a rigorous topological theory of anomalies on the lattice, which are obstructions to gauging global symmetries and the existence of trivial symmetric states. We also construct $\Omega$-spectra of a class of invertible states and quantum cellular automata, which allows us to classify both anomalies and symmetry protected topological phases up to blend equivalence.