Stacking and the triviality of invertible phases

TL;DR

This paper demonstrates that stacking two quantum lattice systems results in a trivial sector structure for invertible states, implying no anyonic quasi-particles can exist in such configurations.

Contribution

It establishes a categorical framework showing that all sectors in a stacked system are equivalent to products of the individual systems' sectors, proving invertible states are trivial.

Findings

All irreducible sectors of a stacked system are equivalent to products of the factors' sectors.

Invertible states support no anyonic quasi-particles.

A categorical equivalence is established for the stacked system.

Abstract

We study the superselection sectors of two quantum lattice systems stacked onto each other in the operator algebraic framework. We show in particular that all irreducible sectors of a stacked system are unitarily equivalent to a product of irreducible sectors of the factors. This naturally leads to a faithful functor between the categories for each system and the category of the stacked system. We construct an intermediate `product' category which we then show is equivalent to the stacked system category. As a consequence, the sectors associated with an invertible state are trivial, namely, invertible states support no anyonic quasi-particles.