A classification of phases of bosonic quantum lattice systems in one dimension

TL;DR

This paper classifies phases of 1D bosonic quantum lattice systems, showing that invertible states are trivial up to ancillas and finite-depth circuits, with symmetry constraints characterized by cohomology indices.

Contribution

It provides a complete classification of invertible 1D bosonic states, including symmetry considerations and cohomological invariants.

Findings

Invertible 1D states are trivial up to ancillas and fuzzy circuits.

Symmetry-preserving disentanglement may be obstructed by cohomology indices.

States with the same index are in the same phase.

Abstract

We study invertible states of 1d bosonic quantum lattice systems. We show that every invertible 1d state is in a trivial phase: after tensoring with some unentangled ancillas it can be disentangled by a fuzzy analog of a finite-depth quantum circuit. If an invertible state has symmetries, it may be impossible to disentangle it in a way that preserves the symmetries, even after adding unentagled ancillas. We show that in the case of a finite unitary symmetry G the only obstruction is an index valued in degree-2 cohomology of $G$. We show that two invertible $G$-invariant states are in the same phase if and only if their indices coincide.