Classifying 2D topological phases: mapping ground states to string-nets

TL;DR

This paper proves the classification of 2D topological phases with gappable boundary, showing they correspond to string-net models and are characterized by unitary modular tensor categories, under certain assumptions.

Contribution

It establishes that all such phases can be transformed into string-net states via constant-depth circuits, confirming the Levin-Wen conjecture under specific conditions.

Findings

All gapped phases with gappable boundary are equivalent to string-net models.

Phases are labeled by unitary modular tensor categories.

States with zero correlation length can be transformed into string-net states.

Abstract

We prove the conjectured classification of topological phases in two spatial dimensions with gappable boundary, in a simplified setting. Two gapped ground states of lattice Hamiltonians are in the same quantum phase of matter, or topological phase, if they can be connected by a constant-depth quantum circuit. It is conjectured that the Levin-Wen string-net models exhaust all possible gapped phases with gappable boundary, and these phases are labeled by unitary modular tensor categories. We prove this under the assumption that every phase has a representative state with zero correlation length satisfying the entanglement bootstrap axioms, or a strict form of area law. Our main technical development is to transform these states into string-net states using constant-depth quantum circuits.