Projector Matrix Product Operators, Anyons and Higher Relative Commutants of Subfactors

TL;DR

This paper establishes a mathematical link between projector matrix product operators used in topological order models and higher relative commutants in subfactor theory, enriching the understanding of anyons and topological phases.

Contribution

It proves that the range of certain projector matrix product operators corresponds to higher relative commutants in subfactor theory, connecting topological order with subfactor structures.

Findings

Range of projector matrix product operator identified with higher relative commutant

Provides a new link between topological order and subfactor theory

Enhances understanding of anyons in 2D topological phases

Abstract

A bi-unitary connection in subfactor theory of Jones producing a subfactor of finite depth gives a 4-tensor appearing in a recent work of Bultinck-Mariena-Williamson-Sahinoglu-Haegemana-Verstraete on 2-dimensional topological order and anyons. In their work, they have a special projection called a projector matrix product operator. We prove that the range of this projection of length k is naturally identified with the k-th higher relative commutant of the subfactor arising from the bi-unitary connection. This gives a further connection between 2-dimensional topological order and subfactor theory.