Detecting Topological Order with Ribbon Operators

TL;DR

This paper presents a numerical approach to detect topological order in 2D quantum models by identifying ribbon operators using matrix product states, applicable to both abelian and nonabelian systems.

Contribution

The authors develop a new numerical method to identify approximate string operators in 2D models, enabling detection of topological order and logical qubits in various quantum spin models.

Findings

Successfully identified ribbon operators in $ ext{Z}_d$ quantum double models.

Detected logical qubits in the Kitaev honeycomb and quantum compass models.

Showed the method can distinguish topologically ordered from trivial phases.

Abstract

We introduce a numerical method for identifying topological order in two-dimensional models based on one-dimensional bulk operators. The idea is to identify approximate symmetries supported on thin strips through the bulk that behave as string operators associated to an anyon model. We can express these ribbon operators in matrix product form and define a cost function that allows us to efficiently optimize over this ansatz class. We test this method on spin models with abelian topological order by finding ribbon operators for $\mathbb{Z}_d$ quantum double models with local fields and Ising-like terms. In addition, we identify ribbons in the abelian phase of Kitaev's honeycomb model which serve as the logical operators of the encoded qubit for the quantum error-correcting code. We further identify the topologically encoded qubit in the quantum compass model, and show that despite this qubit, the model does not support topological order. Finally, we discuss how the method supports generalizations for detecting nonabelian topological order.