Detecting topological order with ribbon operators

2016/03/31 by Jacob C. Bridgeman, Steven T. Flammia, David Poulin
Mathematics · Physics and Astronomy · #Advanced Condensed Matter Physics #Ansatz #Anyon #Combinatorics #Geometry #Ising model #Mathematics #Matrix multiplication #Physics #Physics of Superconductivity and Magnetism #Quantum #Quantum many-body systems #Quantum mechanics #Qubit #Ribbon #String (physics) #Theoretical physics #Topological order #Topological quantum computer #Topology (electrical circuits) #Toric code #cond-mat.str-el #quant-ph

paper · pdf · doi:10.1103/physrevb.94.205123

published as Phys. Rev. B 94, 205123 (2016) · 15 pages, 8 figures, comments welcome

arxiv created 2016/09/07 · openalex publication_date 2016/11/14 · arxiv updated 2016/11/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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 a 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 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 non-Abelian topological order.

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