Vienna Center for Quantum Technology, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria · Department of Physics and Astronomy, Ghent University, Krijgslaan 281 S9, B-9000 Ghent, Belgium
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Abstract
We study symmetry-enriched topological order in two-dimensional tensor network states by using graded matrix product operator algebras to represent symmetry-induced domain walls. A close connection to the theory of graded unitary fusion categories is established. Tensor network representations of the topological defect superselection sectors are constructed for all domain walls. The emergent symmetry-enriched topological order is extracted from these representations, including the symmetry action on the underlying anyons. Dual phase transitions, induced by gauging a global symmetry, and condensation of a bosonic subtheory, are analyzed and the relationship between topological orders on either side of the transition is derived. Several examples are worked through explicitly.
This paper investigates how global symmetries interact with topological order in two-dimensional quantum systems, using a mathematical framework called tensor networks to describe the states. The authors focus on how symmetries create 'domain walls'—boundaries between regions where the symmetry acts differently—and represent these walls using algebraic structures called graded matrix product operator algebras. They show that this framework connects naturally to the mathematical theory of graded unitary fusion categories, providing a rigorous language for describing symmetry-enriched topological phases.
The paper constructs explicit tensor network representations for the 'defects' that arise at these domain walls, and uses them to extract the full structure of symmetry-enriched topological order, including how the symmetry acts on the anyons (the emergent particle-like excitations of the topological phase). The authors also study two related processes—gauging a symmetry (turning it into a dynamical gauge field) and condensing certain anyons—both of which can drive transitions between different topological phases. They derive precise relationships between the topological orders before and after these transitions and illustrate their results with several worked examples.
Plain-language summary by The Gufo; the paper is the authors’ own text.
Key findings
Graded matrix product operator algebras provide a tensor-network representation of symmetry-induced domain walls in two-dimensional topological states.
This framework is closely connected to the mathematical theory of graded unitary fusion categories.
Tensor network representations of topological defect superselection sectors can be constructed for all domain walls, revealing the symmetry-enriched topological order and the symmetry action on anyons.
Gauging a global symmetry and condensing a bosonic anyon subtheory induce dual phase transitions, and the paper derives the relationship between the topological orders on either side of these transitions.
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