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An invitation to topological orders and category theory

Liang Kong, Zhi-Hao Zhang

TL;DR

<3-5 sentence high-level summary> The paper develops a detailed bridge from lattice models of topological orders to the language of category theory, showing that the low-energy excitables form unitary (braided) fusion categories and, in 2d, unitary modular tensor categories. It builds this bridge by starting from physical lattices (e.g., the toric code) and deriving the categorical structures (objects, morphisms, fusion, braiding, twists) as natural consequences of coarse-graining, locality, and defect fusion, while also introducing the conceptual framework of gapped liquids, anomalies, and bulk-boundary relations. The work then surveys a broad set of constructions and examples—Vec, Hilb, Rep(G), Vec_G^ω, D_G, Ising-type categories, and quantum doubles—illuminating how these mathematical structures encode topological data, quantum dimensions, S/T matrices, Frobenius–Perron dimensions, and modular data essential for classifying 2d topological orders. Overall, the paper positions category theory as a practical, unifying calculus for quantum many-body physics, linking microscopic lattice realizations to abstract, computable invariants with wide implications for condensed matter, quantum gravity, and beyond.

Abstract

Although it has been a well-known fact, for more than two decades, that category theory is needed for the study of topological orders, it is still a non-trivial challenge for students and working physicists to master the abstract language of category theory. In this work, for those who have no background in category theory, we explain in great details how the structure of a (braided) fusion category naturally emerges from lattice models and physical intuitions. Moreover, we show that nearly all mathematical notions and constructions in fusion categories and its representation theory, such as (monoidal) functors, Drinfeld center, module categories, Morita equivalence, condensation completion and fusion 2-categories, naturally emerge from lattice models and physical intuitions. In this process, we also introduce some basic notions and important results of topological orders.

An invitation to topological orders and category theory

TL;DR

<3-5 sentence high-level summary> The paper develops a detailed bridge from lattice models of topological orders to the language of category theory, showing that the low-energy excitables form unitary (braided) fusion categories and, in 2d, unitary modular tensor categories. It builds this bridge by starting from physical lattices (e.g., the toric code) and deriving the categorical structures (objects, morphisms, fusion, braiding, twists) as natural consequences of coarse-graining, locality, and defect fusion, while also introducing the conceptual framework of gapped liquids, anomalies, and bulk-boundary relations. The work then surveys a broad set of constructions and examples—Vec, Hilb, Rep(G), Vec_G^ω, D_G, Ising-type categories, and quantum doubles—illuminating how these mathematical structures encode topological data, quantum dimensions, S/T matrices, Frobenius–Perron dimensions, and modular data essential for classifying 2d topological orders. Overall, the paper positions category theory as a practical, unifying calculus for quantum many-body physics, linking microscopic lattice realizations to abstract, computable invariants with wide implications for condensed matter, quantum gravity, and beyond.

Abstract

Although it has been a well-known fact, for more than two decades, that category theory is needed for the study of topological orders, it is still a non-trivial challenge for students and working physicists to master the abstract language of category theory. In this work, for those who have no background in category theory, we explain in great details how the structure of a (braided) fusion category naturally emerges from lattice models and physical intuitions. Moreover, we show that nearly all mathematical notions and constructions in fusion categories and its representation theory, such as (monoidal) functors, Drinfeld center, module categories, Morita equivalence, condensation completion and fusion 2-categories, naturally emerge from lattice models and physical intuitions. In this process, we also introduce some basic notions and important results of topological orders.
Paper Structure (78 sections, 6 theorems, 212 equations, 73 figures, 1 table)

This paper contains 78 sections, 6 theorems, 212 equations, 73 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivations
  3. Significances and impacts
  4. Category theory as a new language
  5. Layout and how to read
  6. Basic notions for topological orders
  7. Gapped quantum phases
  8. Rn-observables
  9. Anomaly-free/anomalous quantum phases
  10. Topological defects and topological skeletons
  11. Topological orders in 2d
  12. Toric code model
  13. Definition of the 2d toric code model
  14. The ground state degeneracy and ground state wave functions
  15. The category of particle-like topological defects
  16. ...and 63 more sections

Key Result

Theorem 3.3.52

A functor $F \colon \EuScript{C} \to \EuScript{D}$ between two categories $\EuScript{C},\EuScript{D}$ is an equivalence if and only if it satisfies the following conditions:

Figures (73)

  • Figure 1: the energy spectrum of a typical gapped system
  • Figure 2: The notion of a phase is local in the sense that it is defined on an open disk.
  • Figure 3: a dimensional reduction process
  • Figure 4: A defect in a lattice model can be given by introducing dislocations, enlarging local Hilbert spaces, or adding local operators to the Hamiltonian. Topological defects are defects in the long wave length limit.
  • Figure 5: The topological defects in various dimensions can be fused together.
  • ...and 68 more figures

Theorems & Definitions (289)

  • Remark 1.2.1
  • Remark 1.3.1
  • Remark 2.1.1
  • Example 2.1.2
  • Example 2.1.3
  • Remark 2.1.4
  • Remark 2.1.5
  • Definition 2.3.1
  • Remark 2.3.3
  • Remark 2.4.1
  • ...and 279 more