[Paper Review] A mathematical theory of anyon condensation

Liang Kong|arXiv (Cornell University)|Jul 31, 2013

Quantum many-body systemsPhysics and Astronomy4 citations

TL;DR

This paper develops a mathematical framework for anyon condensation in 2D topological phases using a bootstrap approach, showing that the condensed phase's anyons (described by modular tensor category D) arise from a connected commutative separable algebra A in the original phase (C). The key result is that D is equivalent to the category of local A-modules, and the condensation process naturally yields a gapped domain wall with A-module excitations.

ABSTRACT

Instead of constructing anyon condensation in various concrete models, we take a bootstrap approach by considering an abstract situation, in which an anyon condensation happens in a 2-d topological phase with anyonic excitations given by a modular tensor category C; and the anyons in the condensed phase are given by another modular tensor category D. By a bootstrap analysis, we derive a relations between anyons in D-phase and anyons in C-phase from natural physical requirements. It turns out that the vacuum (or the tensor unit) A in D-phase is necessary to be a connected commutative separable algebra in C, and the category D is equivalent to the category of local A-modules as modular tensor categories. This condensation also produces a gapped domain wall with wall excitations given by the category of A-modules in C. More general situation is also discussed in this paper. We will also show how to determine such algebra A from the initial and final data. Multi-condensations and 1-d condensations will also be briefly discussed. Examples will be given in the toric code model, Kitaev quantum double models, Levin-Wen types of lattice models and some chiral topological phases.

Motivation & Objective

• To establish a general mathematical framework for anyon condensation in 2D topological phases without relying on specific models.
• To identify the necessary algebraic structure (connected commutative separable algebra) that governs condensation in a modular tensor category.
• To derive the equivalence between the condensed phase's anyon category D and the category of local A-modules in the original category C.
• To show how such condensation gives rise to a gapped domain wall with A-module excitations.
• To provide a systematic method for determining the condensing algebra A from initial and final topological data.

Proposed method

• Using a bootstrap approach based on physical consistency conditions to derive constraints on anyon condensation.
• Identifying the vacuum in the condensed phase as a connected commutative separable algebra A in the original modular tensor category C.
• Establishing that the anyon category D of the condensed phase is equivalent to the category of local A-modules in C.
• Applying categorical duality and module category theory to characterize the structure of the condensation process.
• Deriving the gapped domain wall structure from the A-module category, with wall excitations described by the same A-module category.
• Extending the framework to multi-condensations and 1-dimensional condensations, and illustrating with examples from toric code, Kitaev quantum doubles, and Levin-Wen models.

Experimental results

Research questions

• RQ1 What algebraic structure in the original modular tensor category C must correspond to a valid anyon condensation process?
• RQ2 How is the anyon category D of the condensed phase mathematically related to the original category C?
• RQ3 What is the categorical description of the gapped domain wall formed during condensation?
• RQ4 How can the condensing algebra A be reconstructed from the initial and final topological data?
• RQ5 What generalizations arise in multi-condensation or 1-dimensional condensation scenarios?

Key findings

• The vacuum in the condensed phase is necessarily a connected commutative separable algebra A in the original modular tensor category C.
• The anyon category D of the condensed phase is equivalent to the category of local A-modules in C as modular tensor categories.
• The condensation process naturally produces a gapped domain wall whose excitations are described by the category of A-modules in C.
• The condensing algebra A can be determined from the initial and final topological data using categorical constraints derived from physical consistency.
• The framework generalizes to multi-condensations and 1-dimensional condensations, with consistent categorical descriptions.
• Explicit examples in the toric code, Kitaev quantum double models, Levin-Wen lattice models, and chiral topological phases confirm the theoretical predictions.