Anyon condensation and tensor categories (1307.8244v7)

Abstract: Instead of studying anyon condensation in concrete models, we take an abstract approach. Assume that a system of anyons, which form a modular tensor category D, is obtained via an anyon condensation from another system of anyons (i.e. another modular tensor category C). By a bootstrap analysis, we derive the relation between C and D from natural physical requirements. It turns out that the tensor unit of D can be identified with a connected commutative separable algebra A in C. The modular tensor category D consists of all deconfined particles and can be identified with the category of local $A$-modules in C. If this condensation occurs in a 2d region in the C-phase, then it also produces a 1d gapped domain wall between the C-phase and the D-phase. The confined and deconfined particles accumulate on the wall and form a fusion category that is precisely the category of right A-modules in C. We also consider condensations that are confined to a 1d line. We show how to determine the algebra A from physical macroscopic data. We provide examples of anyon condensation in the toric code model, Kitaev quantum double models and Levin-Wen types of lattice models and in some chiral topological phases. In the end, we briefly discuss Witt equivalence between 2d topological phases. We also attach to this paper an Erratum and Addendum to the original version of "Anyon condensation and tensor categories" published in [Nucl. Phys. B 886 (2014) 436-482].

• The paper introduces an abstract framework using modular tensor categories to model phase transitions in anyonic systems.
• It demonstrates that a connected commutative separable algebra uniquely characterizes the vacuum of the condensed phase.
• The study links anyon condensation to domain wall formation and confined particle behavior, offering insights for topological quantum systems.

An Overview of Anyon Condensation and Tensor Categories

The article under consideration presents a comprehensive paper of anyon condensation in the context of modular tensor categories (MTCs). The authors adopt an abstract framework to investigate how a given MTC, representing a system of anyons, can undergo condensation to yield another MTC. This process is depicted as a phase transition driven by anyon condensation, resulting in a system with fewer anyonic excitations. The paper explores the mathematical structures governing this process, particularly emphasizing the relationship between the original MTC $\mathcal{C}$ and the resulting MTC $\mathcal{D}$ through the introduction of a connected commutative separable algebra $A$ in $\mathcal{C}$.

Key Findings and Methodologies

1. Abstract Approach to Anyon Condensation:
   • Rather than analyzing specific models, the paper takes an abstract approach by utilizing MTCs to describe the anyonic systems before and after condensation. The essence of this analysis is captured by the relation $\mathcal{D} = \mathcal{C}_A^\text{loc}$, where $\mathcal{C}_A^\text{loc}$ denotes the category of local modules of the algebra $A$ in $\mathcal{C}$.
2. Identification of the Tensor Unit:
   • It is determined that the tensor unit of the emergent category $\mathcal{D}$ aligns with a connected commutative separable algebra $A$ in $\mathcal{C}$. This identification is pivotal as it characterizes the vacuum of the condensed phase in $\mathcal{D}$.
3. Domain Wall and Confined Particles:
   • The condensation not only affects the anyonic content of the bulk phase but also engenders a 1-dimensional gapped domain wall separating the $\mathcal{C}$ and $\mathcal{D}$ phases. Particles confined on this domain wall form a fusion category, specifically described as the category of right modules $\mathcal{C}_A$.
4. Physical and Mathematical Implications:
   • The work underscores the power of category theory in understanding phase transitions in two-dimensional topological phases, helping to intuit singular mathematical structures from physical processes. It suggests that mathematical objects, such as algebra $A$, are not merely formal constructions but are deeply tied to the physical characteristics of anyon condensation.
5. Examples and Witt Equivalence:
   • The article discusses examples from the toric code model and other lattice models, showcasing how specific anyon condensation processes relate to modifications in the ground state degeneracy and excitation types. Moreover, Witt equivalence is used to categorize different 2d topological orders, sharing an equivalence relation when connected by gapped domain walls.

Speculative Future Directions

The exploration presented in this paper lays formidable groundwork for further studies in the evolution and transitions of topological phases characterized by anyons. Future research could potentially:

• Extend this analysis to higher-dimensional phases or systems where symmetry considerations play a significant role.
• Investigate the implications of anyon condensation in the dynamics of topologically protected quantum computation and their corresponding error-correcting capabilities.
• Consider the interplay between anyon condensation and gapless phases or phases with nontrivial boundary dynamics.

Conclusion

This work provides an in-depth formal analysis of anyon condensation, illuminating the modular tensor category framework's potential to unravel complex phase transitions in topological quantum systems. By establishing rigorous connections between mathematical categorizations and physical phenomena, it contributes valuable insights into the organization and dynamics of topologically ordered phases. The implications of this paper not only enhance theoretical understanding but also pave the way for practical advancements in the realization and manipulation of anyonic excitations in quantum technology.