Models for gapped boundaries and domain walls (1104.5047v4)

Abstract: We define a class of lattice models for two-dimensional topological phases with boundary such that both the bulk and the boundary excitations are gapped. The bulk part is constructed using a unitary tensor category $\calC$ as in the Levin-Wen model, whereas the boundary is associated with a module category over $\calC$. We also consider domain walls (or defect lines) between different bulk phases. A domain wall is transparent to bulk excitations if the corresponding unitary tensor categories are Morita equivalent. Defects of higher codimension will also be studied. In summary, we give a dictionary between physical ingredients of lattice models and tensor-categorical notions.

• The paper introduces an extension of Levin-Wen models by modeling gapped boundaries as C-module categories hosting unique superselection sectors.
• It demonstrates that domain walls correspond to bimodule categories and can become transparent under Morita equivalence, allowing unobstructed quasiparticle flow.
• The work rigorously maps tensor categorical structures to topological phases, offering valuable insights for designing fault-tolerant quantum computation systems.

Analyzing Models for Gapped Boundaries and Domain Walls

The paper "Models for gapped boundaries and domain walls" by Alexei Kitaev and Liang Kong presents a comprehensive exploration of two-dimensional topological phases of matter using lattice models. It focuses on the intricate relations between bulk and boundary physics in these phases, particularly through the lens of gapped excitations both in the bulk and along boundaries and domain walls. The paper utilizes sophisticated mathematical frameworks, notably involving unitary tensor categories, to characterize these physical systems.

The core of this work is set within the field of Levin-Wen models, a class of exactly solvable lattice models that describe topological phases by employing string-net condensations. These models provide a platform to investigate the properties of bulk quasiparticles, which are shown to correspond to objects in the monoidal center $Z(C)$ of a unitary tensor category $C$.

Key Contributions

1. Gapped Boundaries: The authors extend the Levin-Wen construction by introducing boundaries defined by $C$-module categories, where $C$ is a unitary tensor category. These boundaries are not merely passive; they host unique excitations characterized by module functors over $C$. The work meticulously constructs the algebraic framework necessary to describe these excitations, describing them as superselection sectors of a boundary Hamiltonian.
2. Domain Walls: The paper also deals with the presence of domain walls between distinct topological phases. Such walls are found to correspond to bimodule categories that interpolate between unitary tensor categories associated with different bulk phases. A particularly interesting aspect is the introduction of "transparent" domain walls, which do not scatter quasiparticles — a condition tied to Morita equivalence between the tensor categories representing the bulk regions.
3. Defects of Higher Codimension: The investigation goes further to include defects of higher codimension that offer a richer varied structure, providing a framework that connects complex excitations and transitions across walls and boundaries. These defects are linked to natural transformations between bimodule functors, enriching the interplay between topology and category theory within these systems.
4. Mathematical Structure and Physical Implications: The mappings between physical ingredients of Levin-Wen models and tensor-categorical notions are rigorously established. This dictionary not only clarifies conceptual understanding but also suggests methodologies for constructing new models of topological phases with desired boundary and defect properties.

Implications and Future Directions

This paper's implications stretch across the field of condensed matter physics and quantum computation, offering new tools for constructing and analyzing topologically ordered phases. The identification of transparent domain walls could be pivotal for quantum information protocols that rely on fault-tolerant quantum computation with anyonic excitations. Furthermore, the proposed framework paves the way for exploring higher-dimensional topological quantum field theories, where categorical higher structures become more pronounced.

The findings encourage future exploration in several directions:

• Experimental Realization: Guidance on how the theoretical constructs can be realized in materials or synthetic quantum systems, such as cold atoms or photonic crystals.
• Extension to Other Dimensions: Developing similar frameworks for three-dimensional topological phases, where tensor categories might be replaced by richer algebraic structures, such as higher categories.
• Integration with Quantum Computing: Leveraging the insights for topological quantum computation, especially in designing systems that naturally host non-Abelian anyons with enriched fusion and braiding statistics.

Overall, this paper significantly advances the understanding of how topological phases can exhibit complex boundary behaviors, underpinned by deep mathematical structures.