The Quantum Double Model with Boundary: Condensations and Symmetries (1006.5479v5)

Abstract: Associated to every finite group, Kitaev has defined the quantum double model for every orientable surface without boundary. In this paper, we define boundaries for this model and characterize condensations; that is, we find all quasi-particle excitations (anyons) which disappear when they move to the boundary. We then consider two phases of the quantum double model corresponding to two groups with a domain wall between them, and study the tunneling of anyons from one phase to the other. Using this framework we discuss the necessary and sufficient conditions when two different groups give the same anyon types. As an application we show that in the quantum double model for S_3 (the permutation group over three letters) there is a chargeon and a fluxion which are not distinguishable. This group is indeed a special case of groups of the form of the semidirect product of the additive and multiplicative groups of a finite field, for all of which we prove a similar symmetry.

• The paper presents an extension of Kitaev’s model by introducing boundaries using subgroups and 2-cocycles to enable anyon condensations.
• It rigorously studies ribbon operators that classify excitations as condensed or confined, offering insights for topological quantum error correction.
• The analysis reveals a charge-flux transposition symmetry that may enable robust, fault-tolerant quantum computational systems using anyons.

The Quantum Double Model with Boundary: Condensations and Symmetries

The paper "The Quantum Double Model with Boundary: Condensations and Symmetries" explores the extension of Kitaev's quantum double model to incorporate boundaries and analyze quasi-particle condensations and symmetries. It provides a comprehensive investigation into boundary definitions for quantum double models associated with finite groups and examines the behavior of anyons as they interact with boundaries and domain walls.

Quantum Double Model with Boundary

Originally, Kitaev’s quantum double model was defined for surfaces without boundaries. This paper introduces boundaries by associating a subgroup $K$ of a group $G$ and further extends the model by considering $2$-cocycles, which cluster excitations into condensations near boundary sites.

The quantum double model for a group $G$ assigns a Hilbert space $\mathbb{C}G$ to the edges of a lattice and describes the system's excitations using quasi-particles known as anyons. The Hamiltonian of this model is defined using vertex and face operators that allow the generation and movement of these anyons.

The paper identifies the algebra of ribbon operators, which commutes with certain boundary conditions characterized by $K$ and $2$-cocycles, leading to classification of excitations as condensed or confined. These ribbon operators allow anyons to tunnel to boundary sites and vanish, illustrating the nature of quasi-particle condensations.

Implication of Condensations

Understanding condensations is crucial as certain anyon types are annihilated when moved to boundary sites, indicating a potential for symmetry and simplification in quantum error correction codes. The paper explores the conditions under which these condensations occur and identifies the relationship with modular categories used in mathematical physics.

The representation theory and the character evaluations on the condensed space are pivotal to identifying which anyon types are subject to condensation. This knowledge contributes to the paper of topological quantum computation, where controlled manipulation of anyons can encode and process quantum information.

Symmetry and Tunneling

The paper dives into the interplay between condensation and symmetry, discussing the transitions between different phases characterized by different groups with domain walls. Using the folding approach, it characterizes scenarios where anyons can move between phases without creating additional boundary excitations.

One particularly intriguing outcome is the charge-flux transposition symmetry in systems derived from near-field groups like $\mathbb{F}_q$. This symmetry asserts that certain chargeons and fluxions become indistinguishable, yielding potential applications for realizing Ising anyons from abelian models.

Practical and Theoretical Implications

The results have profound implications for both theoretical studies and practical implementations of topological quantum systems. They aid in developing more refined models of quantum computation, potentially leveraging anyons' unique properties for robust and fault-tolerant information processing.

Exploring boundary applications extends the scope of quantum systems to incorporate complexities like environmental factors and interactions in multi-phase systems, contributing to both the foundational understanding and applied development in quantum technologies.

Future Directions

The paper opens multiple avenues for further research, such as examining more complex boundary configurations on different topological surfaces beyond the plane, understanding higher-order algebras in modular categories, and realizing new quantum phases by controlling anyon-symmetries.

Such studies hold the promise of unveiling new frontiers in quantum computation, bringing into focus the utility of topological methods for innovation and breakthroughs in practical quantum technologies. The interplay between theory and application evident in this paper is emblematic of the dynamic progress in the field of quantum information science.