Condensate induced transitions between topologically ordered phases (0808.0627v1)

Abstract: We investigate transitions between topologically ordered phases in two spatial dimensions induced by the condensation of a bosonic quasiparticle. To this end, we formulate an extension of the theory of symmetry breaking phase transitions which applies to phases with topological excitations described by quantum groups or modular tensor categories. This enables us to deal with phases whose quasiparticles have non-integer quantum dimensions and obey braid statistics. Many examples of such phases can be constructed from two-dimensional rational conformal field theories and we find that there is a beautiful connection between quantum group symmetry breaking and certain well-known constructions in conformal field theory, notably the coset construction, the construction of orbifold models and more general conformal extensions. Besides the general framework, many representative examples are worked out in detail.

• The paper demonstrates that bosonic quasiparticle condensation can induce phase transitions in systems with non-integer quantum dimensions.
• It introduces a framework extending quantum group symmetry breaking to connect with conformal field theory via coset and orbifold constructions.
• Key examples like the SU(2)_4 WZW model and discrete gauge theories highlight implications for fractional quantum Hall systems and topological quantum computation.

Condensate Induced Transitions between Topologically Ordered Phases

This paper investigates transitions between topologically ordered phases in two spatial dimensions, mediated by the condensation of a bosonic quasiparticle. It presents a theoretical framework extending the principles of symmetry breaking phase transitions to phases with topological excitations described by modular tensor categories or quantum groups. The main focus lies on phases whose quasiparticles exhibit non-integer quantum dimensions and obey braid statistics, such as those derived from two-dimensional rational conformal field theories (CFTs). The paper uncovers a profound connection between the quantum group symmetry breaking phenomena and certain constructions in CFT, like the coset construction and orbifold models. A series of representative examples substantiate this conceptual framework.

Key Concepts and Methods

The paper revolves around several crucial concepts:

1. Topologically Ordered Phases: These are quantum phases of matter characterized by ground state degeneracies and quasiparticle excitations with exotic statistics (anyons). Unlike symmetry-breaking phases, they are not described by local order parameters.
2. Bosonic Quasiparticles: The condensation process studied involves quasiparticles having trivial spin and partially trivial self-monodromy, which are essential for inducing the phase transition.
3. Quantum Group Symmetry Breaking: This extends notions of symmetry breaking to quantum group symmetries, integrating representations of modular tensor categories that describe fusion and braiding of anyons.
4. Condensation Framework: The process assumes a system described by quantum group $A$. Condensation leads to symmetry breaking of $A$ down to a subalgebra $T$, and excitations are rerouted into representations of $T$. This framework mandates that fusion operations in the broken phase remain associative, with duals uniquely defined.
5. Confinement: The paper also deals with confinement phenomena in the new phase, where certain excitations become confined entities, pulling strings in the condensate. These excitations are distinguished by their monodromy with the condensate.

Results and Examples

The framework is articulated through various examples, notably:

• $SU(2)_4$ WZW Model: Condensate transitions in this model reveal the connection with constraints in CFTs and demonstrate how the new non-confined phases potentially emerge as $SU(3)_1$ phases.
• Discrete Gauge Theories: These models, described by quantum doubles of finite groups, exemplify how the approach can seamlessly discuss transitions in non-Abelian gauge theories with discrete symmetries, characterized by transformations like the Higgs effect.
• CFT Extensions and Embeddings: The framework successfully connects to conformal extensions and embeddings, offering physical insights into algebra enlargements and understanding how bosonic fields in CFTs can serve as conduits for embedding and phase transitions.

Implications

The research has profound implications for both theoretical understanding and practical exploration of topologically ordered phases:

1. Fractional Quantum Hall Systems (FQHS): The theoretical structure aids in the conceptualization of phase transitions in FQHS, enriching the toolkit available for modeling these systems' complex behaviors.
2. Topological Quantum Computation (TQC): Understanding such transitions is vital for TQC, where manipulation and control of anyonic excitations are foundational to computational schemes designed to be fault-tolerant.
3. Extensions to Other Phases: The versatility of the framework suggests potential applicability to other topologically orchestrated phenomena and transitions, broadening the horizons for future research on exotic phases of matter.

Future Directions

The paper opens avenues for further exploration:

• Mathematical Rigor: Efforts can be directed toward a more meticulous grounding of the proposed frameworks within the formal mathematics of modular tensor categories.
• Experimental Realizations: Investigating these theoretical propositions in experimental settings, particularly in fractional quantum Hall setups or engineered Josephson junction arrays, to possibly manifest and control the predicted transitions.
• Other Constructions: Exploring constructions like doubled Chern-Simons theories and other CFT-based models to understand different manifestations of topological order and their potential phase transitions.

In conclusion, the research presents a robust theoretical model for understanding and characterizing phase transitions in topologically ordered phases induced by bosonic condensates, bridging concepts in quantum groups with phenomena in CFT, thus enriching the discourse on topological phases of matter.