Symmetry Fractionalization, Defects, and Gauging of Topological Phases (1410.4540v4)

Abstract: We examine the interplay of symmetry and topological order in $2+1$ dimensional topological phases of matter. We present a definition of the \it topological symmetry \rm group, which characterizes the symmetry of the emergent topological quantum numbers of a topological phase, and we describe its relation with the microscopic symmetry of the underlying physical system. We derive a general framework to characterize and classify symmetry fractionalization in topological phases, including phases that are non-Abelian and symmetries that permute the quasiparticle types and/or are anti-unitary. We develop a theory of extrinsic defects (fluxes) associated with elements of the symmetry group, which provides a general classification of symmetry-enriched topological phases derived from a topological phase of matter $\mathcal{C}$ with symmetry group $G$. The algebraic theory of the defects, known as a $G$-crossed braided tensor category $\mathcal{C}{G}^{\times}$, allows one to compute many properties, such as the number of topologically distinct types of defects associated with each group element, their fusion rules, quantum dimensions, zero modes, braiding exchange transformations, a generalized Verlinde formula for the defects, and modular transformations of the $G$-crossed extensions of topological phases. We also examine the promotion of the global symmetry to a local gauge invariance, wherein the extrinsic $G$-defects are turned into deconfined quasiparticle excitations, which results in a different topological phase $(\mathcal{C}{G}^{\times})^{G}$. A number of instructive and/or physically relevant examples are studied in detail.

• The paper presents a novel framework linking symmetry actions, fractionalization, and gauging in 2+1D topological phases.
• It establishes an algebraic approach using Aut(C) and H² classifications to rigorously analyze defect fusion and braiding rules.
• Gauging transforms global symmetries into local invariances, leading to new topological phases and insights into quantum phase transitions.

Overview of "Symmetry Fractionalization, Defects, and Gauging of Topological Phases"

In "Symmetry Fractionalization, Defects, and Gauging of Topological Phases," the authors present a rigorous framework for understanding the interplay between symmetry and topological order in 2+1 dimensional topological quantum phases. This manuscript provides a comprehensive mathematical and physical formalism to analyze how symmetries manifest in these systems and how they lead to the emergence of extrinsic defects and symmetry fractionalization. Furthermore, the paper explores the process of gauging symmetries, effectively transforming global symmetries into local gauge invariances, and explores its implications for topological quantum phase transitions.

Key Contributions

1. Symmetry Action on Topological Phases: The work introduces a precise definition of the "topological symmetry" group, denoted as $\text{Aut}(\mathcal{C})$, which reflects the symmetry of the emergent quantum numbers characterizing a topological phase $\mathcal{C}$. It establishes a link between these symmetries and the microscopic symmetries of the system, governed by a global symmetry group $G$. The interplay between $G$ and $\text{Aut}(\mathcal{C})$ serves as the foundation for analyzing symmetry fractionalization.
2. Symmetry Fractionalization and Classification: The authors develop a framework to classify and characterize symmetry fractionalization, where quasiparticles in topological phases acquire fractional quantum numbers of the global symmetry. They demonstrate that symmetry fractionalization is classified by $H^2_{[\rho]}(G, \mathcal{A})$, where $\mathcal{A}$ is the group of Abelian topological charges, given that the obstruction class $[#1{O}] \in H^3_{[\rho]}(G, \mathcal{A})$ vanishes. If the obstruction is non-zero, symmetry fractionalization is prohibited, highlighting an essential obstruction theory within these systems.
3. Algebraic Theory of Defects: The notion of $G$-crossed braided tensor categories is established, which describes the topological properties of extrinsic defects associated with symmetry group elements. This mathematical structure encapsulates both the fusion and braiding of defects, vital for understanding how symmetries can enrich topological phases. The paper provides detailed equations and consistency constraints for these categories, allowing for computations of defect properties like fusion rules, quantum dimensions, and braiding statistics.
4. Gauging of Symmetries: Gauging transforms a global symmetry into a local one, introducing deconfined quasiparticles and resulting in a new topological phase denoted by $#1{C}{G}$. The paper outlines systematic methods to compute the properties of these gauged phases, contributing significantly to our understanding of topological quantum phase transitions and classification of symmetry-enriched topologies.

Implications and Future Directions

The implications of this paper are manifold. Practically, the ability to classify and characterize symmetry fractionalization provides insights into the possible quantum phases of matter that can be realized and understood in condensed matter systems. Theoretical implications shed light on the structure and classification of symmetry-enriched topological phases beyond traditional paradigms. The paper also suggests a trajectory for future research, particularly in exploring the full extent of defect theory and the rich landscape of quantum phase transitions facilitated by symmetry gauging. The insights garnered from the paper could inspire further exploration into higher-dimensional analogs and the interplay with quantum information science.

In conclusion, this manuscript establishes a robust theoretical groundwork for the exploration of symmetry in topological phases, offering tools that could help unravel the complexities of these systems. It sets the stage for deeper explorations into both foundational aspects of quantum theory and practical applications in advanced materials and quantum technologies.