Braiding statistics approach to symmetry-protected topological phases (1202.3120v2)

Abstract: We construct a 2D quantum spin model that realizes an Ising paramagnet with gapless edge modes protected by Ising symmetry. This model provides an example of a "symmetry-protected topological phase." We describe a simple physical construction that distinguishes this system from a conventional paramagnet: we couple the system to a Z_2 gauge field and then show that the \pi-flux excitations have different braiding statistics from that of a usual paramagnet. In addition, we show that these braiding statistics directly imply the existence of protected edge modes. Finally, we analyze a particular microscopic model for the edge and derive a field theoretic description of the low energy excitations. We believe that the braiding statistics approach outlined in this paper can be generalized to a large class of symmetry-protected topological phases.

• The paper demonstrates a 2D quantum spin model exhibiting semionic braiding statistics in a Z2 symmetric SPT phase.
• The study shows that coupling to a gauge field protects gapless edge modes through Ising symmetry.
• The work paves the way for extending the braiding statistics approach to classify broader classes of SPT phases.

Overview of "Braiding Statistics Approach to Symmetry-Protected Topological Phases"

The paper "Braiding statistics approach to symmetry-protected topological phases" by Michael Levin and Zheng-Cheng Gu investigates the theoretical construction and analysis of a novel type of symmetry-protected topological (SPT) phase in two-dimensional systems. The focus is on a specific SPT phase characterized by a $\mathbb{Z}_2$ (Ising-like) symmetry. Unlike conventional paramagnets, this phase is distinguished by its unique braiding statistics — particularly in how it interacts with $\pi$-flux excitations when coupled to a $\mathbb{Z}_2$ gauge field.

Key Concepts and Methodology

The paper constructs a solvable 2D quantum spin model that exhibits gapless edge modes protected by Ising symmetry, serving as a concrete example of an SPT phase. The methodology entails:

• Developing a spin model that supports these edge modes and investigating its properties through coupling it with a $\mathbb{Z}_2$ gauge field.
• Demonstrating that the system exhibits distinct braiding statistics when compared to typical paramagnets, particularly through the interaction of $\pi$-flux excitations that manifest semionic statistics, unlike the bosonic or fermionic statistics in conventional paramagnets.
• Through dualities, relating the spin models to known lattice models such as the toric code and the doubled semion models, thereby linking these SPT phases with established classification schemes for gauge theories.

Results and Analyses

The authors present several significant results:

• Braiding Statistics of Excitations: The paper rigorously shows that upon coupling with a gauge field, the new type of paramagnet exhibits quasiparticles with nontrivial braiding, specifically semionic statistics for flux excitations, which is not observable in conventional paramagnets.
• Protected Edge States: It is demonstrated that the existence of semionic braiding statistics implies the protection of gapless edge modes, robust under the Ising symmetry. This is a critical distinction from conventional paramagnets where edge states can be gapped out without symmetry-breaking.
• Generalization Potential: The paper opens a pathway to generalize this braiding statistics approach to a wide class of SPT phases, indicating that these properties could potentially be employed to explore and classify other SPT phases in higher dimensions or with different symmetry groups.

Implications and Future Directions

Theoretical implications of this work are profound. It provides a framework for understanding and classifying SPT phases through braiding statistics, emphasizing the role of gauge field couplings in uncovering novel topological properties. On the practical side, realizing such systems could lead to advancements in quantum computation, particularly in topological quantum computers where protected edge modes and nontrivial quasiparticle statistics play crucial roles.

The authors conjecture potential extensions of their approach beyond $\mathbb{Z}_2$ symmetric systems to higher dimensions and other symmetry types, both unitary and anti-unitary. If these braiding statistics approaches can be generalized, they might offer an unifying language to discuss broad classes of SPT phases, especially in how such phases behave and transition in multidimensional spaces.

Finally, the application of this theoretical framework to real material systems or quantum simulations could provide experimental validation, thereby enhancing our understanding of SPT phases and their potential utility in future technologies.