Classification of topological phases in periodically driven interacting systems

Abstract: We consider topological phases in periodically driven (Floquet) systems exhibiting many-body localization, protected by a symmetry $G$. We argue for a general correspondence between such phases and topological phases of undriven systems protected by symmetry $\mathbb{Z} \rtimes G$, where the additional $\mathbb{Z}$ accounts for the discrete time translation symmetry. Thus, for example, the bosonic phases in $d$ spatial dimensions without intrinsic topological order (SPT phases) are classified by the cohomology group $H^{d+1}(\mathbb{Z} \rtimes G, \mathrm{U}(1))$. For unitary symmetries, we interpret the additional resulting Floquet phases in terms of the lower-dimensional SPT phases that are pumped to the boundary during one time step. These results also imply the existence of novel symmetry-enriched topological (SET) orders protected solely by the periodicity of the drive.

Overview of the Classification of Topological Phases in Periodically Driven Interacting Systems

The paper focuses on the complex classification of topological phases in periodically driven systems, specifically addressing those with many-body localization (MBL). The authors, Else and Nayak, propose a comprehensive framework to understand these phases in Floquet systems characterized by a symmetry ( G ). They assert a connection between these phases and topological phases in undriven systems safeguarded by the symmetry ( Z \rtimes G ), utilizing the cohomology group ( H^{d+1}(Z \rtimes G, U(1)) ).

Key Contributions

1. Classification Schema for Floquet-MBL Systems:

   • The paper lays out a method for categorizing symmetry-protected topological (SPT) phases in driven systems. Through detailed theoretical analysis, it corresponds these to phases in systems with discrete time-translation symmetry.

2. General Correspondence:

   • The authors postulate a more comprehensive correspondence between driven and stationary topological phases, including those in higher dimensions. They propose a classification schema for bosonic systems, demonstrating its applicability to further spatial dimensions and suggesting a potential alignment with ( H^d(G, U(1)) ) for stationary systems.

3. Implications of Periodicity:

   • A novel suggestion posited in the paper is that periodicity alone could protect certain SPT orders, encouraging further investigation into Floquet systems that may stabilize such novel phases.

Theoretical and Practical Implications

• Extension to Higher Dimensions:

  • The results can be extended to higher-dimensional systems, posing intriguing questions about the behavior and classification of topological phases in such scenarios.

• Enhanced Understanding of SET Orders:

  • The work implies new manifestations of symmetry-enriched topological (SET) orders that arise specifically in periodically driven settings.

• Impact on Future Research:

  • The provided conjecture regarding the correspondence between Floquet-driven and stationary topological phases lays the foundation for future exploration into complex quantum behavior and the broadened applicability in designing experiments focused on MBL systems.

Future Directions and Speculation

This comprehensive analysis could motivate further inquiries into understanding topological phases beyond the realm of conventional undriven systems, challenging established paradigms in quantum physics. As the classification of driven systems becomes increasingly intricate, burgeoning theories may lead to advancements in manipulating quantum states deliberately in applied physics, potentially creating novel quantum computing methodologies.

Furthermore, the exploration of intrinsic properties linked to periodicity suggests areas for experimental realization that might confirm the theorized existence of unique driven phases. Collaborative efforts between theoretical physicists and experimentalists will be pivotal in translating these insights into tangible technology breakthroughs.

This paper marks a progressive step towards decoding the multifaceted characteristics of periodically driven systems, thereby enriching the field of quantum topological phases and projecting new trajectories for academic endeavor.