Topology and Broken Symmetry in Floquet Systems (1905.01317v2)

Abstract: Floquet systems are governed by periodic, time-dependent, Hamiltonians. Prima facie they should absorb energy from the external drives involved in modulating their couplings and heat up to infinite temperature. However this unhappy state of affairs can be avoided in many ways. Instead, as has become clear from much recent work, they can exhibit a variety of nontrivial behavior---some of it impossible in undriven systems. In this review we describe the main ideas and themes of this work: novel Floquet drives which exhibit nontrivial topology in single-particle systems, the existence and classification of exotic Floquet drives in interacting systems, and the attendant notion of many-body Floquet phases and arguments for their stability to heating.

Overview of "Topology and Broken Symmetry in Floquet Systems"

This paper provides a comprehensive review of the paper of topology and broken symmetry in periodically driven quantum systems, known as Floquet systems. The authors, Harper, Roy, Rudner, and Sondhi, aim to elucidate the various theoretical developments that have emerged in understanding these systems, which exhibit behaviors not typically possible in static, time-independent scenarios.

Key Concepts and Findings

1. Floquet Systems and Basics: The paper discusses Floquet systems characterized by time-dependent, periodic Hamiltonians. Despite the expectation that these systems should absorb energy and heat to infinite temperature, certain conditions prevent this. Understanding these systems involves the unitary time-evolution operator, capturing the temporal dynamics induced by the periodic driving forces.
2. Single-Particle Topological Phenomena: The paper explores the topological properties of single-particle Floquet systems in both space and time. A major focus is on the emergence of nontrivial phases in these driven systems that mirror topological insulators but can exhibit unique characteristics, such as the 'winding number' not found in static systems. Importantly, the structure of micromotion phases contributes to new forms of bulk-boundary correspondence.
3. Classification and Symmetries: A systematic classification of Floquet topological insulators and superconductors (FTIs) is presented, extending known static topological invariants through the inclusion of dynamical symmetries. The periodic table for FTIs is expanded, incorporating dimensions and symmetry classes via K-theory methods and phase band analysis, unveiling a rich structure of possible topological phases.
4. Interacting Many-Body Systems: The paper transitions to interacting systems, where challenges such as heating are addressed through mechanisms like many-body localization (MBL), which stabilizes nontrivial Floquet phases. Introducing disorder can significantly impact the behavior, allowing for the persistence of exotic orders like 'Floquet time crystals' that manifest broken symmetry in time.
5. Floquet Symmetry-Protected Topological Phases (FSPTs): Similar in spirit to static symmetry-protected topological phases, these systems exhibit robust topological phenomena preserved under symmetry constraints. The dynamical nature of Floquet phases leads to classifications involving group cohomology, allowing for novel edge phenomena such as SPT pumping.
6. Chiral Floquet Phases and Future Directions: A distinct class without symmetry protection is the so-called chiral Floquet phases, which transport information at system boundaries. These unique phases invite speculation on possible nontrivial bulk invariants that correlate with observable phenomena.

Implications and Future Prospects

This review synthesizes a theoretical framework that opens pathways for experimental exploration, particularly where disorder and periodic driving offer control over otherwise complex quantum behaviors. The implications for cold atoms, solid-state systems, and other platforms are profound, suggesting that engineered Floquet systems could innovate in quantum control and computation.

Looking forward, the challenge remains to translate theoretical classifications into tangible experimental signatures. Further exploration into the experimental stabilization of many-body Floquet phases, improved understanding of edge state phenomena under dynamical conditions, and the potential interference with realistic environmental factors will continue to be fruitful areas of research in the domain of driven quantum systems. The authors conclude by inviting further inquiry into the fundamental and practical aspects of these intriguing phases of matter.