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arXiv:1701.01440 · Physical review. B./Physical review. B

Radical chiral Floquet phases in a periodically driven Kitaev model and beyond

Hoi Chun Po, Lukasz Fidkowski, Ashvin Vishwanath, Andrew C. Potter

Published Dec 12, 2017• 91 citations
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Abstract

Chiral topological phases, like quantum Hall states, are characterized by anomalous unidirectional edge transport arising from a nontrivial insulating bulk. Recent developments suggest that they can be extended to periodically driven (Floquet) systems, where quantum information is pumped in lieu of charge or energy. The authors uncover a new chiral Floquet phase of quantum dynamics, whose edge pumps quantized units of fractionalized excitations. These phases are ``radical,'' in that the quantum dimension pumped is the square root of an integer. Unexpectedly, such edge borders a bulk with dynamical anyon transmutation, demonstrating a novel form of bulk-boundary correspondence.

DOI
10.1103/physrevb.96.245116

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Chiral topological phases, like quantum Hall states, are characterized by anomalous unidirectional edge transport arising from a nontrivial insulating bulk. Recent developments suggest that they can be extended to periodically driven (Floquet) systems, where quantum information is pumped in lieu of charge or energy. The authors uncover a new chiral Floquet phase of quantum dynamics, whose edge pumps quantized units of fractionalized excitations. These phases are ``radical,'' in that the quantum dimension pumped is the square root of an integer. Unexpectedly, such edge borders a bulk with dynamical anyon transmutation, demonstrating a novel form of bulk-boundary correspondence.
We construct a many-body quantized invariant that sharply distinguishes among two-dimensional nonequilibrium driven phases of interacting fermions. This is an interacting generalization of a band-structure Floquet quasienergy winding number and describes chiral pumping of quantum information along the edge. In particular, our invariant sharply distinguishes between a trivial and anomalous Floquet Anderson insulator in the interacting, many-body localized setting. It also applies more generally to models where only fermion parity is conserved, where it differentiates between trivial models and ones that pump Kitaev Majorana chains to the boundary, such as ones recently introduced in the context of emergent fermions arising from eigenstate Z2{\mathbb{Z}}_{2}Z2​ topological order. We evaluate our invariant for the edge of such a system with eigenstate Z2{\mathbb{Z}}_{2}Z2​ topological order, and show that it is necessarily nonzero when the Floquet unitary exchanges electric and magnetic excitations, proving a connection between bulk anyonic symmetry and edge chirality that was recently conjectured.
Chiral Parafermions in Floquet ZN\mathbb{Z}_NZN​ Topological Order
Floquet topological order (FTO) extends topology beyond equilibrium, giving rise to phenomena, such as the dynamical transmutation of anyons. Here, we realize ZN\mathbb{Z}_NZN​ FTO in a generalized Kitaev honeycomb model with clock degrees of freedom using a three-step periodic drive. The driven system admits an emergent description in terms of constrained parafermions coupled to conserved ZN\mathbb{Z}_NZN​ gauge fields. At the boundary, the drive generates chiral parafermionic Floquet edge modes. We resolve their propagation through correlation functions, characterize their fractional quantum-information transport through the radical chiral unitary index, and use their dynamics to probe exchange statistics. In the bulk, we explore the dynamical anyon transmutation with a non-equilibrium loop order parameter. At a Floquet sweet spot, where the drive becomes Clifford, these phenomena can be accessed using stabilizer simulations. Short-time matrix-product-state simulations away from this point show that the chiral edge response persists at accessible times. Our results establish ZN\mathbb{Z}_NZN​ Kitaev models as a setting for exploring intrinsically non-equilibrium topological phenomena on qudit-based quantum processors.
Topologically protected edge states exactly at topological phase boundaries challenge the conventional belief that topological states must be associated with a bulk energy gap. Because periodically driven (Floquet) systems host unusually intricate topological phase boundaries, topological edge states can be prolific at such Floquet quantum criticality. Working on a class of chiral-symmetric, Floquet-driven Majorana fermion chains, we analytically and computationally show that the precise boundaries between different Floquet topological gapped phases can accommodate topological edge modes, including the so-called critical Majorana π modes. We also identify a general bulk-edge correspondence formula to predict and understand the emergence of topological edge modes at Floquet quantum criticality. Of direct interest to quantum simulation experiments, our work opens an avenue for exploring topological physics intrinsic to nonequilibrium phase boundaries. The authors develop a theoretical framework to characterize the topology and edge states at the nonequilibrium critical points of periodically driven (Floquet) quantum systems. The rule of bulk-edge correspondence applicable to both gapped and gapless Floquet topological phases in 1D chiral symmetric driven systems is proposed.
We study two-terminal transport through two-dimensional periodically driven systems in which all bulk Floquet eigenstates are localized by disorder. We focus on the anomalous Floquet-Anderson insulator (AFAI) phase, a topologically nontrivial phase within this class, which hosts topologically protected chiral edge modes coexisting with its fully localized bulk. We show that the unique properties of the AFAI yield remarkable far-from-equilibrium transport signatures: for a large bias between leads, a quantized amount of charge is transported through the system each driving period. Upon increasing the bias, the chiral Floquet edge mode connecting source to drain becomes fully occupied and the current rapidly approaches its quantized value.
We construct and classify chiral topological phases in driven (Floquet) systems of strongly interacting bosons, with finite-dimensional site Hilbert spaces, in two spatial dimensions.The construction proceeds by introducing exactly soluble models with chiral edges, which in the presence of many-body localization (MBL) in the bulk are argued to lead to stable chiral phases.These chiral phases do not require any symmetry and in fact owe their existence to the absence of energy conservation in driven systems.Surprisingly, we show that they are classified by a quantized many-body index, which is well defined for any MBL Floquet system.The value of this index, which is always the logarithm of a positive rational number, can be interpreted as the entropy per Floquet cycle pumped along the edge, formalizing the notion of quantum-information flow.We explicitly compute this index for specific models and show that the nontrivial topology leads to edge thermalization, which provides an interesting link between bulk topology and chaos at the edge.We also discuss chiral Floquet phases in interacting fermionic systems and their relation to chiral bosonic phases.
The manifestations of topology are ubiquitous in condensed matter physics. One of the most striking examples is the integer quantum Hall effect, which served as a theoretical basis for developing many classes of new materials in the last years such as topological insulators. It has been realized that topology can manifest itself also in time-dependent quantum systems. The question that we try to address in this thesis is how much of these topological structures can be harnessed in the realistic dynamics of a periodically driven closed quantum system. We investigate two kinds of topological periodically driven systems Thouless pumping and Floquet Chern insulators. Our analysis shows that the nonequilibrium occupations of the eigenvalues of the Floquet Hamiltonian, the so called quasienergies, play a crucial role. In the case of Thouless pumping we analysed the pumped charge averaged over many cycles and discovered that the corrections to perfect quantization of the pumped charge are actually quadratic in the frequency of the driving for a suddenly switched-on driving. In the case of Floquet Chern insulators, we studied fermions on the honeycomb lattice subjected to a circularly polarised field. We found that while the quasienergy bulk states can be populated in a controlled way, edge states get unavoidable excitations due to exponentially small gaps preventing adiabaticity. These defects reflect in nonequilibrium currents flowing at the edges, that can be manipulated by tailoring the parameters of the periodic driving. Nevertheless, the nonequilibrium state undergoes a generalized form of topological transition, thanks to the controlled population of bulk states. We describe it with a generalized version of the Chern marker of Bianco and Resta