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arXiv:1804.02004 · Physical review. B./Physical review. B

Tracking the quantized information transfer at the edge of a chiral Floquet phase

Blake Duschatko, Philipp T. Dumitrescu, Andrew C. Potter

Published Aug 30, 2018• 17 citations
arXivHTMLPDF

Abstract

Two-dimensional arrays of periodically driven qubits can host inherently dynamical topological phases with anomalous chiral edge dynamics. These chiral Floquet phases are formally characterized by a dynamical topological invariant, the chiral unitary index. Introducing a quantity called the chiral mutual information, we show that this invariant can be precisely interpreted in terms of a quantized chiral transfer of quantum information along the edge of the system, and devise a physical setup to measure it.

DOI
10.1103/physrevb.98.054309

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Two-dimensional arrays of periodically driven qubits can host inherently dynamical topological phases with anomalous chiral edge dynamics. These chiral Floquet phases are formally characterized by a dynamical topological invariant, the chiral unitary index. Introducing a quantity called the chiral mutual information, we show that this invariant can be precisely interpreted in terms of a quantized chiral transfer of quantum information along the edge of the system, and devise a physical setup to measure it.
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.
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.
Periodically driven noninteracting systems may exhibit anomalous chiral edge modes, despite hosting bands with trivial topology. We find that these drives have surprising many-body analogs, corresponding to class A, which exhibit anomalous charge and information transport at the boundary. Drives of this form are applicable to generic systems of bosons, fermions, and spins, and may be characterized by the anomalous unitary operator that acts at the edge of an open system. We find that these operators are robust to all local perturbations and may be classified by a pair of coprime integers. This defines a notion of dynamical topological order that may be applied to general time-dependent systems, including many-body localized phases or time crystals.
Discrete-time quantum walks have been shown to simulate all known topological phases in one and two dimensions. Being periodically driven quantum systems, their topological description, however, is more complex than that of closed Hamiltonian systems. We map out the topological phases of the particle-hole symmetric one-dimensional discrete-time quantum walk. We find that there is no chiral symmetry in this system: its topology arises from the particle-hole symmetry alone. We calculate the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ topological invariant in a simple way that is consistent with a general definition for one-dimensional periodically driven quantum systems. These results allow for a transparent interpretation of the edge states on a finite lattice via the the bulk-boundary correspondence. We find that the bulk Floquet operator does not contain all the information needed for the topological invariant. As an illustration to this statement, we show that in the split-step quantum walk, the edges between two bulks with the same Floquet operator can host topologically protected edge states.
Abstract Out-of-equilibrium phases in many-body systems constitute a new paradigm in quantum matter—they exhibit dynamical properties that may otherwise be forbidden by equilibrium thermodynamics. Among these non-equilibrium phases are periodically driven (Floquet) systems1–5, which are generically difficult to simulate classically because of their high entanglement. Here we realize a Floquet topologically ordered state theoretically proposed in ref. 6, on an array of superconducting qubits. We image the characteristic dynamics of its chiral edge modes and characterize its emergent anyonic excitations. Devising an interferometric algorithm allows us to introduce and measure a bulk topological invariant to probe the dynamical transmutation of anyons for system sizes up to 58 qubits. Our work demonstrates that quantum processors can provide key insights into the thus-far largely unexplored landscape of highly entangled non-equilibrium phases of matter.