We propose a scheme to implement Kitaev's honeycomb model with cold atoms, based on a periodic (Floquet) drive, in view of realizing and probing non-Abelian chiral spin liquids using quantum simulators.We derive the effective Hamiltonian to leading order in the inverse-frequency expansion, and show that the drive opens up a topological gap in the spectrum without mixing the effective Majorana and vortex degrees of freedom.We address the challenge of probing the physics of Majorana fermions, while having access only to the original composite spin degrees of freedom.Specifically, we propose to detect the properties of the chiral spin liquid phase using gap spectroscopy and edge quenches in the presence of the Floquet drive.The resulting chiral edge signal, which relates to the thermal Hall effect associated with neutral Majorana currents, is found to be robust for realistically prepared states.By combining strong interactions with Floquet engineering, our work paves the way for future studies of non-Abelian excitations and quantized thermal transport using quantum simulators.
We propose a scheme to implement Kitaev's honeycomb model with cold atoms, based on a periodic (Floquet) drive, in view of realizing and probing non-Abelian chiral spin liquids using quantum simulators.We derive the effective Hamiltonian to leading order in the inverse-frequency expansion, and show that the drive opens up a topological gap in the spectrum without mixing the effective Majorana and vortex degrees of freedom.We address the challenge of probing the physics of Majorana fermions, while having access only to the original composite spin degrees of freedom.Specifically, we propose to detect the properties of the chiral spin liquid phase using gap spectroscopy and edge quenches in the presence of the Floquet drive.The resulting chiral edge signal, which relates to the thermal Hall effect associated with neutral Majorana currents, is found to be robust for realistically prepared states.By combining strong interactions with Floquet engineering, our work paves the way for future studies of non-Abelian excitations and quantized thermal transport using quantum simulators.
Understanding topological matter is an outstanding challenge across several disciplines of physical science.Programmable quantum simulators have emerged as a powerful approach to studying such systems.While quantum spin liquids of paradigmatic toric code type have recently been realized in the laboratory, controlled exploration of topological phases with non-Abelian excitations remains an open problem.We introduce and analyze a new approach to simulating topological matter based on periodic driving.Specifically, we describe a model for a so-called Floquet spin liquid, obtained through a periodic sequence of parallel quantum gate operations that effectively simulates the Hamiltonian of the non-Abelian spin liquid in Kitaev's honeycomb model.We show that this approach, including the toolbox for preparation, control, and readout of topological states, can be efficiently implemented in state-of-the-art experimental platforms.One specific implementation scheme is based on Rydberg atom arrays and utilizes recently demonstrated coherent qubit transport combined with controlled-phase gate operations.We describe methods for probing the non-Abelian excitations, and the associated Majorana zero modes, and simulate possible fusion and braiding experiments.Our analysis demonstrates the potential of programmable quantum simulators for exploring topological phases of matter.Extensions including simulation of Kitaev materials and lattice gauge theories are also discussed.
In condensed matter, Chiral Spin Liquids (CSL) are quantum spin analogs of electronic Fractional Quantum Hall states (in the continuum) or Fractional Chern Insulators (on the lattice). As the latter, CSL are remarkable states of matter, exhibiting topological order and chiral edge modes. Preparing CSL on quantum simulators like cold atom platforms is still an open challenge. Here we propose a simple setup on a finite cluster of spin-1/2 located at the sites of a square lattice. Using a Resonating Valence Bond (RVB) non-chiral spin liquid as initial state on which fast time-modulations of strong nearest-neighbor Heisenberg couplings are applied, following different protocols (out-of-equilibrium quench or semi-adiabatic ramping of the drive), we show the slow emergence of such a CSL phase. An effective Floquet dynamics, obtained from a high-frequency Magnus expansion of the drive Hamiltonian, provides a very accurate and simple framework fully capturing the out-of-equilibrium dynamics. An analysis of the resulting prepared states in term of Projected Entangled Pair states gives further insights on the topological nature of the chiral phase. Finally, we discuss possible applications to quantum computing.
Chiral Parafermions in Floquet ZN 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 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 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 Kitaev models as a setting for exploring intrinsically non-equilibrium topological phenomena on qudit-based quantum processors.
This thesis reports on the realization of Floquet topological systems with ultracold atoms in an optical honeycomb lattice. Using periodically driven quantum systems, topological phases of matter can be simulated by an effective, static Hamiltonian related to the time-evolution at integer multiples of the driving period, which is known as Floquet engineering. However, periodic driving can also give rise to genuinely time-dependent settings without static counterparts. One example is the anomalous Floquet phase in two dimensions, in which all bulk bands have a Chern number of zero but nevertheless robust chiral edge modes appear, which would be precluded by the bulk-edge correspondence in a static system. In an optical honeycomb lattice, anomalous Floquet systems can be created by continuous, periodic modulation of the laser intensities. This driving scheme results in different topological regimes, three of which are investigated, including the anomalous Floquet phase and a Haldane-like phase. Periodically driven systems feature periodic quasienergies and can be characterized in terms of winding numbers, which count the number of chiral edge modes in each quasienergy gap. By interferometric measurements of the quasienergy gaps between the first two Floquet bands, the topological phase transitions, emerging as gap closings, are located. For periodically driven systems, a modified bulk-edge correspondence can be formulated. In particular, the change of the winding number at a phase transition is related to the sign change of the local Berry curvature. The Berry curvature at the gap closing points is probed by Hall deflection measurements to obtain the winding numbers in each of the topological regimes, revealing the existence of chiral edge modes also in a setting with smooth boundaries as used in the experiments. The measured quasienergy gaps and transverse deflections are quantitatively well described by a numerical calculation of the Floquet bandstructure that includes coupling to higher bands during the driving period. To derive the spectrum of the modulated lattice in a geometry with edges, a tight-binding description of the system is discussed. Circular phase modulation of the honeycomb lattice can also give rise to an anomalous Floquet system when inversion symmetry is broken. The topological phase diagrams and the experimental feasibility are compared for both modulation schemes. Due to the vanishing Chern numbers, the bulk states in the anomalous Floquet regime can be fully localized by a disorder potential. This could prevent heating in interacting, periodically driven systems, resulting in a many-body localized bulk coexisting with thermalizing edge states. The experimental realization of disorder and strong interactions, as well as independent probes of the edge states are investigated. Bloch band geometry can be extended to multiband systems, using Wilson lines. The possible symmetry protection of their eigenvalues is discussed, along with measurements in the optical honeycomb lattice. Moreover, the onset of heating in weakly-interacting, periodically driven systems, triggered by parametric instabilities, is studied experimentally in a one-dimensional lattice.
We show that non-Abelian anyons can emerge from an Abelian topologically ordered system subject to local time-periodic driving. This is illustrated with the toric-code model, as the canonical representative of a broad class of Abelian topological spin liquids. The Abelian anyons in the toric code include fermionic and bosonic quasiparticle excitations which see each other as π fluxes; namely, they result in the accumulation of a π phase if wound around each other. Non-Abelian behavior emerges because the Floquet modulation can engineer a nontrivial band topology for the fermions, inducing their fractionalization into Floquet-Majorana modes bound to the bosons. The latter then develop non-Abelian character akin to vortices in topological superconductors, realizing Ising topological order. Our findings shed light on the nonequilibrium physics of driven topologically ordered quantum matter and may facilitate the observation of non-Abelian behavior in engineered quantum systems.