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arXiv:2610.07156

Chiral Parafermions in Floquet $\mathbb{Z}_N$ Topological Order

Melissa Will, Harald Schmid, Jan Egger, Adam Gammon-Smith, Michael Knap, Frank Pollmann

cond-mat.str-elquant-phSubmitted Oct 5, 2026• Announced Oct 7, 2026• 0 citations
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Abstract

Floquet topological order (FTO) extends topology beyond equilibrium, giving rise to phenomena, such as the dynamical transmutation of anyons. Here, we realize $\mathbb{Z}_N$ 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 $\mathbb{Z}_N$ 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 $\mathbb{Z}_N$ Kitaev models as a setting for exploring intrinsically non-equilibrium topological phenomena on qudit-based quantum processors.

Comments
7+7 pages, 5+3 figures

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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.
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.
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.
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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.
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.