From Approximate Floquet Engineering to Full Floquet Theory: Coherent Control of Chiral Spin Systems in Spintronics

Abstract: Coherent control of interacting spin systems under time-periodic driving is a central challenge in spin-based quantum technologies. Here we demonstrate the applicability of a full Floquet-space formalism, adapted from Nuclear Magnetic Resonance (NMR) methodologies, to model the dynamics of driven coupled electron spins in the presence of a static magnetic field B0 and a transverse oscillating field B1. The framework explicitly includes isotropic exchange coupling J and the chiral Dzyaloshinskii-Moriya antisymmetric exchange interaction (DMI), and its numerical convergence is systematically validated with respect to Fourier-space truncation. In the non-interacting limit, the expected driven-spin dynamics is recovered, with the oscillation periodicity governed by B1. Exchange coupling alone does not modify the collective spin expectation values under the chosen initial condition, consistent with symmetry considerations. In contrast, increasing DMI generates a finite expectation value of Sy, suppresses the expectation value of Sz, and produces tilted, elliptical Bloch-sphere trajectories, reflecting the emergence of chiral spin-spin correlations. These effects are pronounced for open boundary conditions, while remaining nearly negligible in the periodic boundary case. When exchange coupling and DMI coexist, the dynamics becomes strongly perturbed and multi-frequency in nature. Together, these results demonstrate that full Floquet-space modeling provides a robust and predictive framework for analyzing and engineering coherent dynamics in driven interacting spin systems beyond simple coherent-rotation regimes.

• The paper establishes a full Floquet-space formalism that accurately captures multi-frequency dynamics in chiral spin systems beyond high-frequency approximations.
• It demonstrates that the interplay of isotropic exchange and Dzyaloshinskii-Moriya interactions leads to boundary-sensitive, complex spin trajectories in both open and periodic systems.
• The work provides actionable insights for designing precise pulse sequences and mitigating decoherence in quantum spintronics devices.

Floquet-Space Formalism for Driven Chiral Spin Dynamics in Spintronics

Introduction and Context

Coherent control of interacting electronic spin systems subjected to time-periodic driving is a strategically central topic in quantum technologies and spintronics. The work presents a rigorous Floquet-space formalism—transferred from NMR Hamiltonian engineering methodologies—applied to the modeling and control of coupled electron spins, focusing on the interplay of isotropic Heisenberg exchange and the antisymmetric Dzyaloshinskii-Moriya interaction (DMI). This formalism is executed in both minimal (two- and three-spin) Ising chain models. The approach critically advances over approximate Floquet engineering, which dominates the spintronics literature, by conducting controlled convergence of the Floquet-Hilbert space, thus capturing multi-frequency, strongly correlated, and chiral dynamic regimes inaccessible to perturbative or effective-Hamiltonian models.

Theoretical Framework

The core formalism is the operator-based Hilbert-Floquet representation: the time-periodic Hamiltonian, including static ($B_0$) and oscillating ($B_1$) magnetic fields, isotropic exchange $J$, and vector DMI $\vec{D}$, is expanded in a Fourier basis. Mapping the Schrödinger equation into this extended Sambe space establishes a time-independent eigenproblem whose solution includes all modulation frequencies and their couplings. Numerical convergence is enforced via systematic truncation of the Floquet index, with observables checked for stability. By iteratively including higher Floquet modes, the Fourier space is truncated only when all physically relevant dynamical features (quasienergies, expectation values, Bloch-sphere trajectories) stabilize—contrasting with high-frequency approximations that neglect essential off-resonant contributions.

The approach is computationally scalable and, unlike the common Magnus expansion or rotating-wave-like reductions, remains valid for arbitrary ratios of driving, exchange, and DMI strengths. The formalism incorporates both open and periodic boundary conditions, enabling a detailed examination of finite-size, topological, and boundary-induced phenomena.

Results: Isolated and Combined Interaction Effects

1. Non-Interacting and Exchange-Only Regimes

In the regime where both $J$ and DMI vanish, Floquet dynamics trivially reproduces Rabi-type oscillations, with collective magnetization exhibiting single-frequency, perfectly periodic motion governed solely by $B_1$. Introducing isotropic exchange coupling $J$ (with DMI absent) does not affect the collective spin observables, as Heisenberg exchange commutes with the total spin operator for the specified initial conditions. Thus, spin expectation values and Bloch trajectories are strictly invariant to $J$ in this regime—manifesting the underlying symmetry (conservation of total spin) and confirming analytical predictions.

2. Dzyaloshinskii-Moriya Interaction (DMI) Effects

Including finite DMI, with $J=0$, fundamentally alters dynamics. The DMI term, breaking inversion symmetry, mixes transverse and longitudinal spin components, redistributes magnetization into correlated two-spin (and higher) channels, and leads to the emergence of finite transverse (Sy) components. This is evidenced by the transformation of Bloch-sphere trajectories from circular to tilted elliptical and, at higher DMI, to complex multi-loop orbits indicative of multi-frequency entanglement and chiral correlation. The dynamical impact is strongly sensitive to boundary conditions: open chains show pronounced DMI-induced distortions, while periodic boundary systems exhibit partial restoration of regularity due to higher symmetry and translational invariance.

3. Competing Exchange and DMI

The coexistence of finite $J$ and DMI generates highly nontrivial driven dynamics. For open boundary conditions, this competition leads to strongly perturbed, non-Bloch-like, multi-frequency evolution characterized by the breakdown of simple coherent rotations. Magnetization trajectories become increasingly aperiodic and fill significant fractions of the Bloch sphere as $B_1$0 increases—a direct signature of competing symmetric and chiral correlations destabilizing collective modes. For periodic boundary conditions, coherent trajectories are robust and regular, demonstrating that topological closure suppresses boundary-induced symmetry breaking and stabilizes the collective Floquet motion, even in the presence of strong anisotropic interactions.

Limitations of Approximate Floquet Engineering

Direct comparison with the frequently used first-order Floquet-Magnus (high-frequency) expansion shows that approximate Floquet engineering substantially fails to capture multi-frequency, chiral-induced, or resonance-enhanced effects when the driving strength is comparable to intrinsic interactions. Only full Floquet theory reproduces all dynamical observables accurately outside the strict high-frequency regime. Therefore, for engineering quantum gates or control schemes involving realistic chiral spin textures, perturbative approaches are insufficient and potentially misleading.

Implications for Spintronics, Magnonics, and Quantum Control

This full Floquet methodology enables the systematic engineering of complex quantum dynamics in nanoscale chiral spin systems. Control landscape analyses—mapping the longitudinal spin response over the driving-parameter space—demonstrate that arbitrary target rotations (such as $B_1$1-pulses analogous to NMR) remain feasible in interacting, nonintegrable systems. The parameter regions supporting robust, high-fidelity rotations can be precisely predicted and optimized, providing direct guidance for pulse design, parameter calibration, and device operation in spintronic and quantum technological settings.

Theoretical implications include a deeper understanding of coherence loss mechanisms in topological qubits: DMI, while stabilizing skyrmionic and noncollinear quantum states, also constitutes a dominant, geometry- and boundary-conditioned decoherence channel during gate operations. The demonstrated methodology is extendable to higher dimensions and larger spin networks. Critically, boundary and topological effects are shown to be decisive in dictating the coherent control landscape—implying that experimental design in low-dimensional devices must incorporate detailed microscopic modeling.

Conclusion

This work unifies the operator-based Floquet-space formalism of NMR with models of driven chiral spin systems in quantum spintronics, addressing the key limitation of existing approaches through controlled convergence and explicit treatment of both symmetric and antisymmetric interactions. The results clarify the dynamical consequences of DMI and exchange, resolve the influence of system topology and boundaries, and establish a predictive, scalable toolset for quantum coherent control well beyond the reach of standard Floquet Hamiltonian engineering. This foundation supports future developments in designing pulse sequences, gate operations, and robust quantum logic in complex magnetic architectures and lays the groundwork for rigorous theoretical analysis of interacting spin platforms underpinning emerging quantum technologies.

Reference: "From Approximate Floquet Engineering to Full Floquet Theory: Coherent Control of Chiral Spin Systems in Spintronics" (2606.27183)