Floquet dynamical quantum phase transitions in periodically flux-quenched systems

Floquet Dynamical Quantum Phase Transitions in Periodically Flux-Quenched Systems

Introduction

The study rigorously analyzes Floquet dynamical quantum phase transitions (FDQPTs) within periodically driven quantum many-body systems, specifically via a piecewise constant flux-quench protocol implemented on an extended XY spin chain. The context leverages Floquet engineering, exploiting time-periodicity to explore nonequilibrium quantum critical phenomena inaccessible under equilibrium conditions. The work addresses how the discrete switching of a control parameter—in this case, a magnetic flux—modifies system dynamics, and systematically characterizes the conditions under which FDQPTs are observable, extending and challenging the prevailing understanding established through studies with time-independent or smoothly driven Hamiltonians.

Model and Periodic Flux-Quench Protocol

The physical setup involves an integrable, extended XY spin chain with Hamiltonian parameters modulated by piecewise-constant magnetic flux within each Floquet period. The protocol divides the driving period $T$ into two segments of duration $T_1$ and $T_2$; during each, the system evolves under $H(\phi_1)$ or $H(\phi_2)$, with the flux difference $\Delta\phi = \phi_1 - \phi_2$ as a principal control parameter. Importantly, the spectrum of the system remains invariant under flux modulation, as only the orientation, not the magnitude, of the "Bloch vector" in the effective Bogoliubov-de Gennes (BdG) description is altered. This geometric character is critical: the primary effect of the protocol is to rotate the BdG field without gap closures, distinguishing FDQPTs in this setting from conventional DQPTs tied directly to energy gap closing.

The analytical tractability of the protocol enables explicit construction of the time evolution operator and Floquet effective Hamiltonian for each $k$-mode, ensuring a transparent mapping between segment Hamiltonians and the quasiparticle basis. Experimental realization is feasible with solid-state electronic spin platforms such as NV centers in diamond using phase-engineered microwave control.

Diagnostics of FDQPTs: Loschmidt Echo, Rate Function, and DTOP

FDQPTs are detected through nonanalyticities in the time-evolved Loschmidt echo and associated rate function. In the thermodynamic limit, zeros of the mode-resolved Loschmidt echo $\mathcal{L}_k(t)$ indicate critical times where evolved many-body wavefunctions become orthogonal to their initial state. The emergence and periodic recurrence of such zeros are direct signatures of FDQPTs in Floquet systems.

The dynamical topological order parameter (DTOP), constructed from the Pancharatnam geometric phase (PGP), provides a robust and quantized topological diagnostic. Jumps in the DTOP match the nonanalyticities in the rate function, unambiguously marking FDQPT events. This correspondence persists across both symmetric ($T_1 = T_2$) and asymmetric driving protocols.

Floquet Quench Fidelity and Necessary/Sufficient Conditions

A notable theoretical advancement is the formal extension of the concept of quench fidelity to periodically driven systems—Floquet quench fidelity—defined as the many-body ground state overlap between the effective Floquet Hamiltonian and each segmental quench Hamiltonian. Critically, the familiar correspondence from single-quench DQPTs, wherein the condition $F_{\alpha,k_c} = 1/\sqrt{2}$ signals a critical mode $T_1$0, is shown to be no longer sufficient in the presence of temporal segmentation.

The paper makes a bold claim: for the piecewise periodic protocol, the necessary and sufficient condition for FDQPTs at mode $T_1$1 in segment $T_1$2 is twofold:

1. Existence of a critical $T_1$3 such that $T_1$4.
2. The corresponding critical time $T_1$5 (deduced from the Fisher zeros formalism) must fall within the actual evolution window $T_1$6 or $T_1$7 of segment $T_1$8.

Thus, even when the fidelity condition is met, if the segment duration is too short, the orthogonality event is dynamically inaccessible, and FDQPTs do not manifest. This temporal constraint, absent in single-quench or continuous-driving scenarios, constitutes a fundamental distinction.

Floquet FDQPTs as Dynamics on the Bloch Sphere

The geometric content is elucidated by mapping the micromotion onto evolution trajectories on the Bloch sphere for each $T_1$9-mode. The occurrence of FDQPTs is visualized as the trajectory of the Bloch vector becoming antiparallel to its initial direction within the allowed segment duration—an event susceptible to interruption by the protocol's switching. Thus, the quench protocol not only determines critical points through parameter differences but also restricts their dynamical realization through fine-tuning of time intervals.

Implications and Prospects

The formalism presented generalizes to arbitrary parameter-controlled periodic quench protocols. It delivers a complete set of conditions for the observation of FDQPTs that incorporate both state-space overlap (Floquet fidelity) and accessible dynamical trajectories (time window constraints), transcending the paradigm of equilibrium phase transitions as universal boundary determiners for quantum criticality.

From an experimental perspective, this underscores the importance of temporal engineering—not simply the choice of control parameters—in designing driven quantum simulators for realizing and detecting nonequilibrium critical phenomena. The results also constrain possible mappings between static and dynamic critical manifolds, with implications for the classification of nonequilibrium universality classes.

In the broader context of Floquet systems, the work provides a template for systematic analysis of FDQPTs across a wide array of quantum many-body models, invites exploration of topological dynamical invariants under complex protocol structures, and motivates future investigation into interplay with dissipation, integrability breaking, and multi-segment or higher-dimensional driving schemes.

Conclusion

This study establishes a rigorous framework for the characterization and control of FDQPTs in periodically flux-quenched spin chains, pinpointing the essential role of both flux differences and temporal segmentation. By introducing Floquet quench fidelity and identifying temporal accessibility as a limiting factor, the analysis advances the theoretical description of nonequilibrium critical phenomena in periodically driven quantum systems and positions protocol engineering as a central theme for future research in quantum dynamics and control (2604.14946).