Subsystem Thermalization and Work Statistical Characterizations of Floquet Dynamics

Abstract: We study Floquet thermalization in a periodically driven quantum non-integrable Ising chain by combining two operational diagnostics: subsystem thermalization and work statistics. For generic interacting Floquet systems, stroboscopic dynamics lead to heating toward an infinite-temperature state at low driving frequencies, to a prethermal state during the crossover regime, and then to a finite-temperature state at high driving frequencies. We show that reduced density matrices of small subsystems provide a precise measure of local equilibration and clearly resolve prethermal plateaus. In parallel, we analyze the statistics of work performed over a Floquet cycle using both the characteristic function of work with and without the two-point measurements, and the related fluctuation theorem, which captures coherent contributions and deviations from thermal equilibrium. By comparing these two diagnostics within the same Floquet setting, we demonstrate that work statistics encode the same dynamical crossover that governs subsystem thermalization. Our results establish a unified and experimentally accessible framework for characterizing Floquet thermalization, prethermal regimes, and coherent energy absorption in interacting quantum systems.

• The paper presents a novel framework combining subsystem reduced density matrices and work statistics to distinguish prethermal, thermal, and infinite-temperature regimes.
• The analysis shows that low-frequency drives lead to full dephasing while high-frequency drives sustain prethermal plateaus with varying local and global equilibration.
• The study underscores the importance of using effective (time-averaged) Hamiltonians to validate fluctuation theorems and guide experimental verification in driven quantum systems.

Subsystem Thermalization and Work Statistical Characterizations in Floquet Dynamics

Overview

This work, "Subsystem Thermalization and Work Statistical Characterizations of Floquet Dynamics" (2607.00381), presents a comprehensive operational framework for diagnosing thermalization in periodically driven, non-integrable quantum spin chains. The authors combine diagnostics from two domains: subsystem thermalization (rooted in the ETH and reduced density matrix analysis) and work statistics (encompassing both TPM-based and quantum-coherent fluctuation analyses). Their systematic study elucidates the interplay between local equilibrium, global energy absorption, and quantum coherence destruction, ultimately demonstrating the equivalence and complementarity of these diagnostics in revealing dynamical crossovers between prethermal, thermal, and infinite-temperature regimes in Floquet systems.

Floquet Thermalization: Regimes and Diagnostics

The paper uses a periodically driven quantum Ising chain as a canonical non-integrable system, with driving protocols ranging across frequency regimes. Stroboscopic Floquet evolution naturally divides system behavior into three distinct heating regimes:

• Low-frequency driving: Rapid heating to the infinite-temperature state, characterized by complete dephasing and global scrambling.
• High-frequency driving: Emergence of long-lived prethermal plateaus; heating is suppressed, and the system approaches a finite-temperature quasi-equilibrium determined by an effective (Floquet) Hamiltonian.
• Intermediate frequencies: A crossover regime exhibiting transient prethermalization and partial loss of coherence before thermalization is eventual.

Two key operational diagnostics are developed and compared within this setting:

1. Subsystem Thermalization: Validation through relative entropy between subsystem reduced density matrices and those of an appropriately defined Gibbs state. This approach explicitly detects local equilibration and enables fine-grained distinction of prethermal and fully thermalized regimes for subsystems of varying spatial extents.
2. Work Statistics: Characterization through the work distribution function and its Fourier characteristic function using both TPM and coherence-preserving (full counting statistics, FCS) protocols. The fluctuation theorem, particularly the Jarzynski equality, is used as an operational test for global thermalization, while the difference between TPM and FCS protocols quantifies residual quantum coherence.

Strong Results and Contrasting Behaviors

Subsystem Thermalization Metrics

The authors show that the relative entropy between evolving subsystem states and candidate thermal states exhibits sharply distinct asymptotics in different regimes:

• For initial thermal states and at low driving frequency, both small and large subsystems become locally and globally thermalized (relative entropy approaches zero for half-chain subsystems).
• At high frequencies, while small subsystems (single site) thermalize, larger subsystems do not, reflecting the persistence of long-range correlations and incomplete global thermalization.
• Pure initial states, such as the Néel state, cannot become globally thermalized under unitary evolution; relative entropy reveals thermalization only at the smallest subsystems and strong fluctuations for larger $A$.

Work Statistics and Quantum Coherence

The detailed analysis of work statistics uncovers:

• Quantum coherence measures $C_n^W$ and $C_n^\chi$, constructed as the deviation between TPM and FCS averages and characteristic functions, reveal a non-monotonic dependence on driving frequency. Both metrics collapse to zero at low frequencies (full dephasing in the infinite-temperature regime) and remain finite at higher frequencies, with a pronounced maximum at intermediate frequencies (prethermal plateau).
• Fluctuation theorem deviations (quantified via $C_n^{\mathrm{FT}}$) vanish only when the effective thermal state is constructed with the time-averaged Hamiltonian $H_{\mathrm{ave}}$, not with the static $H_J$. This underscores the operational necessity of selecting the correct modular Hamiltonian when diagnosing thermalization in driven systems.

Unified Characterization and Crossover

Both diagnostics—local (subsystem) and global (work statistics)—converge on the same dynamical crossovers. Notably, in regimes where subsystem thermalization up to half-chain subsystems is achieved, fluctuation-theorem deviations also vanish, indicating robust global thermalization under periodic drive. Conversely, for pure initial states, global measures persistently indicate significant deviations even as local entropies for small subsystems drop, accentuating the difference between local and global equilibration in unitary Floquet dynamics.

Implications and Theoretical Significance

This work establishes a rigorous and operationally accessible route for detecting and quantifying prethermalization, heating, and the fate of quantum coherence in periodically driven many-body systems. The use of both local and global diagnostics, grounded in reduced density matrix comparison and advanced work statistics, moves beyond earlier ETH-centric or purely energetic approaches, integrating insights from quantum thermodynamics and nonequilibrium statistical mechanics.

Experimental Accessibility

Importantly, both diagnostics developed here—relative entropy of reduced density matrices and TPM-based/quantum-coherent work statistics—are experimentally accessible via modern quantum simulation platforms. Their practical applicability supports experimental verification of predicted prethermal and heating plateaus, as well as fundamental tests of quantum fluctuation relations in dynamical regimes.

Broader Theoretical Impact

• The demonstration that the correct modular Hamiltonian for the late-time ensemble in Floquet systems is generally not the undriven Hamiltonian but the time-averaged (or effective Floquet-Magnus) one, with substantial dependence on frequency and driving protocol, motivates a more nuanced treatment for constructing generalized Gibbs ensembles in driven systems.
• The nuanced relation found between local thermalization, global passivity/fluctuation theorems, and the destruction of quantum coherence can further inform the quantum control and engineering of non-equilibrium steady states, including in contexts of Floquet engineering and time crystal formations.

Directions for Future Research

The operational framework presented can be extended to:

• Systems with disorder and many-body localization, probing the interplay of localization-protected phases and Floquet-induced heating.
• Models with additional conserved quantities or symmetry constraints, addressing non-Abelian ETH and generalized thermalization [Murthy et al., (Murthy et al., 2022)].
• Larger system sizes or higher dimensions, exploiting the scalability of the diagnostics for future experimental and numerical explorations.

Conclusion

Through a systematic juxtaposition of subsystem and work-statistical diagnostics, this study provides a unified, operational approach to tracking thermalization, coherence loss, and energy absorption in driven quantum systems. The findings underline the deep connections between local reduced-state equilibration, global fluctuation theorems, and the structure of late-time steady states under periodic driving. These results constitute a solid methodological advance in the characterization of non-equilibrium phases and transitions in quantum many-body physics (2607.00381).