Many-Body Dynamical Localization

• Many-body dynamical localization is the arrest of energy absorption and thermalization in clean quantum systems via periodic driving.
• Models such as the kicked rotor and Bose-Hubbard chains utilize Floquet protocols to induce an Anderson-type localization mechanism in momentum space.
• Experimental and theoretical analyses reveal nonergodic, multifractal states with robust entanglement dynamics, offering promising routes for quantum information protection.

Many-body dynamical localization (MBDL) is the emergent arrest of energy absorption, diffusion, and thermalization in interacting quantum systems subjected to periodic driving. Unlike conventional many-body localization (MBL), which relies on quenched disorder and yields localization in real space, MBDL arises in clean (non-disordered) systems through coherent quantum interference or special Floquet protocols. The phenomenon is observed especially in paradigmatic models such as the kicked rotor, Bose-Hubbard chains, long-range spin models, and one-dimensional quantum gases, where interacting particles fail to heat to infinite temperature and display frozen states in Hilbert space for experimentally relevant timescales. MBDL is associated with ergodicity breaking, exotic entanglement dynamics, and non-thermal steady states, and has profound implications for quantum thermalization boundaries, memory retention, and quantum information protection.

1. Model Systems and Floquet Construction

MBDL is most frequently realized in periodically driven, interacting quantum systems where the drive alternates between non-trivial Hamiltonian evolution and a pulsed (often spatially periodic) potential. A canonical example is the many-body quantum kicked rotor realized via a one-dimensional Bose gas (Lieb–Liniger model) under periodic kicking (Guo et al., 2023, Yang et al., 6 Mar 2025, Rylands et al., 2019). The time-dependent Hamiltonian is

$$H(t) = \int dz\,\hat\Psi^\dagger(z)\left[-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial z^2} + V_{\mathrm{ext}}(z) + \hbar\kappa\cos(2k_L z)\sum_n \delta_{T_p}(t-nT)\right]\hat\Psi(z) + \frac{g_{1D}}{2}\int dz\,\hat\Psi^\dagger(z)\hat\Psi^\dagger(z)\hat\Psi(z)\hat\Psi(z).$$

Here, $g_{1D}$ is the interaction strength, $\kappa$ sets the single-particle kick energy, and the periodic $\delta$-kicks model a stroboscopic drive. The system is analyzed in terms of its Floquet operator $U_F$ describing evolution over one period, which allows mapping the long-time dynamics to an effective tight-binding model in many-body momentum space.

Other clean models include kicked Bose-Hubbard chains (Fava et al., 2019), long-range spin models like the Lipkin–Meshkov–Glick (LMG) Hamiltonian under time-periodic transverse fields (Rahaman et al., 2023), and non-integrable XY spin chains with quadratic kicks (Tang et al., 2024). In these systems, the Floquet operator typically factorizes into an "interaction" part and a "kicking" part, giving rise to an emergent Anderson-type model in a suitably chosen basis (momentum, Fock, or spin).

2. Anderson-Type Localization Mechanism and Mapping

The underlying mechanism for MBDL generalizes Anderson localization from single-particle physics to the many-body context. In the quantum kicked rotor case, Floquet evolution in momentum space is mapped onto a one-dimensional tight-binding chain with pseudo-random on-site phases arising from quadratic dispersion: $$\sum_\ell \left[\epsilon_\ell c_\ell + V (c_{\ell+1} + c_{\ell-1})\right] = E c_\ell,$$ where the site energies $\epsilon_\ell$ are pseudo-random, and $V$ gives the hopping amplitude (Guo et al., 2023). For $N$ interacting particles, the mapping becomes a lattice in the space of momentum configurations, with each site representing a multi-particle state.

Interactions, such as contact terms in the Lieb–Liniger model, introduce off-diagonal ("hopping") couplings between many-body configurations. Extended analytical mappings (Yang et al., 6 Mar 2025) and Bethe–Ansatz analyses reveal:

• On-site energies remain pseudo-random (Lorentzian distributed) for all interaction strengths.
• Off-diagonal couplings decay algebraically with a critical exponent that crosses over from $-4$ to $-3$, or from $-2$ to $-3/2$, as interactions are increased.
• Localization is maintained due to sufficiently fast decay of hopping amplitudes, which ensure the many-body lattice stays on the "localized" side of the Anderson phase diagram for all $g$. This universality persists at both weak and strong interactions, with intermediate regimes exhibiting multifractal, nonergodic extended states.

3. Diagnostics: Momentum Distribution, Energy, Entropy, and Entanglement

Quantitative measures for MBDL employ stroboscopic and long-time averages of observables:

• Momentum distribution $$n(p, t) = \langle \hat a^\dagger(p, t) \hat a(p, t)\rangle$$, which saturates after hundreds of kicks, indicating arrested transport.
• Mean energy $E(t) = \sum_p p^2 n(p, t) / 2m$, which plateaus in the localized regime.
• Shannon entropy $S(t) = -\sum_p n(p, t)\ln n(p, t)$, quantifying momentum-space complexity and also remains bounded after initial growth.
• First-order spatial correlations $g^{(1)}(z)$, extracted via Fourier transforms of $n(p)$, show distinct decay laws: Lorentzian in the non-interacting case, exponential for strong interactions (Tonks–Girardeau limit) (Guo et al., 2023).

In the kicked Bose-Hubbard model (Fava et al., 2019), the entanglement entropy $S(t)$ for a bipartition grows linearly in time, consistent with ballistic spread within dynamically frozen subsectors (Hilbert space fragmentation), but never reaches infinite-temperature values due to block-diagonal structure in the Floquet operator. This distinguishes MBDL from MBL, where entanglement growth follows $S(t)\sim \log t$.

Inverse participation ratios (IPR) measured in the relevant basis (momentum, Fock, spin) provide direct evidence: IPR remains finite in localized phases (indicating confinement within subsectors) and vanishes in ergodic regimes.

4. Role of Interactions, Phase Boundaries, and Multifractality

Interactions play a dual role in MBDL:

• Weak interactions ($g \ll 1$): System behaves as a single-particle localized model; Floquet eigenstates are exponentially localized in momentum configurations.
• Intermediate interactions: Delocalization can arise as resonance overlap enables hopping among Anderson-localized sectors, leading to a regime of multifractal, nonergodic extended states (generalized fractal dimensions $D_q$ satisfy $0 < D_q < 1$) (Yang et al., 6 Mar 2025).
• Strong interactions ($g \gg 1$, Tonks–Girardeau limit): Fermionization restores effective integrability; Floquet eigenstates relocalize, with $D_q$ shrinks toward zero.

Numerical measures such as the level-spacing ratio $\langle r \rangle$ track transitions from Poissonian statistics (integrable/localized) to Circular Orthogonal Ensemble (ergodic, time-reversal symmetric), then returning to Poissonian at large $g$. Critical couplings $g_c, g_c'$ mark transitions between diffusive/delocalized and arrested/localized dynamics (Guo et al., 2023).

5. Comparison with Conventional Many-Body Localization (MBL) and Floquet Protocols

Conventional MBL relies on spatial disorder—random fields or couplings—to break ergodicity and promote localization. Floquet-MBDL is distinct:

• No quenched disorder needed: Emergent pseudo-randomness and set periodic driving suffice.
• Suppression of Eigenstate Thermalization Hypothesis (ETH): Floquet eigenstates fail to thermalize, retaining initial-state memory, yet remain spatially extended within dynamically frozen sectors.
• Entanglement properties: Linear growth and volume-law scaling within subsectors, area law for localization.
• Protocols: Fine-tuning of drive parameters, such as amplitude and frequency, can shift the system between delocalized and MBDL regimes, enabling potential MBL engine cycles in clean systems (Rahaman et al., 2023, Tang et al., 2024).
• Extensions: Many models generalize MBDL to long-range interactions (Rahaman et al., 2023), gauge field settings (Yao et al., 2020), and non-integrable chains with local kicks (Tang et al., 2024).

6. Experimental Observation and Quantum Information Applications

Empirical verification was achieved via cold atom experiments on 1D Bose gases subjected to hundreds of periodic kicks (Guo et al., 2023), with signatures including:

• Arrested spread of $n(k)$ beyond several hundred kicks.
• Saturation of $E(t)$ and $S(t)$ at interaction-dependent plateaus.
• Distinct decay behavior of $g^{(1)}(z)$ associated with the dynamical localization regime.

Proposals for quantum information protection exploit the exponential scaling of coherence times in MBDL settings (Tang et al., 2024), encoding quantum memory in localized states at high temperature without disorder. At special kick strengths, "dynamical decoupling" enables persistent Rabi oscillations of boundary spins, offering practical schemes for robust quantum storage.

Experimental platforms extend to Josephson-junction arrays (realizing periodically kicked rotors (Keser et al., 2015)), trapped-ion chains, and programmable quantum simulators. Manipulation of drive amplitudes and frequencies provides direct control over localization–delocalization transitions, while measurement of local and global observables, energy, and entropy supplies diagnostic access.

7. Theoretical Open Questions and Future Directions

While MBDL is now firmly established in one-dimensional settings and for moderate interaction strengths, outstanding challenges include:

• Full characterization of scaling laws and crossover exponents for large $N$, strong driving, and many-body phase transitions.
• Classification of multifractal spectra and nonergodic extended regimes (Yang et al., 6 Mar 2025).
• Role of integrability-breaking, quantum scarring, and Hilbert space fragmentation in defining the limits and stability of MBDL (Gunawardana et al., 2021).
• Generalization to higher-dimensional systems, long-range interactions, dissipative environments, and particle–spin–gauge coupled models.

MBDL provides a fertile domain for probing the boundaries of quantum thermalization, the interplay of coherence and strong correlations, and the emergence of unconventional stationary states in non-equilibrium quantum matter. Its robustness against delocalization and heating in clean, periodically driven systems holds promise for future applications in quantum control and information science.