Origin and emergent features of many-body dynamical localization

This paper resolves the long-standing debate on whether interactions break dynamical localization in quantum kicked rotor systems by introducing an extended mapping from the kicked Lieb-Liniger model to a high-dimensional lattice, which reveals universal features like hybrid exponential-algebraic decay couplings and explains the emergence of many-body dynamical localization in strongly correlated quantum gases.

The Gist

Imagine a crowded dance floor where everyone is trying to learn a new, chaotic dance routine. In the quantum world, this "dance floor" is a ring of atoms, and the "routine" is a series of rhythmic kicks (like a DJ dropping a beat) that try to make the dancers spin faster and faster, absorbing energy.

Usually, if you keep kicking a system, it heats up, gets chaotic, and eventually reaches a state of maximum disorder (thermalization). It's like a room full of people eventually getting so tired and sweaty that they stop dancing in sync and just wander around randomly.

However, in the quantum world, something magical happens: Dynamical Localization. Even though the DJ keeps kicking, the dancers suddenly freeze. They stop gaining energy. They get "stuck" in their spots. This is like a dancer who, despite the music getting louder and faster, suddenly locks their knees and refuses to move an inch.

The Big Question: Can Friends Break the Freeze?

For decades, physicists have debated a crucial question: What happens if the dancers start interacting with each other?

In the real world, particles (dancers) bump into each other. If they are just one lonely dancer, they freeze perfectly. But if they are a crowd of interacting friends, does the chaos of their interactions break the freeze? Do they eventually start dancing wildly again, or do they stay frozen together?

This paper answers that question by building a new "map" of the quantum world.

The "High-Dimensional Lattice" Map

The authors took a complex quantum system (the kicked Lieb-Liniger model) and translated it into a simpler, visual concept: a giant, multi-dimensional grid (like a 3D chessboard, but with many more dimensions).

• The Grid Points: Each spot on this grid represents a specific way the atoms can arrange their energy.
• The "On-Site" Noise: Imagine every spot on the grid has a slightly different, random height. This randomness is what usually causes the "freeze" (localization). It's like trying to walk on a bumpy, uneven floor; you get stuck in the holes.
• The Connections (The Kicks): The rhythmic kicks create "bridges" between these grid spots, allowing the system to jump from one state to another.

The Discovery: The "Algebraic Tail"

The team discovered something fascinating about these bridges.

1. The Short Bridges: For low-energy states, the bridges are very short and weak. The system stays frozen.
2. The Long Bridges: For high-energy states, the bridges don't just disappear; they stretch out like long, thin vines. The authors call this an "algebraic tail."

Think of it like this:

• No Interaction (Free Bosons): The vines are short and snap quickly. The system stays frozen.
• Strong Interaction (Tonks-Girardeau): The vines are also short, but for a different reason (the dancers are so crowded they can't pass each other). The system stays frozen.
• Medium Interaction (The Sweet Spot): Here is the surprise. As the interaction strength increases to a medium level, the vines stretch out the longest. They become strong, long bridges that connect distant parts of the grid.

The "Goldilocks" Zone of Chaos

This stretching of the bridges creates a "Goldilocks Zone" (or Regime II in the paper).

• Weak Interaction: The system is frozen (localized).
• Strong Interaction: The system is frozen again (localized).
• Medium Interaction: The bridges are so long and strong that the system can finally "walk" across the grid. The freeze breaks! The system becomes chaotic and starts absorbing energy again.

It's like a crowd of people:

• If they don't talk to each other, they all stand still.
• If they are screaming and pushing so hard they can't move, they stand still.
• But if they are talking and nudging each other just right, they start a chaotic mosh pit!

Why Does This Matter?

The paper explains that the "freeze" (Dynamical Localization) isn't just a simple on/off switch. It's a delicate balance. The interactions between particles can actually help the system break free from the freeze, but only if the interaction strength is in a specific range.

They also found that the system isn't just "frozen" or "chaotic." In the middle zone, it has a weird, fractal structure (like a snowflake or a coastline)—it's partially frozen and partially chaotic at the same time. This is called multifractality.

The Takeaway

The authors have shown us that in the quantum world, interaction is a double-edged sword.

• It can help maintain order (localization) in extreme cases.
• But in the middle, it acts like a catalyst that breaks the order, allowing the system to thermalize.

This helps us understand how quantum gases behave in experiments. If you want to keep a quantum system stable and frozen (useful for quantum computers), you need to be careful about how strongly the particles interact. If they interact just a little bit too much, the whole system might suddenly wake up and start heating up, ruining your experiment.

In short: The paper reveals that the "freeze" of quantum particles is fragile. A little bit of socializing (interaction) between the particles can actually make them break free from the freeze and start dancing again, but only if the party isn't too wild or too quiet.

Technical Summary

1. Problem Statement

The paper addresses a long-standing debate in quantum many-body physics: Can dynamical localization (DL) survive in the presence of interparticle interactions?

• Context: The Quantum Kicked Rotor (QKR) is a paradigmatic model for DL, where non-resonant periodic driving leads to the suppression of energy absorption (Anderson localization in momentum space).
• The Conflict: While single-particle DL is well-established, the introduction of interactions (many-body effects) raises the question of whether the system thermalizes (delocalizes) or remains localized.
• Specific Gap: Previous studies on the kicked Lieb-Liniger (LL) model (bosons with contact interactions) yielded contradictory results. Mean-field approximations predicted delocalization, while non-perturbative limits (Tonks-Girardeau, $g \to \infty$) predicted persistence of localization (Many-Body Dynamical Localization, MBDL). The microscopic mechanism driving MBDL at finite interaction strengths remained unclear.

2. Methodology

The authors employ a combination of analytical mapping, exact diagonalization (ED), and statistical analysis of eigenstates.

• Model: They study the 1D kicked Lieb-Liniger model, consisting of $N$ bosons on a ring subject to a pulsed sinusoidal potential (kicks) and contact interactions ($g$).
• Core Innovation: Extended Lattice Mapping:
  • Instead of analyzing the time evolution directly, they map the Floquet eigenvalue problem of the kicked LL model onto a high-dimensional ($N$-dimensional) lattice model.
  • Diagonal Terms ($V_I$): Represent on-site potentials derived from the quasi-energy spectrum. These are shown to be pseudorandom, analogous to the disorder in Anderson localization.
  • Off-Diagonal Terms ($W_{II'}$): Represent hopping couplings between Bethe eigenstates in the quasimomentum space.
• Analytical Tools:
  • Bethe Ansatz: Used to analytically derive the asymptotic behavior of the hopping terms ($W_{II'}$) for $N=2$ and approximate for $N>2$.
  • Exact Diagonalization (ED): Performed on the mapped lattice model to compute off-diagonal terms and eigenstate properties for $N=2$ and $N=3$.
• Diagnostic Metrics:
  • Generalized Fractal Dimensions ($D_\beta$): To characterize the multifractality of eigenstates.
  • Level-Spacing Ratio ($\langle r \rangle$): To distinguish between integrable (Poisson) and chaotic/ergodic (Wigner-Dyson) statistics.
  • Inverse Participation Ratio (IPR): To measure the extent of eigenstate localization.

3. Key Contributions and Results

A. Discovery of Hybrid Decay Couplings

The mapping reveals that the off-diagonal hopping terms ($W_{II'}$) exhibit a hybrid decay structure:

1. Exponential Decay: Dominates at low energies (or for free bosons/fermions limits), ensuring localization in the center-of-mass (CM) momentum sector.
2. Algebraic Decay: Emerges at high energies due to interactions. The hopping amplitude decays as a power law: $W \sim E^{-\lambda/2}$.

B. Interaction-Driven Crossover

A critical finding is the non-monotonic crossover in the algebraic tail as interaction strength ($g$) increases:

• Exponent ($\lambda$): Transitions from $\lambda = 4$ (weak interaction regime) to $\lambda = 3$ (strong interaction regime).
• Amplitude ($A$): Exhibits a non-monotonic behavior, peaking at intermediate interaction strengths.
• Mechanism: The kicks induce exponential decay in the CM momentum, while interactions generate the algebraic tail in the relative momentum. The crossover reflects a change in how the system handles excitations (from two-particle excitations in weak coupling to fermionization in strong coupling).

C. Phase Diagram and Localization Breakdown

The analysis of the mapped model and eigenstate statistics reveals three distinct regimes in the $(g, K)$ parameter space:

1. Regime I (Weak Interaction, Low $K$): The system is Many-Body Dynamically Localized (MBDL). Eigenstates are localized ($D_\beta \approx 0$), and the system is nearly integrable ($\langle r \rangle \approx 0.386$).
2. Regime II (Intermediate Interaction, Moderate $K$): The algebraic tail amplitude is maximized, and the decay exponent is $\lambda \approx 3$. This leads to a breakdown of localization. Eigenstates become multifractal ($0 < D_\beta < 1$) and extended but non-ergodic. The level-spacing ratio increases toward the ergodic value ($\langle r \rangle \approx 0.527$), indicating a transition toward chaos.
3. Regime III (Strong Interaction/Tonks-Girardeau, High $g$): The system re-enters a localized phase (MBDL) due to fermionization. The algebraic tail weakens, and the system returns to being nearly integrable ($\langle r \rangle \approx 0.386$).

D. Experimental Signatures

The paper proposes a specific experimental observable to detect this crossover:

• By quenching a cold-atom system from a non-interacting state to a finite-interaction regime, the momentum distribution $n(m)$ should exhibit a crossover in its algebraic tail.
• Prediction: The tail should shift from $n(m) \propto m^{-4}$ (characteristic of kicking the ground state or weak interactions) to $n(m) \propto m^{-2}$ (characteristic of kicking a non-equilibrium state in the intermediate interaction regime).

4. Significance

• Resolution of the MBDL Debate: The paper provides a microscopic explanation for why MBDL persists at finite interaction strengths, identifying the algebraic tail in the high-dimensional lattice mapping as the key factor.
• Universality: The findings suggest that the algebraic tail and the associated crossover are universal features for arbitrary particle numbers ($N$), not just limited to $N=2$.
• Integrability and Chaos: It highlights a complex interplay where finite interactions can temporarily destroy integrability and induce multifractality (a "pre-thermal" or intermediate chaotic phase) before the system re-integrates in the strong-coupling limit.
• Experimental Feasibility: The proposed crossover in momentum distribution tails ($m^{-4} \to m^{-2}$) offers a concrete, testable prediction for current cold-atom experiments, bridging the gap between theoretical mapping and observable phenomena.

In summary, the authors demonstrate that many-body dynamical localization is not a simple binary state but a rich phase structure governed by the interplay between pseudorandom on-site potentials and interaction-induced algebraic hopping, leading to a distinct multifractal phase at intermediate interaction strengths.