Krylov complexity of a tilted extended Bose-Hubbard chain with Rydberg-dressed interactions

Krylov complexity analysis reveals state-dependent information scrambling in a tilted extended Bose-Hubbard chain with Rydberg-dressed interactions, providing complementary dynamical insights beyond traditional spectral diagnostics. The study establishes that this complexity captures unique features of many-body chaos driven by competing interactions, offering a valuable diagnostic tool for experimentally accessible quantum simulators.

The Model and Setup

The research investigates the dynamics of a one-dimensional extended Bose-Hubbard model described by Hamiltonian (1), which includes nearest-neighbor hopping (J), on-site interaction (U), and nearest-neighbor interaction (V). This extended model is realized using Rydberg dressing in optical lattices, allowing for tunable long-range interactions. The study focuses on the fixed particle number sector, specifically considering the case where N = L = 6 to ensure a Hilbert space dimension of D = 462 for tractable exact diagonalization. To probe state dependence, three physically motivated Fock states are chosen as initial seeds: ψl⟩ (maximally localized), ψh⟩ (homogeneous Mott-like product state), and ψs⟩ (period-two staggered density wave state).

Krylov Complexity Diagnostics

The Krylov complexity, CK(t), is defined as the mean position on the one-dimensional tight-binding chain generated by the Lanczos recursion applied to the time evolution of an initial quantum state. The recurrence relations are given by (2a) and (2b), where Kn⟩ is constructed iteratively. The complexity itself is quantified as CK(t) = K/X−1 n=0 n φn(t)2 (5). Its long-time average, C sat, is governed by Q0n, which depends on the overlap between the initial state and the exact many-body eigenstates Em⟩. At short times, CK(t) grows quadratically with time: CK(t) = b21 t2 + O(t4), where b21 = σ squared (7).

Key Findings in Interaction Regimes

The analysis yields three main findings regarding the competition between U and V:

1. Enhanced Krylov spreading is observed at intermediate interactions when the hopping J is comparable to the nearest-neighbor interaction V, distinguishing chaotic from integrable regimes.

2. A pronounced diagonal ridge near U ≃ V in the saturation of the Krylov complexity that is not accompanied by an equally sharp spectral signature, indicating strong state-dependent information scrambling due to competing interactions.

3. The Krylov state complexity provides complementary, state-sensitive information about many-body chaos that is not captured by the spectral diagnostics alone.

Comparison with Spectral Diagnostics

The study contrasts Krylov complexity with standard spectral probes:

(i) Spectral Measures:

(ii) Lanczos Fluctuations:

The paper evaluates consecutive-gap ratios (10), where Poisson (POI)-like distributions indicate integrable spectra, and Gaussian Orthogonal Ensemble (GOE)-like distributions suggest chaotic regimes. Furthermore, the Shannon information entropy of state intensities in Fock space is probed using finite-size generalized fractal dimensions (GFDs), De(m)1 (12).

Regime Analysis

The results are analyzed across different interaction regimes:

(a) Bose-Hubbard Regime (V ∼ 0):

In this limit, the Krylov complexity exhibits a pronounced dependence on the on-site interaction. At short times, the weak interaction U/J = 0.562 reaches a much larger Krylov complexity than intermediate and strong cases (U/J = 2.512, 14.125).

(b) Nearest-Neighbor Interaction Dominant Regime:

When V dominates, the dynamics show that the nearest-neighbor interaction alone plays largely similar roles as the on-site interaction in the spectral statistics. The long-time saturation shows that the nearest-neighbor interaction enhances the accessible Hilbert-space connectivity, with complexity saturating at a larger value compared to the Bose-Hubbard case.

(c) Comparable On-Site and Nearest-Neighbor Interactions (U ∼ V):

When U and V are comparable, the Krylov diagnostics identify a pronounced enhancement of many-body spreading. A crucial finding is the diagonal ridge near U ∼ V, where complexity is dramatically enhanced due to the generation of a dense manifold of near-resonant many-body configurations with other eigenstates. This regime demonstrates that Krylov complexity captures state-dependent information scrambling and can identify strong dynamical mixing.

Conclusion

The investigation concludes that Krylov state complexity offers complementary information to level-spacing statistics and eigenstate delocalization, establishing it as a valuable diagnostic for studying thermalization and information scrambling in experimentally accessible quantum simulators. The study highlights that Krylov diagnostics can diverge from spectral chaos indicators in finite interacting lattice systems, capturing unique features of state complexity over spectral diagnostics.

As a fastidious researcher, I have analyzed this paper titled Krylov complexity of a tilted extended Bose-Hubbard chain with Rydberg-dressed interactions. The key findings revolve around using Krylov complexity as a state-sensitive diagnostic for quantum chaos and thermalization in interacting many-body systems.

Here are the specific improvements to AI systems based on this research, along with what those improved systems can achieve:

The primary improvement lies in enhancing AI's ability to predict and characterize complex, non-equilibrium dynamics in strongly correlated quantum many-body systems, particularly near integrable-to-chaotic crossovers.

1. The AI system will be enhanced with a module capable of calculating and analyzing Krylov complexity metrics (e.g., time evolution of the mean position on the Krylov basis, saturation values, and fluctuation statistics).

2. The AI will be trained to correlate these Krylov complexity signatures with specific Hamiltonian parameters (like the competition between on-site interaction U and nearest-neighbor interaction V) across different initial states.

This improved AI system can perform the following specific tasks:

1. Predicting Dynamical Regimes: The system can accurately predict whether a given configuration of interacting particles in an optical lattice will exhibit integrable or chaotic dynamics based on calculated Krylov complexity patterns, rather than relying solely on spectral diagnostics (like level spacing statistics).

2. Identifying State-Dependent Scrambling Pathways: The AI can pinpoint the specific initial states (e.g., maximally localized vs. Mott-like vs. density wave) that lead to distinct information scrambling pathways, allowing for targeted study of how different experimental preparations influence quantum chaos.

3. Diagnosing Non-Classical Crossovers: The system can detect subtle dynamical crossovers, such as the pronounced diagonal ridge near U ≈ V where Krylov complexity saturates sharply while spectral signatures remain relatively smooth. This capability is crucial for identifying regimes where traditional diagnostics fail, providing a more robust indicator of strong state-dependent information scrambling.

4. Guiding Experimental Control: By mapping Hamiltonian parameters to Krylov complexity saturation values, the AI can provide feedback to experimentalists on which interaction strengths (U/J vs. V/J) are most likely to induce desired chaotic or thermalizing behavior in an experimentally accessible quantum simulator (like a Rydberg-dressed chain).

5. Characterizing Thermalization Efficiency: The long-time average of Krylov complexity provides a complementary measure to spectral diagnostics for studying eigenstate thermalization hypothesis (ETH). The AI can use this metric to quantify how quickly and effectively quantum information is scrambled across the many-body Hilbert space.

We investigate Krylov state complexity in a tilted extended Bose-Hubbard chain in which both the on-site interaction U and nearest-neighbor interaction V are present. The tilted extended Bose-Hubbard model can be realized with Rydberg-dressed interactions in optical lattices. Using exact diagonalization and Lanczos recursion from three physically motivated Fock states, we compute the time-dependent complexity, its long-time saturation value, and the fluctuations of the Lanczos coefficients for an open chain at unit filling. Crucially, we identify a pronounced quasi-chaotic diagonal regime near U V in which the Krylov complexity saturation exhibits a sharp ridge while the spectral level-spacing statistics remain partially chaotic. This dissociation arises from the competition between the on-site and nearest-neighbor interaction, which generates a dense manifold of near-resonant many-body configurations that is efficiently coupled by the state-dependent dynamics but retains residual spectral structure. Our results show that Krylov state complexity provides complementary, state-sensitive information about many-body chaos that is not captured by the spectral diagnostics alone, establishing Krylov complexity as a useful diagnostic for studying thermalization and information scrambling in experimentally accessible quantum simulators.