Alberto Rodríguez

Quantum many-body systems pose a formidable computational challenge due to the exponential growth of their Hilbert space. While deep learning (DL) has shown promise as an alternative paradigm, many applications in these systems are still to mature further, beyond the proof-of-concept stages of energy estimation at the lower end of the spectrum. Here, we extend DL beyond this frontier by extensively testing HubbardNet , a deep neural network architecture for the Bose–Hubbard model. Pushing improvements in the optimizer and learning rates, and introducing physics-informed output activations that can resolve extremely small wave-function amplitudes, we achieve ground-state energy errors reduced by orders of magnitude and wave-function fidelities exceeding 99%. We further assess physical relevance by analyzing generalized inverse participation ratios and multifractal dimensions for ground and excited states in one and two dimensions, demonstrating that optimized DL models reproduce localization, delocalization, and multifractality trends across the spectrum. Crucially, these qualitative predictions remain robust across four decades of the interaction strength, e.g. spanning across superfluid, Mott-insulating, as well as quantum chaotic regimes. Together, these results suggest DL as a viable qualitative predictor of many-body structure, complementing the quantitative strengths of exact diagonalization and tensor-network methods.

We investigate the propagation of two-point density correlations in the one-dimensional Bose–Hubbard Hamiltonian in the thermodynamic limit in terms of the correlation transport distance (CTD), an experimentally measurable magnitude that characterizes the spatial spreading of correlations in time. We confirm that the integrable limits of the model exhibit CTD ballistic growth, while the onset of the chaotic phase leads to the emergence of a pronounced sub-ballistic regime, in agreement with previous results for finite systems. By a meticulous analysis of the spatio-temporal correlation profiles, we show that the correlation front nonetheless propagates ballistically for all interaction strengths, and that the chaos-induced slowdown of the CTD originates from the emergence of long-time distance-dependent correlation tails, together with an enhanced decay of the correlation front amplitude. Our results thus provide a detailed characterization of correlation transport that goes beyond a simple light-cone picture.

The chaotic phase of the tilted Bose-Hubbard model is identified as a function of energy, tilt strength, and particle interaction, from the eigenstate structure and the statistical features of the energy spectrum. Our analysis reveals that the chaotic phase of the bare Bose-Hubbard Hamiltonian can be enhanced by the presence of a moderate tilt. We further unveil the development and scaling of the chaotic regime from the perspective of a homogeneous density configuration typically used in cold atom experiments, providing a valuable phase diagram for future theoretical and experimental studies of this system.

We demonstrate how the initial state of ultracold atoms in an optical lattice controls the emergence of ergodic dynamics as the underlying spectral structure is tuned into the quantum chaotic regime. Distinct initial states’ chaos threshold values in terms of tunneling as compared to interaction strength are identified, as well as dynamical signatures of the chaos transition, on the level of experimentally accessible observables and time scales.

We analyze the propagation of experimentally relevant two-particle correlations for one-dimensional interacting bosons, and give evidence that many-body chaos induces the emergence of an effective diffusive regime for the fully coherent correlation dynamics, characterized by an interaction dependent diffusion coefficient, which we estimate. This result supports very recent experimental observations, and paves the way toward an efficient description of the dynamical behavior of nonintegrable complex many-body systems. Furthermore, we show that the dynamical features within experimentally accessible time scales of a conveniently defined two-particle correlation transport distance provide a direct and unambiguous characterization of many-body quantum chaos in perfect agreement with its spectral identification.

We unveil the signature of many-body interference across dynamical regimes of the Bose-Hubbard model. Increasing the particles’ indistinguishability enhances the temporal fluctuations of few-body observables, with a dramatic amplification at the onset of quantum chaos. By resolving the exchange symmetries of partially distinguishable particles, we explain this amplification as the fingerprint of the initial state’s coherences in the eigenbasis.

The dependence of the chaotic phase of the Bose–Hubbard Hamiltonian on particle number N , system size L and particle density is investigated in terms of spectral and eigenstate features. We analyse the development of the chaotic phase as the limit of infinite Hilbert space dimension is approached along different directions, and show that the fastest route to chaos is the path at fixed density n ≲ 1. The limit N → ∞ at constant L leads to a slower convergence of the chaotic phase towards the random matrix theory benchmarks. In this case, from the distribution of the eigenstate generalized fractal dimensions, the chaotic phase becomes more distinguishable from random matrix theory for larger N , in a similar way as along trajectories at fixed density.

We investigate the chaotic phase of the Bose-Hubbard model [L. Pausch et al, Phys. Rev. Lett. 126, 150601 (2021)] in relation to the bosonic embedded random matrix ensemble, which mirrors the dominant few-body nature of many-particle interactions, and hence the Fock space sparsity of quantum many-body systems. The energy dependence of the chaotic regime is well described by the bosonic embedded ensemble, which also reproduces the Bose-Hubbard chaotic eigenvector features, quantified by the expectation value and eigenstate-to-eigenstate fluctuations of fractal dimensions. Despite this agreement, in terms of the fractal dimension distribution, these two models depart from each other and from the Gaussian orthogonal ensemble as Hilbert space grows. These results provide further evidence of a way to discriminate among different many-body Hamiltonians in the chaotic regime.

We investigate the chaotic phase of the Bose–Hubbard model (Pausch et al 2021 Phys. Rev. Lett. 126 150601) in relation to the bosonic embedded random matrix ensemble, which mirrors the dominant few-body nature of many-particle interactions, and hence the Fock space sparsity of quantum many-body systems. The energy dependence of the chaotic regime is well described by the bosonic embedded ensemble, which also reproduces the Bose–Hubbard chaotic eigenvector features, quantified by the expectation value and eigenstate-to-eigenstate fluctuations of fractal dimensions. Despite this agreement, in terms of the fractal dimension distribution, these two models depart from each other and from the Gaussian orthogonal ensemble as Hilbert space grows. These results provide further evidence of a way to discriminate among different many-body Hamiltonians in the chaotic regime.