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arXiv:2205.04209 · Journal of Physics A Mathematical and Theoretical

Optimal route to quantum chaos in the Bose–Hubbard model

Lukas Pausch, Andreas Buchleitner, Edoardo G. Carnio, Alberto Rodríguez

Published Jul 4, 2022• 19 citations
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Abstract

Abstract 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.

DOI
10.1088/1751-8121/ac7e0b

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Abstract 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.
Abstract 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.
We identify the chaotic phase of the Bose-Hubbard Hamiltonian by the energy-resolved correlation between spectral features and structural changes of the associated eigenstates as exposed by their generalized fractal dimensions. The eigenvectors are shown to become ergodic in the thermodynamic limit, in the configuration space Fock basis, in which random matrix theory offers a remarkable description of their typical structure. The distributions of the generalized fractal dimensions, however, are ever more distinguishable from random matrix theory as the Hilbert space dimension grows.
In the large-NNN, classical limit, the Bose-Hubbard dimer undergoes a transition to chaos when its tunneling rate is modulated in time. We use exact and approximate numerical simulations to determine the features of the dynamically evolving state that are correlated with the presence of chaos in the classical limit. We propose the statistical distance between initially similar number distributions as a reliable measure to distinguish regular from chaotic behavior in the quantum dynamics. Besides being experimentally accessible, number distributions can be efficiently reconstructed numerically from binned phase-space trajectories in a truncated Wigner approximation. Although the evolving Wigner function becomes very irregular in the chaotic regions, the truncated Wigner method is nevertheless able to capture accurately the beyond-mean-field dynamics.
We study the time-evolution operator in a family of local quantum circuits with random fields in a fixed direction. We argue that the presence of quantum chaos implies that at large times the time-evolution operator becomes effectively a random matrix in the many-body Hilbert space. To quantify this phenomenon, we compute analytically the squared magnitude of the trace of the evolution operator-the generalized spectral form factor-and compare it with the prediction of random matrix theory. We show that for the systems under consideration, the generalized spectral form factor can be expressed in terms of dynamical correlation functions of local observables in the infinite temperature state, linking chaotic and ergodic properties of the systems. This also provides a connection between the many-body Thouless time τ_{th}-the time at which the generalized spectral form factor starts following the random matrix theory prediction-and the conservation laws of the system. Moreover, we explain different scalings of τ_{th} with the system size observed for systems with and without the conservation laws.
According to the eigenstate thermalization hypothesis (ETH), the eigenstate-to-eigenstate fluctuations of expectation values of local observables should decrease with increasing system size. In approaching the thermodynamic limit-the number of sites and the particle number increasing at the same rate-the fluctuations should scale as ∼D^{-1/2} with the Hilbert space dimension D. Here, we study a different limit-the classical or semiclassical limit-by increasing the particle number in fixed lattice topologies. We focus on the paradigmatic Bose-Hubbard system, which is quantum-chaotic for large lattices and shows mixed behavior for small lattices. We derive expressions for the expected scaling, assuming ideal eigenstates having Gaussian-distributed random components. We show numerically that, for larger lattices, ETH scaling of physical midspectrum eigenstates follows the ideal (Gaussian) expectation, but for smaller lattices, the scaling occurs via a different exponent. We examine several plausible mechanisms for this anomalous scaling.
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.