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arXiv:2307.07208 · Physical review. E

Signatures of quantum chaos and fermionization in the incoherent transport of bosonic carriers in the Bose-Hubbard chain

Pavel S. Muraev, Дмитрий Н. Максимов, Andrey R. Kolovsky

Published Mar 22, 2024• 3 citations
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Abstract

We analyze the stationary current of Bose particles across the Bose-Hubbard chain connected to a battery, focusing on the effect of interparticle interactions. It is shown that the current magnitude drastically decreases as the strength of interparticle interactions exceeds the critical value which marks the transition to quantum chaos in the Bose-Hubbard Hamiltonian. We found that this transition is well reflected in the nonequilibrium many-body density matrix of the system. Namely, the level-spacing distribution for eigenvalues of the density matrix changes from Poisson to Wigner-Dyson distributions. With the further increase of the interaction strength, the Wigner-Dyson spectrum statistics change back to the Poisson statistics which now marks fermionization of the Bose particles. With respect to the stationary current, this leads to the counter-intuitive dependence of the current magnitude on the particle number.

DOI
10.1103/physreve.109.l032107

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We analyze the stationary current of bosonic carriers in the Bose-Hubbard chain of length L where the first and the last sites of the chain are attached to reservoirs of Bose particles acting as a particle source and sink, respectively. The analysis is carried out by using the pseudoclassical approach which reduces the original quantum problem to the classical problem for L coupled nonlinear oscillators. It is shown that an increase of oscillator nonlinearity (which is determined by the strength of interparticle interactions) results in a transition from the ballistic transport regime, where the stationary current is independent of the chain length, to the diffusive regime, where the current is inversely proportional to L.
We study the level spacing statistics P(s)P(s)P(s) in many-body Fermi systems and determine a critical two-body interaction strength Uc{U}_{c}Uc​ at which a crossover from Poisson to Wigner-Dyson statistics takes place. Near the Fermi level, the results allow one to find a critical temperature Tch{T}_{\mathrm{ch}}Tch​ above which quantum chaos and thermalization set in.
We study the nonequilibrium steady state of the one-dimensional Bose-Hubbard model coupled to Lindblad reservoirs, using the density-matrix renormalization group in operator space. We observe a transition from a flat particle density profile in the noninteracting limit to the linearly increasing profile with onset of the interparticle interaction. Analyzing the effect of coherent pumping on the nonequilibrium steady state, we find a subspace which remains unaffected by the pumping in the noninteracting limit with the protection gradually diminishing due to interparticle interaction. In the equilibrium situation with one or more symmetric reservoirs, we show analytically that the steady state of the system is a product state for any interaction strength. We also provide analytical results in the noninteracting limits, using the method of the third quantization in operator space.
The emergence of quantum chaos for interacting Fermi systems is investigated by numerical calculation of the level spacing distribution P(s)P(s)P(s) as a function of interaction strength U and the excitation energy $\ensuremath{\epsilon}$ above the Fermi level. As U increases, P(s)P(s)P(s) undergoes a transition from Poissonian (nonchaotic) to Wigner-Dyson (chaotic) statistics and the transition is described by a single scaling parameter given by $Z=(U{\ensuremath{\epsilon}}^{\ensuremath{\alpha}}\ensuremath{-}{u}_{0}){\ensuremath{\epsilon}}^{1/(2\ensuremath{\nu})},$ where u0{u}_{0}u0​ is a constant. While the exponent $\ensuremath{\alpha},$ which determines the global change of the chaos border, is indecisive within a broad range of $0.9\ensuremath{\sim}2.0,$ the finiteness of $\ensuremath{\nu},$ which comes from the increase of the Fock space size with $\ensuremath{\epsilon},$ suggests that the transition becomes sharp as $\ensuremath{\epsilon}$ increases.
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.