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arXiv:2202.10109 · New Journal of Physics

Self-consistent description of Bose–Bose droplets: modified gapless Hartree–Fock–Bogoliubov method

Paweł Zin, Maciej Pylak, Zbigniew Idziaszek, Mariusz Gajda

Published Nov 1, 2022• 7 citations
arXivHTMLPDF

Abstract

Abstract We define a formalism of a self-consistent description of the ground state of a weakly interacting Bose system, accounting for higher order terms in expansion of energy in the diluteness parameter. The approach is designed to be applied to a Bose–Bose mixture in a regime of weak collapse where quantum fluctuations lead to stabilization of the system and formation of quantum liquid droplets. The approach is based on the generalized Gross–Pitaevskii equation accounting for quantum depletion and renormalized anomalous density terms. The equation is self-consistently coupled to modified Bogoliubov equations. We derive well defined procedure to calculate the zero temperature renormalized anomalous density—the quantity needed to correctly describe the formation of quantum liquid droplet. We pay particular attention to the case of droplets harmonically confined in some directions. The method allows to determine the Lee–Huang–Yang-type contribution to the chemical potential of inhomogeneous droplets when the local density approximation fails.

DOI
10.1088/1367-2630/aca175

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cond-mat.quant-gasCross-listedarXiv:2604.00889•Apr 2, 2026•0 citations
Abstract We define a formalism of a self-consistent description of the ground state of a weakly interacting Bose system, accounting for higher order terms in expansion of energy in the diluteness parameter. The approach is designed to be applied to a Bose–Bose mixture in a regime of weak collapse where quantum fluctuations lead to stabilization of the system and formation of quantum liquid droplets. The approach is based on the generalized Gross–Pitaevskii equation accounting for quantum depletion and renormalized anomalous density terms. The equation is self-consistently coupled to modified Bogoliubov equations. We derive well defined procedure to calculate the zero temperature renormalized anomalous density—the quantity needed to correctly describe the formation of quantum liquid droplet. We pay particular attention to the case of droplets harmonically confined in some directions. The method allows to determine the Lee–Huang–Yang-type contribution to the chemical potential of inhomogeneous droplets when the local density approximation fails.
Abstract The static and dynamic properties of self-bound quantum droplets in a one-dimensional Bose-Bose mixture are discussed in the spirit of the Hartree–Fock-Bogoliubov theory. This latter enables us to provide beyond the Lee-Huang-Yang (LHY) quantum corrections to the equation of state at both zero and finite temperatures. In the uniform case our results for the ground-state energy and the critical temperature are confirmed through comparison with Quantum Monte-Carlo simulation and with available theoretical results. The density profiles are supported by numerical simulations of the generalized Gross-Pitaevskii equation which selfconsistently includes higher-order terms originating from the normal and anomalous fluctuations under the local density approximation. We show that the density exhibits a dip near its center in the flat-top plateau region for large interspecies interactions. We exemplify the impact of the beyond LHY corrections on the spatiotemporal evolution of the self-bound droplet in the presence of excitation induced by periodic density modulation. It is found that higher-order corrections may lead to the formation of a train of small droplets. We then extend our study for the case of inhomogeneous droplets in quasi one-dimensional Bose mixtures.
We systematically study the effects of higher-order quantum and thermal fluctuations on the stabilization of self-bound droplets in Bose mixtures employing the time-dependent Hartree-Fock-Bogoliubov theory. We calculate the ground-state energy, the droplet equilibrium density, the depletion and anomalous density of the droplets as well as the critical temperature as a function of the relevant parameters. Our findings are compared with previous analytical predictions and diffusion Monte Carlo simulations. We employ our theory together with the local density approximation for quantum and thermal fluctuations to obtain an extended finite-temperature Gross-Pitaevskii equation. The density profiles and breathing modes of the droplet are deeply examined in terms of the interaction strength and the temperature by numerically solving the developed generalized Gross-Pitaevskii equation.
We study the equilibrium properties of self-bound droplets in two-dimensional Bose mixtures employing the time-dependent Hartree-Fock-Bogoliubov theory. This theory allows one to understand both the many-body and temperature effects beyond the Lee-Huang-Yang description. We calculate higher-order corrections to the excitations, the sound velocity, and the energy of the droplet. Our results for the ground-state energy are compared with the diffusion Monte Carlo data and good agreement is found. The behavior of the depletion and anomalous density of the droplet is also discussed. At finite temperature, we show that the droplet emerges at temperatures well below the Berezinskii-Kosterlitz-Thouless transition temperature. The critical temperature strongly depends on the interspecies interactions. Our study is extended to the finite size droplet by numerically solving the generalized finite-temperature Gross-Pitaevskii equation which is obtained self-consistently from our formalism in the framework of the local density approximation.
We study the properties of self-bound liquid droplets of three-dimensional Bose mixtures in a weak random potential with Gaussian correlation function at both zero and finite temperatures. Using the Bogoliubov theory, we derive useful formulas for the ground-state energy, the equilibrium density, the depletion, and the anomalous density of the droplet. The quantum fluctuation induced by the disorder known as the glassy fraction is also systematically computed. At finite temperature, we calculate the free energy, the thermal equilibrium density, and the critical temperature in terms of the disorder parameters. We show that when the strength and the correlation length of the disorder potential exceed a certain critical value, the droplet evaporates and is eventually entirely destroyed. We calculate the density profiles of this exotic state by means of numerical simulations of the corresponding generalized disorder Gross-Pitaevskii equation. Our predictions reveal that as the strength of the disorder gets larger, the atomic density varies rapidly in the plateau region. We point out in addition that the peculiar interplay of the disorder and the repulsive Lee-Huang-Yang corrections play a pivotal role in the collective modes of the self-bound droplet.
We describe a quantum droplet of a Bose-Bose mixture squeezed by an external harmonic force in one spatial direction. Our approach is based on the self-consistent method formulated in Zin et al. (arXiv:2202.10109) The true spatial droplet profile in the direction of confinement is accounted for; however, the local density approximation is assumed in the free directions. We define a numerical approach to find the beyond-mean-field contribution to the chemical potential (Lee-Huang-Yang chemical potential)---the quantity that determines the droplet's profile. In addition to the numerical approach, we find the Lee-Huang-Yang potential in the analytic form in two limiting cases: a perturbative result for a strong confinement and a semiclassical expression when confinement is very weak.
In this paper, we present a systematic study on the ground state computation of quantum droplets in homonuclear Bose-Bose mixtures, governed by the extended Gross-Pitaevskii equations (eGPEs) with Lee-Huang-Yang (LHY) corrections. This model captures the formation of self-bound droplets stabilized by the delicate balance between the attractive mean-field interaction and the repulsive quantum fluctuations. We formulate dimensionless energy functionals for both the general two-component system and the reduced single-component density-locked model. To compute the ground states efficiently, we adapt and benchmark various gradient flow discretization schemes, identifying a backward-forward sine-pseudospectral scheme based on the gradient flow with Lagrange multiplier method (GFLM-BFSP) as the robust solver for our simulations. Utilizing this method, we report three main numerical observations: (i) the density-locked model is quantitatively validated as a reliable approximation for ground state properties; (ii) the dimension-dependent convergence rates of the Thomas-Fermi approximation are established in the strong-coupling regime; and (iii) the critical particle number for self-binding in free space is numerically determined, providing a precise correction to the analytical prediction by Petrov [Phys. Rev. Lett. 115, 155302 (2015)].