Self-consistent Description of Bose-Bose Droplets: Modified Gapless Hartree-Fock-Bogoliubov Method

TL;DR

This paper introduces a self-consistent theoretical framework for describing Bose-Bose droplets, incorporating higher order quantum fluctuation effects to accurately predict their properties and excitation spectra.

Contribution

It develops a modified gapless Hartree-Fock-Bogoliubov method that resolves phonon excitation issues and accurately models quantum droplets in Bose-Bose mixtures.

Findings

Ensures a gapless phonon spectrum in the presence of higher order terms

Provides a way to compute Lee-Huang-Yang corrections in inhomogeneous droplets

Applicable to droplets confined in harmonic traps

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 anomalous density terms. The equation is self-consistently coupled to modified Bogoliubov equations. The modification we introduce resolves the longstanding issue of missing phonon-branch excitations when higher order terms are included. Our method ensures a gapless phononic low-energy excitation spectrum, crucial to correctly account for quantum fluctuations. 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.