Singular Mean-Field States: A Brief Review of Recent Results

Elad Shamriz,Zhaopin Chen,Hidetsugu Sakaguchi,Boris A Malomed

doi:10.3390/condmat5010020

Elad Shamriz, Zhaopin Chen + Show 2 more

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https://doi.org/10.3390/condmat5010020

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Journal: Condensed Matter	Publication Date: Mar 20, 2020
Citations: 14	License type: CC BY 4.0

Affiliation: Tel Aviv University, Kyushu University

Abstract

This article provides a focused review of recent findings which demonstrate, in some cases quite counter-intuitively, the existence of bound states with a singularity of the density pattern at the center; the states are physically meaningful because their total norm converges. One model of this type is based on the 2D Gross–Pitaevskii equation (GPE), which combines the attractive potential ∼ r − 2 and the quartic self-repulsive nonlinearity, induced by the Lee–Huang–Yang effect (quantum fluctuations around the mean-field state). The GPE demonstrates suppression of the 2D quantum collapse, driven by the attractive potential, and emergence of a stable ground state (GS), whose density features an integrable singularity ∼ r − 4 / 3 at r → 0 . Modes with embedded angular momentum exist too, but they are unstable. A counter-intuitive peculiarity of the model is that the GS exists even if the sign of the potential is reversed from attraction to repulsion, provided that its strength is small enough. This peculiarity finds a relevant explanation. The other model outlined in the review includes 1D, 2D, and 3D GPEs, with the septimal (seventh-order), quintic, and cubic self-repulsive terms, respectively. These equations give rise to stable singular solitons, which represent the GS for each dimension D, with the density singularity ∼ r − 2 / ( 4 − D ) . Such states may be considered the results of screening a “bare” delta-functional attractive potential by the respective nonlinearities.

Highlights

Much interest was drawn to quasi-2D and 3D self-trapped states in Bose–Einstein condensates (BECs) in the form of “quantum droplets”, filled by a nearly incompressible two-component condensate, which is considered as an ultradilute quantum fluid
The condition of the convergence of the integral norm is relevant for self-trapped states, which are predicted, as localized solutions, by models such as the nonlinear Schrödinger equation (NLSE) for a complex wave function, ψ (x, y, z, t): iψt + (1/2)∇2ψ − σ |ψ|2ν ψ = 0, (10)
The numerical scheme for producing singular solitons as solutions of the NLSEs with the self-repulsive nonlinearity must be adjusted to the fact that, in the analytical form, the solutions take infinite values at the origin

Summary

Introduction

A usual example of the relevance of this condition is provided by the quantum-mechanical Schrödinger equation with the attractive Coulomb potential, U (r) ∼ −r−1: the stationary real wave function of states with integer azimuthal quantum number (alias vorticity) l ≥ 0 has expansion u(r) = u0 + u1r + O r2 rl (1). With U0 > 0, which gives rise to the quantum collapse, alias “fall onto the center” [1]. This well-known phenomenon means the nonexistence of the ground state (GS) in the 3D and 2D Schrödinger equations with potential (2).

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Conclusion