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arXiv:1508.05449 · Physical Review A

Quantum versus mean-field collapse in a many-body system

G. E. Astrakharchik, Boris A. Malomed

Published Oct 29, 2015• 20 citations
arXivHTMLPDF

Abstract

The recent analysis, based on the mean-field approximation (MFA), has predicted that the critical quantum collapse of the bosonic wave function, pulled to the center by the inverse-square potential in the three-dimensional space, is suppressed by the repulsive cubic nonlinearity in the bosonic gas, the collapsing ground state being replaced by a regular one. We demonstrate that a similar stabilization acts in a quantum many-body system, beyond the MFA. While the collapse remains possible, repulsive two-particle interactions give rise to a metastable gaseous state, which is separated by a potential barrier from the collapsing regime. The stability of this state improves with the increase of the number of particles. The results are produced by calculations of the variational energy, with the help of the Monte Carlo method.

DOI
10.1103/physreva.92.043632

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The recent analysis, based on the mean-field approximation (MFA), has predicted that the critical quantum collapse of the bosonic wave function, pulled to the center by the inverse-square potential in the three-dimensional space, is suppressed by the repulsive cubic nonlinearity in the bosonic gas, the collapsing ground state being replaced by a regular one. We demonstrate that a similar stabilization acts in a quantum many-body system, beyond the MFA. While the collapse remains possible, repulsive two-particle interactions give rise to a metastable gaseous state, which is separated by a potential barrier from the collapsing regime. The stability of this state improves with the increase of the number of particles. The results are produced by calculations of the variational energy, with the help of the Monte Carlo method.
It is known that attractive potential ~− 1 / r 2 gives rise to the critical quantum collapse in the framework of the three-dimensional (3D) linear Schrödinger equation. This article summarizes theoretical analysis, chiefly published in several original papers, which demonstrates suppression of the collapse caused by this potential, and the creation of the otherwise missing ground state in a 3D gas of bosonic dipoles pulled by the same potential to the central charge, with repulsive contact interactions between them, represented by the cubic term in the respective Gross–Pitaevskii equation (GPE). In two dimensions (2D), quintic self-repulsion is necessary for the suppression of the collapse; alternatively, this may be provided by the effective quartic repulsion produced by the Lee–Huang–Yang correction to the GPE. 3D states carrying angular momentum are constructed in the model with the symmetry reduced from spherical to cylindrical by an external polarizing field. Interplay of the collapse suppression and miscibility–immiscibility transition is considered in a binary condensate. The consideration of the 3D setting in the form of the many-body quantum system, with the help of the Monte Carlo method, demonstrates that, although the quantum collapse cannot be fully suppressed, the self-trapped states predicted by the GPE exist in the many-body setting as metastable modes protected against the collapse by a tall potential barrier.
Attraction of the quantum particle to the center in three-dimensional space with potential $\ensuremath{-}{V}_{0}{r}^{\ensuremath{-}2}$ gives rise to the quantum collapse, i.e., nonexistence of the ground state (GS) when the attraction strength exceeds a critical value [(V0(cr))1=1/8{({V}_{0}^{(\mathrm{cr})})}_{1}=1/8(V0(cr)​)1​=1/8, in the present notation]. Recently, we have demonstrated that the quantum collapse is suppressed and the GS is restored if repulsive interactions between particles in the quantum gas are taken into account in the mean-field approximation. This setting can be realized in a gas of dipolar molecules attracted to the central charge, with dipole-dipole interactions taken into regard as well. Here we analyze this problem for a binary gas. GSs supported by the repulsive interactions are constructed in a numerical form, as well as by means of analytical approximations for both miscible and immiscible binary systems. In particular, the Thomas-Fermi approximation is relevant if V0{V}_{0}V0​ is large enough. It is found that the GS of the miscible binary gas, both balanced and imbalanced, features a weak phase transition at another critical value, ${({V}_{0}^{(\mathrm{cr})})}_{2}=1/2\ensuremath{\equiv}4{({V}_{0}^{(\mathrm{cr})})}_{1}$. The transition is characterized by an analyticity-breaking change in the structure of the wave functions at small rrr. To illustrate the generic character of the present phenomenology, we also consider the binary system with attraction between the species (rather than repulsion) in the case when the central potential pulls a single component only.
It is known that the attractive singular inverse-square potential gives rise to the critical quantum collapse in the framework of the three-dimensional (3D) linear Schroedinger equation. This article summarizes theoretical results which demonstrate suppression of the collapse, caused by this singular potential, and the creation of the otherwise missing ground state (GS) in a 3D gas of bosonic particles, carrying an electric dipole moment, which are pulled to the central electric charge, with repulsive contact interactions between the particles. In the mean-field approximation, the repulsive interactions are represented by the cubic term in the respective Gross-Pitaevskii (GP) equation. In addition to the GS, excited states with angular momentum are briefly considered too. Another topic considered in the article is 1D and 2D bound states in the linear Schroedinger and GP equations with the repulsive potential, which demonstrates a singularity at r --> infinity. A very recent result is that such a potential, growing faster than the negative harmonic-oscillator potential, produces a full spectrum of counter-intuitive normalizable (localized) bound states. The article puts forward perspectives for further studies of linear and nonlinear bound states existing under the action of the potentials with the singularity at r --> 0 or r --> infinity.
Quantum collapse in three and two dimensions (3D and 2D) is induced by attractive potential $\ensuremath{\sim}\ensuremath{-}{r}^{\ensuremath{-}2}$. It was demonstrated that the mean-field (MF) cubic self-repulsion in the 3D bosonic gas suppresses the collapse and creates the missing ground state (GS). However, the cubic nonlinearity is not strong enough to suppress the 2D collapse. We demonstrate that the Lee-Hung-Yang (LHY) quartic term, induced by quantum fluctuations around the MF state, is sufficient for the stabilization of the 2D gas against the collapse. By means of numerical solution of the Gross-Pitaevskii equation including the LHY term, as well as with the help of analytical methods, such as expansions of the wave function at small and large distances from the center and the Thomas-Fermi approximation, we construct stable GS, with a singular density $\ensuremath{\sim}{r}^{\ensuremath{-}4/3}$ but convergent integral norm. Counterintuitively, the stable GS exists even if the external potential is repulsive, with the strength falling below a certain critical value. An explanation to this finding is given. Along with the GS, singular vortex states are produced too, and their stability boundary is found analytically. Unstable vortices spontaneously transform into the stable GS, expelling the vorticity to the periphery.
The quantum-mechanical collapse (alias fall onto the center of particles attracted by potential $\ensuremath{-}{r}^{\ensuremath{-}2}$) is a well-known issue in quantum theory. It is closely related to the quantum anomaly, i.e., breaking of the scaling invariance of the respective Hamiltonian by quantization. We demonstrate that the mean-field repulsive nonlinearity prevents the collapse and thus puts forward a solution to the quantum-anomaly problem that differs from that previously developed in the framework of the linear quantum-field theory. This solution may be realized in the 3D or 2D gas of dipolar bosons attracted by a central charge and in the 2D gas of magnetic dipoles attracted by a current filament. In the 3D setting, the dipole-dipole interactions are also taken into regard, in the mean-field approximation, resulting in a redefinition of the scattering length which accounts for the contact repulsion between the bosons. In lieu of the collapse, the cubic nonlinearity creates a 3D ground state (GS), which does not exist in the respective linear Schr\"odinger equation. The addition of the harmonic trap gives rise to a tristability, in the case when the Schr\"odinger equation still does not lead to the collapse. In the 2D setting, the cubic nonlinearity is not strong enough to prevent the collapse; however, the quintic term does it, creating the GS, as well as its counterparts carrying the angular momentum (vorticity). Counterintuitively, such self-trapped 2D modes exist even in the case of a weakly repulsive potential ${r}^{\ensuremath{-}2}$. The 2D vortical modes avoid the phase singularity at the pivot (r=0r=0r=0) by having the amplitude diverging at $r\ensuremath{\rightarrow}0$ instead of the usual situation with the amplitude of the vortical mode vanishing at $r\ensuremath{\rightarrow}0$ (the norm of the mode converges despite of the singularity of the amplitude at $r\ensuremath{\rightarrow}0$). In the presence of the harmonic trap, the 2D quintic model with a weakly repulsive central potential ${r}^{\ensuremath{-}2}$ gives rise to three confined modes, the middle one being unstable, spontaneously developing into a breather. In both the 3D and 2D cases, the GS wave functions are found in a numerical form and in the form of an analytical approximation, which is asymptotically exact in the limit of the large norm.
A three-dimensional attractive Bose-Einstein condensate is expected to collapse when the number of the particles NNN in the ground state or the interaction strength ${\ensuremath{\lambda}}_{0}$ exceeds a critical value. We study systems of different particle numbers and interaction strength and find that even if the overall ground state is collapsed there is a plethora of fragmented excited states that are still in the metastable region. Utilizing the configuration interaction expansion we determine the spectrum of the ground (``yrast'') and excited many-body states with definite total angular-momentum quantum numbers $0\ensuremath{\leqslant}L\ensuremath{\leqslant}N$ and $\ensuremath{-}L\ensuremath{\leqslant}{M}_{L}\ensuremath{\leqslant}L$, and we find and examine states that survive the collapse. This opens up the possibility of realizing a metastable system with overcritical numbers of bosons in a ground state with angular momentum $L\ensuremath{\ne}0$. The multiorbital mean-field theory predictions about the existence of fragmented metastable states with overcritical numbers of bosons are verified and elucidated at the many-body level. The descriptions of the total angular momentum within the mean-field and the many-body approaches are compared.