#### 1.1.1. Outline of the Topic

A commonly adopted condition which distinguishes physically relevant states produced by various two- and three-dimensional (2D and 3D) models originating in quantum mechanics, studies of Bose–Einstein condensates (BECs), nonlinear optics, plasma physics, etc., is that the respective fields, such as wave functions in quantum mechanics and the mean-field (MF) description of BEC, or local amplitudes of optical fields, must avoid singularities at

$r\to 0$, where

r is the radial coordinate. A usual example of the relevance of this condition is provided by the quantum-mechanical Schrödinger equation with the attractive Coulomb potential,

$\mathcal{U}\left(r\right)\sim -{r}^{-1}$: the stationary real wave function of states with integer azimuthal quantum number (alias vorticity)

$l\ge 0$ has expansion

at

$r\to 0,$ thus avoiding a singularity, despite the fact that the trapping potential is singular [

1].

In quantum mechanics, a critical role is played by a more singular attractive potential;

viz.,

with

${U}_{0}>0$, 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). In 3D, the collapse occurs when

${U}_{0}$ exceeds a finite critical value,

${\left({U}_{0}^{\left(3\mathrm{D}\right)}\right)}_{\mathrm{cr}}$ (in the notation adopted below in Equation (

8),

${\left({U}_{0}^{\left(3\mathrm{D}\right)}\right)}_{\mathrm{cr}}=1/4$), whereas in two dimensions

${\left({U}_{0}^{\left(2\mathrm{D}\right)}\right)}_{\mathrm{cr}}=0$; i.e., the 2D collapse happens at any

${U}_{0}>0$.

In both 3D and 2D cases, potential (

2) may be realized as the electrostatic pull of a particle (small molecule), carrying a permanent electric dipole moment, to a charge placed at the center, assuming that the local orientation of the dipole is fixed by the minimization of its energy in the central field [

2]. In addition to that, in the 2D case the same potential (

47) may be realized as the attraction of a magnetically polarizable atom to a thread carrying electric current (e.g., an electron beam) transversely to the system’s plane, or the attraction of an electrically polarizable atom to a uniformly charged transverse thread (other 2D settings in Bose–Einstein condensates (BECs) under the action of similar fields were considered in [

3,

4]).

A fundamental issue is the stabilization of the 3D and 2D quantum-mechanical settings with pulling potential (

2) against the collapse, with the aim of creating a missing GS. A solution was proposed in [

5,

6,

7], which replaced the original quantum-mechanical problem with one based on a linear quantum-field theory. While such a model produces a GS, it does not answer a natural question: what is an effective radius of the GS for given parameters of the setting, such as

${U}_{0}$ in (

2) and the mass of the quantum particle,

m. Actually, the field-theory solution defines the GS size as an arbitrary spatial scale, which varies as a parameter of the respective field-theory renormalization group. A related problem is the definition of self-adjoint Hamiltonians in the linear quantum theory, including the interaction of a particle with singular potentials [

8,

9].

Another solution was proposed in [

2], which replaced the 3D linear Schrödinger equation by a nonlinear Gross–Pitaevskii equation (GPE) [

10] for a gas of particles attracted to the center by potential (

47), with repulsive inter-particle collisions. In the framework of the mean-field (MF) approximation, the 3D GPE gives rise to the missing GS at all values of

${U}_{0}>{\left({U}_{0}^{\left(3\mathrm{D}\right)}\right)}_{\mathrm{cr}}$. The radius of the GS is fully determined by model’s constants, i.e.,

${U}_{0}$,

m, the scattering length of the inter-particle collisions, and the number of particles,

N. For typical values of the physical parameters, an estimate for the radius is a few microns. Beyond the framework of the MF, it was demonstrated that the many-body quantum theory, applied to the same setting, does not, strictly speaking, create GS, but the interplay of the attraction to the center and inter-particle repulsion gives rise to a metastable state. For sufficiently large

N, the metastable state is nearly tantamount to GS, being separated from the collapse regime by a very tall potential barrier [

11]. Further, the mean-field GS was also constructed in the 3D gas embedded in a strong uniform electric field, which reduces the symmetry of the effective pulling potential from spherical to cylindrical [

12], as well as in the two-component 3D gas [

13].

In 2D, the problem is more difficult, as the repulsive cubic term in GPE, which represents inter-atomic collisions in the MF approximation [

10], is not strong enough to suppress the 2D quantum collapse and create the GS. The main issue is that, in 3D and 2D settings alike, the MF wave function,

$\psi \left(r\right)$, produced by the respective GPE, has density

${\left|\psi \left(r\right)\right|}^{2}$ diverging

$\sim {r}^{-2}$ at

$r\to 0$. In terms of the integral norm,

(

$D=3$ or 2 is the dimension), the density singularity

${r}^{-2}$ is integrable in 3D, but not in 2D, where it gives rise to a logarithmic divergence of the norm:

where

${r}_{\mathrm{cutoff}}$ is a cutoff (smallest) radius, which may be determined by the size of particles in the condensate. Actually, the cubic self-repulsion is

critical in 2D, as any stronger nonlinear term is sufficient to stabilize the 2D setting. In 3D, the critical value of the repulsive-nonlinearity power, which also leads to the logarithmic divergence of

N, is

$7/3$; it is relevant to mention that the respective nonlinear term,

${\left|\psi \right|}^{4/3}\psi $, represents the effective repulsion in the density-functional model of the Fermi gas [

14,

15,

16,

17].

A solution for the 2D setting is offered by the quintic defocusing nonlinearity [

2], which may account for three-body repulsive collisions in the bosonic gas [

18,

19]. However, a difficulty in the physical realization of the quintic term is the fact that three-body collisions give rise to effective losses, kicking out particles from BEC to the thermal component of the gas [

20,

21,

22].

In addition to original papers [

2,

11,

12,

13], the above-mentioned results were summarized in a short review [

23].

#### 1.1.2. New Results Included in the Review

Recently, much interest was drawn to quasi-2D and 3D self-trapped states in BEC in the form of “quantum droplets”, filled by a nearly incompressible two-component condensate, which is considered as an ultradilute quantum fluid. This possibility was predicted in the frameworks of the 3D [

24] and 2D [

25,

26,

27] GPEs, which include the Lee–Huang–Yang (LHY) corrections to the MF approximation [

28]. They represent effects of quantum fluctuations around the MF states. The two-component structure of the condensate makes it possible to provide nearly complete cancellation between the inter-component MF attraction and intra-component repulsion (which, in turn, can be adjusted by means of the Feshbach-resonance technique [

29]), and thus create stable droplets through the balance of the relatively weak residual MF attraction and LHY-induced quartic self-repulsion. The quantum droplets of oblate (quasi-2D) [

30,

31] and fully 3D (isotropic) [

32,

33] shapes were created in a mixture of two different spin states of

${}^{39}$K atoms, and in a mixture of

${}^{41}$K and

${}^{87}$Rb atoms [

34]. Further, it was predicted that 2D [

35,

36,

37] and 3D [

38] droplets with

embedded vorticity have their stability regions too. The LHY effect has also opened the way to the creation of stable 3D droplets in single-component BEC with long-range interactions between atoms carrying magnetic dipole moments [

39,

40,

41,

42,

43], although dipolar-condensate droplets with embedded vorticity are unstable [

44].

The LHY effect in the 3D model of BEC pulled to the center by potential (

2) was considered in [

23], with the conclusion that the LHY term gives rise to a quantum phase transition at

${U}_{0}=2/9$ (note that it is close to but smaller than the above-mentioned critical value for the linear Schrödinger equation,

${\left({U}_{0}^{\left(3\mathrm{D}\right)}\right)}_{\mathrm{cr}}=1/4$). The phase transition manifests itself in the change of the asymptotic form of the GS stationary wave function at

$r\to 0$:

This phase transition may be categorized as one of the first kind, as power

$\alpha $ in terms

${r}^{-\alpha}$ in Equation (

5) undergoes a finite jump at

${U}_{0}=2/9$, from

$1/3$ to

$2/3$.

In

Section 2 of the present review, we summarize recent results which demonstrate that the stabilization of the GS in the quasi-2D bosonic gas pulled to the center by potential (

47) may be provided by the LHY correction to the GPE [

45]. This possibility is essential because, as mentioned above, the alternative, in the form of the quintic self-repulsion, is problematic in the BEC setting. The underlying three-dimensional GPE, which includes the LHY quartic defocusing term, is

where

$\mathsf{\Psi}$ stands for equal wave functions of two components of the BEC;

$W\left(\mathbf{r}\right)$ is the general trapping potential;

$a>0$ is the scattering length of inter-particle collisions,

$\delta a\gtrless 0$, with

$\left|\delta a\right|\ll a$, representing the above-mentioned small imbalance of the inter-component attraction and intra-component repulsion; and the last term in Equation (

6) is the LHY correction to the MF equation [

24].

The reduction of Equation (

6) to the 2D form, with coordinates

$\left(x,y\right)$, under the action of tight confinement applied in the

z direction, was derived in [

25], producing the GPE with a cubic term multiplied by an additional logarithmic factor,

However, this limit implies extremely strong confinement in the

z direction, with the transverse size

${a}_{\perp}\ll \xi $, where the healing length is

$\xi =\left(32\sqrt{2}/3\pi \right){\left(a/\left|\delta a\right|\right)}^{3/2}a\approx 5{\left(a/\left|\delta a\right|\right)}^{3/2}a$ [

24]. For experimentally relevant parameters [

30,

31,

32,

33], an estimate is

$\xi \simeq 30$ nm. On the other hand, a realistic size of the confinement length in the experiment is a few

$\mathsf{\mu}$m, implying relation

${a}_{\perp}\gg \xi $, opposite to the above-mentioned one necessary for the derivation of Equation (

7). Therefore, it is relevant to reduce Equation (

6) to the 2D form, keeping the same nonlinearity as in Equation (

6).

To complete the derivation of the effective 2D equation, we first rescale three-dimensional Equation (

6), measuring the density, length, time, and trapping potential in units of

$\left(36/25\right){n}_{0}$,

$\xi $,

$\tau \equiv \left(m/\hslash \right){\xi}^{2}$, and

$\hslash /\tau $, respectively:

where

$\mathsf{\sigma}=\pm 1$ is the sign of

$\delta a$; the potential is a sum of term (

2) and a transverse-confinement term,

$\left(1/2\right){a}_{\perp}^{-4}{z}^{2}$, with sufficiently small

${a}_{\perp}^{2}$. Then, the 3D → 2D reduction is performed by means of the usual substitution [

46,

47],

$\mathsf{\Psi}\left(x,y,z,t\right)=\psi \left(x,y,t\right)exp\left(-{z}^{2}/2{a}_{\perp}^{2}\right)$, followed by averaging in the transverse direction,

z. Additional rescaling,

$\psi \to \left(2/\sqrt{5}\right)\psi $,

$\left(x,y\right)\to \left(\sqrt{5}/2\right)\left(x,y\right)$, and

$t\to \left(5/4\right)t$, casts the effective 2D equation, written in polar coordinates

$\left(r,\theta \right)$, in the final form:

which includes potential (

2).

In the framework of Equation (

9), it is also relevant to consider the case of

$\delta a=0$, which implies the exact compensation of the inter-component attraction and intra-component repulsion. In this case, one should set

$\mathsf{\sigma}=0$ in Equation (

9), keeping the nonlinearity which originates as the LHY correction to the MF field theory (cf. [

48]).

Results for both GS and vortex states, produced by the analysis of the 2D Equation (

9) [

45], are summarized, as a part of the present review, in

Section 2. Essential conclusions are that all the GS solutions (with zero vorticity) are stable. States with embedded vorticity are constructed too, but they are unstable.