  Quantum collapse in three and two dimensions (3D and 2D) is induced by
attractive potential ~ -1/r^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, ~ 1/r^{4/3},
but convergent integral norm. Counter-intuitively, 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 periphery.
