  BEC is a quantum phenomenon, where a macroscopic number of bosons occupy the
lowest energy state and acquire coherence at low temperatures. It is realized
not only in $^4$He and dilute atomic gases, but also in quantum magnets, where
hardcore bosons, introduced by the Matsubara-Matsuda transformation of spins,
condense. In 3D antiferromagnets, an XY-type long-range ordering (LRO) occurs
near a magnetic-field-induced transition to a fully polarized state (FP) and
has been successfully described as a BEC in the last few decades. An attractive
extension of the BEC in 3D magnets is to make their 2D analogue. For a strictly
2D system, BEC cannot take place due to the presence of a finite density of
states at zero energy, and a Berezinskii-Kosterlitz-Thouless (BKT) transition
may instead emerge. In a realistic quasi-2D magnet consisting of stacked 2D
magnets, a small but finite interlayer coupling stabilizes marginal LRO and
BEC, but such that 2D physics, including BKT fluctuations, is still expected to
dominate. A few systems were reported to show such 2D-limit BEC, but at very
high magnetic fields that are difficult to access. The honeycomb $S$ = 1/2
Heisenberg antiferromagnet YbCl$_3$ with an intra-layer coupling $J\sim$ 5 K
exhibits a transition to a FP state at a low in-plane magnetic field of $H_{\rm
s}$ = 5.93 T. Here, we demonstrate that the LRO right below $H_{\rm s}$ is a
BEC in the 2D-limit stabilized by an extremely small interlayer coupling
$J_{\perp}$ of 10$^{-5}J$. At the quantum critical point Hs, we capture
2D-limit quantum fluctuations as the formation of a highly mobile, interacting
2D Bose gas in the dilute limit. A much-reduced effective boson-boson repulsion
Ueff as compared with that of a prototypical 3D system indicates the presence
of a logarithmic renormalization of interaction unique to 2D.
