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BEC is a quantum phenomenon, where a macroscopic number of bosons occupy thelowest energy state and acquire coherence at low temperatures. It is realizednot only in $^4$He and dilute atomic gases, but also in quantum magnets, wherehardcore bosons, introduced by the Matsubara-Matsuda transformation of spins,condense. In 3D antiferromagnets, an XY-type long-range ordering (LRO) occursnear a magnetic-field-induced transition to a fully polarized state (FP) andhas been successfully described as a BEC in the last few decades. An attractiveextension of the BEC in 3D magnets is to make their 2D analogue. For a strictly2D system, BEC cannot take place due to the presence of a finite density ofstates at zero energy, and a Berezinskii-Kosterlitz-Thouless (BKT) transitionmay instead emerge. In a realistic quasi-2D magnet consisting of stacked 2Dmagnets, a small but finite interlayer coupling stabilizes marginal LRO andBEC, but such that 2D physics, including BKT fluctuations, is still expected todominate. A few systems were reported to show such 2D-limit BEC, but at veryhigh magnetic fields that are difficult to access. The honeycomb $S$ = 1/2Heisenberg antiferromagnet YbCl$_3$ with an intra-layer coupling $J\sim$ 5 Kexhibits a transition to a FP state at a low in-plane magnetic field of $H_{\rms}$ = 5.93 T. Here, we demonstrate that the LRO right below $H_{\rm s}$ is aBEC in the 2D-limit stabilized by an extremely small interlayer coupling$J_{\perp}$ of 10$^{-5}J$. At the quantum critical point Hs, we capture2D-limit quantum fluctuations as the formation of a highly mobile, interacting2D Bose gas in the dilute limit. A much-reduced effective boson-boson repulsionUeff as compared with that of a prototypical 3D system indicates the presenceof a logarithmic renormalization of interaction unique to 2D.

Quantum critical Bose gas in the two-dimensional limit in the honeycomb antiferromagnet YbCl$_3$ under magnetic fields