  The Berezinskii-Kosterlitz-Thouless (BKT) mechanism, building upon
proliferation of topological defects in 2D systems, is the first example of
phase transition beyond the Landau-Ginzburg paradigm of symmetry breaking. Such
a topological phase transition has long been sought yet undiscovered directly
in magnetic materials. Here, we pin down two transitions that bound a BKT phase
in an ideal 2D frustrated magnet TmMgGaO$_4$, via nuclear magnetic resonance
under in-plane magnetic fields, which do not disturb the low-energy electronic
states and allow BKT fluctuations to be detected sensitively. Moreover, by
applying out-of-plane fields, we find a critical scaling behaviour of the
magnetic susceptibility expected for the BKT transition. The experimental
findings can be explained by quantum Monte Carlo simulations applied on an
accurate triangular-lattice Ising model of the compound which hosts a BKT
phase. These results provide a concrete example for the BKT phase and offer an
ideal platform for future investigations on the BKT physics in magnetic
materials.
