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Recently, there have been increasingly hot debates on whether there exists aquantum spin liquid in the Kitaev honeycomb magnet $\alpha$-RuCl$_3$ in highmagnetic field. To investigate this issue, we perform the ultralow-temperaturethermal conductivity measurements on the single crystals of $\alpha$-RuCl$_3$down to 80 mK and up to 9 T. Our experiments clearly show a field-induced phasetransition occurring at $H_c$ $\approx$ 7.5 T, above which the zigzag magneticorder is completely suppressed. The minimum of thermal conductivity at 7.5 T isattributed to the strong scattering of phonons by the magnetic fluctuations.Most importantly, above 7.5 T, we do not observe any significant contributionof thermal conductivity from gapless magnetic excitations, which puts a strongconstraint on the nature of the high-field phase of $\alpha$-RuCl$_3$.

Ultralow-temperature thermal conductivity of the Kitaev honeycomb magnet $\alpha$-RuCl$_3$ across the field-induced phase transition