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

Recently, there have been increasingly hot debates on whether there exists a quantum spin liquid in the Kitaev honeycomb magnet $\alpha$-RuCl$_3$ in high magnetic field. To investigate this issue, we perform the ultralow-temperature thermal 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 phase transition occurring at $H_c$ $\approx$ 7.5 T, above which the zigzag magnetic order is completely suppressed. The minimum of thermal conductivity at 7.5 T is attributed to the strong scattering of phonons by the magnetic fluctuations. Most importantly, above 7.5 T, we do not observe any significant contribution of thermal conductivity from gapless magnetic excitations, which puts a strong constraint on the nature of the high-field phase of $\alpha$-RuCl$_3$.