Robustness of the thermal Hall effect close to half-quantization in a field-induced spin liquid state

Thermal signatures of fractionalized excitations are a fingerprint of quantum spin liquids (QSLs). In the $J_{eff}=1/2$ honeycomb magnet $alpha$-RuCl$_3$, a QSL state emerges upon applying an in-plane magnetic field $H_{||}$ greater than the critical field $H_{C2} approx$ 7 T along the a-axis, where the thermal Hall conductivity ($k_{XY}/T$) was reported to take on the half-quantized value $k_{HQ}/T$. This finding was discussed as a signature of an emergent Majorana edge mode predicted for the Kitaev QSL. The $H_{||}$- and $T$-range of the half-quantized signal and its relevance to a Majorana edge mode are, however, still under debate. Here we present a comprehensive study of $k_{XY}/T$ in $alpha$-RuCl$_3$ with $H_{||}$ up to 13 T and $T$ down to 250 mK, which reveals the presence of an extended region of the phase diagram with $k_{XY}/T approx k_{HQ}/T$ above $H_{C2}$, in particular across a plateau-like plane for $H_{||}$ > 10 T and $T$ < 6.5 K. From 7 T up to $sim$10 T, $k_{XY}/T$ is suppressed to zero upon cooling to lowest temperature without any plateau-like behavior and exhibits correlations with complex anomalies in the longitudinal thermal conductivity ($k_{XX}$) and magnetization around 10 T. The results are in support of a topological state with a half-quantized $k_{XY}/T$ and suggest an interplay with crossovers or weak phase transitions beyond $H_{C2}$ in RuCl$_3$.