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High-resolution thermal expansion and magnetostriction measurements wereperformed on single crystals of $\alpha$-RuCl$_3$ in magnetic fields appliedparallel to the Ru-Ru bonds. The length changes were measured in the directionperpendicular to the honeycomb planes. Our data show clear thermodynamiccharacteristics for the field-induced phase transition at the critical field$\mu_0H_{c1} = 7.8(2)$ T where the antiferromagnetic zigzag order issuppressed. At higher fields, a kink in the magnetostriction coefficientsignals an additional phase transition around $\mu_0H_{c2} \approx 11$ T. Theextracted Gr\"uneisen parameter shows typical hallmarks for quantum criticalitynear $H_{c1}$, but also displays anomalous behavior above $H_{c1}$. We compareour experimental data with linear spin-wave calculations employing a minimalKitaev-Heisenberg model in the semiclassical limit. Most of the salientfeatures are in agreement with each other, however, the peculiar features inthe high-field region above $H_{c1}$ cannot be accounted for in our modellingand hence suggest a genuine quantum nature. We construct a phase diagram for$\alpha$-RuCl$_3$ showing two low-temperature transitions induced by anin-plane field along the Ru-Ru bonds.

Field-induced phase transitions of the Kitaev material $\alpha$-RuCl$_3$ probed by thermal expansion and magnetostriction