High-resolution thermal expansion and magnetostriction measurements were performed on single crystals of $alpha$-RuCl$_3$ in magnetic fields applied parallel to the Ru-Ru bonds. The length changes were measured in the direction perpendicular to the honeycomb planes. Our data show clear thermodynamic characteristics for the field-induced phase transition at the critical field $mu_0H_{c1} = 7.8(2)$ T where the antiferromagnetic zigzag order is suppressed. At higher fields, a kink in the magnetostriction coefficient signals an additional phase transition around $mu_0H_{c2} approx 11$ T. The extracted Gruneisen parameter shows typical hallmarks for quantum criticality near $H_{c1}$, but also displays anomalous behavior above $H_{c1}$. We compare our experimental data with linear spin-wave calculations employing a minimal Kitaev-Heisenberg model in the semiclassical limit. Most of the salient features are in agreement with each other, however, the peculiar features in the high-field region above $H_{c1}$ cannot be accounted for in our modelling and hence suggest a genuine quantum nature. We construct a phase diagram for $alpha$-RuCl$_3$ showing two low-temperature transitions induced by an in-plane field along the Ru-Ru bonds.
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