Thermodynamic perspective on the field-induced behavior of $\alpha$-RuCl$_3$

Measurements of the magnetic Gr\"uneisen parameter ($\Gamma_B$) and specific heat on the Kitaev material candidate $\alpha$-RuCl$_3$ are used to access in-plane field- and temperature-dependence of the entropy up to 12 T and down to 1 K. No signatures corresponding to phase transitions are detected beyond the boundary of the magnetically ordered region, but only a shoulder-like anomaly in $\Gamma_B$, involving an entropy increment as small as $10^{-5} R\log 2$. These observations put into question the presence of a thermodynamic phase transition between the purported quantum spin liquid and the field-polarized state of $\alpha$-RuCl$_3$. We show theoretically that at low temperatures $\Gamma_B$ is sensitive to crossings in the lowest excitations within gapped phases, and identify the measured shoulder-like anomaly as being of such origin. Exact diagonalization calculations demonstrate that the shoulder-like anomaly can be reproduced in extended Kitaev models that gain proximity to an additional phase at finite field without entering it. We discuss manifestations of this proximity in other measurements.