Thermodynamic Perspective on Field-Induced Behavior of $\alpha \text{−}{\mathrm{RuCl}}_3$

Measurements of the magnetic Grüneisen parameter (${\mathrm{\Gamma }}_B$) and specific heat on the Kitaev material candidate $\alpha \text{−}{\mathrm{RuCl}}_3$ 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 shoulderlike anomaly in ${\mathrm{\Gamma }}_B$, involving an entropy increment as small as ${10}^{-5}R\log 2$. These observations put into question the presence of a phase transition between the purported quantum spin liquid and the field-polarized state of $\alpha \text{−}{\mathrm{RuCl}}_3$. We show theoretically that at low temperatures ${\mathrm{\Gamma }}_B$ is sensitive to crossings in the lowest excitations within gapped phases, and identify the measured shoulderlike anomaly as being of such origin. Exact diagonalization calculations demonstrate that the shoulderlike 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.

Figure 1

Magnetic field dependencies of (a) the specific heat $C$, (b) the magnetic Grüneisen parameter ${\mathrm{\Gamma }}_B$, (c) the field derivative of the entropy $\partial S/\partial B$ and entropy increment $\mathrm{\Delta }S$ (see text) for $\alpha \text{−}{\mathrm{RuCl}}_3$ at 2 K. The inset in (a) shows a temperature-field phase diagram derived from our data, where solid lines stand for thermodynamic transitions and the dotted line for a crossover at $B^{*}$, and in (b) the field direction of our experiment.

Figure 2

Magnetic field dependence of the specific heat (a) and field derivative of the entropy (b), as well as the entropy change (inset) for $\alpha \text{−}{\mathrm{RuCl}}_3$ at 2 K between 8 and 12 T. The black solid lines indicate $C=\beta (B-B_c{)}^{\epsilon }$ with $B_c^{\mathrm{AF}2}=7.4\mathrm{T}$, $\beta =22.7(9){\mathrm{mJK}}^{-1}{\mathrm{mol}}^{-1}{\mathrm{T}}^{-\epsilon }$, and $\epsilon =0.64(5)$ (a), and $\partial S/\partial B=\beta (B-B_c{)}^{\epsilon }{G}_r/(B-B_c)$ (b), with $G_r=-0.157(2)$, respectively [46]. The inset shows the difference between the measured entropy and the integration of the above function for $\partial S/\partial B$ for fields between 8.6 and 11 T.

Figure 3

$T=2\mathrm{K}$ exact diagonalization results of the magnetic Grüneisen parameter ${\mathrm{\Gamma }}_B$ for the minimal model of $\alpha \text{−}{\mathrm{RuCl}}_3$ [26]. The inset shows the employed periodic cluster.

Figure 4

(a) Schematic of a discrete spectrum evolving with a control parameter $\lambda$. It contains an avoided level crossing at ${\lambda }_c$ and a level crossing in the lowest excited states at ${\lambda }^{*}$. These lead, respectively, to a sign change (${\lambda }_c$) and a shoulder (${\lambda }^{*}$) in the low-temperature Grüneisen parameter ${\mathrm{\Gamma }}_{\lambda }$ (b). If ${\lambda }_c$ is a quantum critical point, the gap would vanish at ${\lambda }_c$ in the thermodynamic limit, leading to a diverging ${\mathrm{\Gamma }}_{\lambda }$ at ${\lambda }_c$ [46, 59, 60]. The low-temperature specific heat $C$ [dashed curve] is nearly unaffected by the level crossing at ${\lambda }^{*}$.