Angle-dependent thermodynamics of $\alpha \text{−}\mathrm{Ru}{\mathrm{Cl}}_3$

Thermodynamics of the Kitaev honeycomb magnet $\alpha \text{−}\mathrm{Ru}{\mathrm{Cl}}_3$ is studied for different directions of in-plane magnetic field using measurements of the magnetic Grüneisen parameter ${\mathrm{\Gamma }}_B$ and specific heat $C$. We identify two critical fields $B_c^{\mathrm{AF}1}$ and $B_c^{\mathrm{AF}2}$ corresponding, respectively, to a transition between two magnetically ordered states and the loss of magnetic order toward a quantum paramagnetic state. The $B_c^{\mathrm{AF}2}$ phase boundary reveals a narrow region of magnetic fields where inverse melting of the ordered phase may occur. No additional transitions are detected above $B_c^{\mathrm{AF}2}$ for any direction of the in-plane field, although a shoulder anomaly in ${\mathrm{\Gamma }}_B$ is observed systematically at $8–10\mathrm{T}$ . Large field-induced entropy effects imply additional low-energy excitations at low fields and/or strongly field-dependent phonon entropies. Our results establish universal features of $\alpha \text{−}\mathrm{Ru}{\mathrm{Cl}}_3$ in high magnetic fields and challenge the presence of a field-induced Kitaev spin liquid in this material.

Figure 1

(a) Rotation of Grüneisen and specific heat setup by using brass wedges ($\overrightarrow{c^{*}}$ perpendicular to sample platform, magnetic field orientation indicated in the sketch). Note that $\alpha$ is not equivalent to $\varphi$ which is assigned by comparing the critical fields to ac-susceptibility measurements (Fig. 2). (b) Cell background subtraction for $\vec{B}\parallel \left[110\right]$ at 1 K following Eq. (1). Raw data denote the Grüneisen parameter from both cell and sample. For the background measurement, ${\mathrm{\Gamma }}_{B,\mathrm{Cell}}$ was measured independently with magnetic field applied perpendicular to the cell platform. Since no field dependency is expected for the cell, all subtractions use the same background measurement. Despite the subtraction, the positions of $B_c^{\mathrm{AF}1}$ and $B_c^{\mathrm{AF}2}$ are unaffected. (c) At high fields far beyond $B_c^{\mathrm{AF}2}$, the background is responsible for the negative ${\mathrm{\Gamma }}_B$ values. The characteristics of ${\mathrm{\Gamma }}_B\left(B\right)$, however, are again untouched by the background subtraction.

Figure 2

Angle dependence of the magnetic Grüneisen parameter ${\mathrm{\Gamma }}_B$ at 2 K (raw data). The critical fields of the phase transitions $B_c^{\mathrm{AF}1}$ and $B_c^{\mathrm{AF}2}$ are shifted for different in-plane field orientations, from (a) $\vec{B}\parallel \left[110\right]$ to (d) $\vec{B}\parallel \left[010\right]$; the data for $\varphi ={20}^{\circ }$ are from Ref. [23]. The inset in (c) shows the excellent agreement with the critical fields determined from ac-susceptibility measurements [25]. No further phase transitions are observed above $B_c^{\mathrm{AF}2}$ for all field directions up to at least 14 T. The measurement perpendicular to the honeycomb plane ($\vec{B}\parallel \overrightarrow{c^{*}}$) in (a) excludes any out-of-plane contribution.

Figure 3

Field dependence of the magnetic Grüneisen parameter ${\mathrm{\Gamma }}_B$ (raw data) for several temperatures with the in-plane magnetic field $\vec{B}$ applied parallel (a) to [110] and (b) to [010]. $B_c^{\mathrm{AF}2}$ is the point where ${\mathrm{\Gamma }}_B$ changes sign upon the transition between long-range-ordered and quantum paramagnetic states of $\alpha \text{−}\mathrm{Ru}{\mathrm{Cl}}_3$ [23]. At higher temperatures, $B_c^{\mathrm{AF}2}$ shifts towards lower fields due to thermal fluctuations. However, below 2 K an opposite behavior is observed, suggesting that the phase boundary is nonmonotonic [(c),(d)]. The phase transition inside the AF state at $B_c^{\mathrm{AF}1}$ is most likely of first order and, thus, manifests itself in a maximum of ${\mathrm{\Gamma }}_B$, which is clearly visible for $\vec{B}\parallel \left[110\right]$. By increasing the temperature the maximum is suppressed and gradually shifted toward lower fields. A very similar behavior is observed for $\vec{B}\parallel \left[010\right]$, yet the determination of $B_c^{\mathrm{AF}1}$ is significantly more difficult due to the proximity to $B_c^{\mathrm{AF}2}$ (see Appendix pp1).

Figure 4

Field dependence of specific heat $C$ and entropy increment $\mathrm{\Delta }S$ for in-plane fields parallel to (a) [110] and (b) [010] at 1 and 2 K, scaled by $T^2$ for better comparison. No signature for any phase transition beyond $B_c^{\mathrm{AF}2}$ is visible up to 14 T. The most prominent part arises from the peak at the phase transition $B_c^{\mathrm{AF}2}$ for both field directions. While $B_c^{\mathrm{AF}1}$ can also be identified in (a) $C\left(B\right)$ and $\mathrm{\Delta }S\left(B\right)$ by another peak, respectively, it is not pronounced in the measurements for (b), most likely due to the closeness to $B_c^{\mathrm{AF}2}$.

Figure 5

Closeup of high-field part of ${\mathrm{\Gamma }}_B$ at 1 K. Any purported phase transition is absent above $B_c^{\mathrm{AF}2}$ up to 14 T, and ${\mathrm{\Gamma }}_B$ goes to zero due to very tiny remaining entropy (Fig. 4). The shoulder reported in Ref. [23] for $\varphi ={20}^{\circ }$ indicative for excited level crossing is also visible for $\vec{B}\parallel \left[010\right]$, roughly setting in at the same position of $\sim 9.3\mathrm{T}$. Additionally, a weaker second kink is present at $8.6\mathrm{T}$. For $\vec{B}\parallel \left[110\right]$ a similar change of slope can be identified at $\sim 8.2\mathrm{T}$ as the only anomaly above $B_c^{\mathrm{AF}2}$. Shown in the inset, the measurements at $0.5\mathrm{K}$ confirm the aforementioned results with the lacking phase transition and the more clearly visible anomalies. Note that the negative values of ${\mathrm{\Gamma }}_B$ are caused by the cell background, which could not be subtracted at $0.5\mathrm{K}$.

Figure 6

Numerical results for ${\mathrm{\Gamma }}_B$ in two extended Kitaev models and different field directions. Blue vertical lines mark sign changes in ${\mathrm{\Gamma }}_B$; orange lines mark shoulder anomalies. (a),(b) Results for the model of Refs. [43, 44] (“model A”). (c),(d) Results for the model of Ref. [38] (“model B”), whose small second-neighbor $J_2$ and Dzyaloshinskii-Moriya interactions were neglected.

Figure 7

Field dependence of the magnetic Grüneisen parameter ${\mathrm{\Gamma }}_B$ over the whole measured range from 0 to 14 T for the different field orientations (a) [110] and (b) [010], respectively. The main signatures are the transitions at $B_c^{\mathrm{AF}2}$ and $B_c^{\mathrm{AF}1}$. The sign change at $\sim 2\mathrm{T}$ from Ref. [23] is reproduced and likely results from domain reconstruction [11].

Figure 8

Detailed ${\mathrm{\Gamma }}_B$ measurements used for the determination of $B_c^{\mathrm{AF}1}$ and $B_c^{\mathrm{AF}2}$ for (a), (b) $\vec{B}\parallel \left[110\right]$ and (c), (d) $\vec{B}\parallel \left[010\right]$. Raw data without cell background subtraction are shown, because corresponding specific heat data were not always available. This does not affect the positions of $B_c^{\mathrm{AF}1}$ and $B_c^{\mathrm{AF}2}$, as explained in the text. The transition at $B_c^{\mathrm{AF}2}$ is easily identified as the sign change from negative to positive for all temperatures and field directions. (a), (b) [110] direction. The maximum indicative of $B_c^{\mathrm{AF}1}$ is clearly visible at temperatures below 2.5 K (black and orange arrow) and broadens toward higher temperatures transforming into a kink at 4 K (purple arrow), and finally vanishes. (c), (d) [010] direction. Here, the determination of $B_c^{\mathrm{AF}1}$ is hindered by the proximity to the dominant transition at $B_c^{\mathrm{AF}2}$. The 2 K data show a maximum (red arrow) that develops into a broad shoulder at 1 K (black arrow), and becomes fully screened by the increasingly sharp feature at $B_c^{\mathrm{AF}2}$ toward lower temperatures. Above 2.5 K, the maximum becomes a kink (orange and purple arrow) and vanishes, similarly to the [110] orientation.

Figure 9

Field dependence of ${\mathrm{\Gamma }}_B$ beyond $B_c^{\mathrm{AF}2}$ at the lowest measured temperatures. (a) More field directions at 1 K. (b) At 500 mK, no background subtraction is available due to the lack of $C\left(B\right)$. Nevertheless, both $B_c^{\mathrm{AF}1}$ and $B_c^{\mathrm{AF}2}$ are very well resolved for all field directions. As already shown previously, the subtraction of the background does not change the position of any anomaly in ${\mathrm{\Gamma }}_B\left(B\right)$. (c) Zoom-in into ${\mathrm{\Gamma }}_B$ above $B_c^{\mathrm{AF}2}$. Shoulderlike anomalies are visible being most pronounced for $\vec{B}\parallel \left[010\right]$.

Figure 10

(a) Field dependence of the total heat capacity $C_{\mathrm{tot}}$ (including background) up to 14 T for several field orientations. They perfectly match the 1 K data from the main text, undoubtedly resolving $B_c^{\mathrm{AF}2}$ and, in the case $\vec{B}\parallel \left[110\right]$, also $B_c^{\mathrm{AF}1}$. Far above $B_c^{\mathrm{AF}2}$, fitting with a single exponential function is not adequate anymore resulting in the unphysical increase towards higher fields due to shortcomings of the single exponential function analysis. Nevertheless, we speculate that a phase transition beyond $B_c^{\mathrm{AF}2}$ would give rise to a deviation from the monotonic behavior, e.g., in form of another peak. As a consequence, from these data, we suspect that a phase transition seems to be unlikely. (b) Heat capacity for $\vec{B}\parallel \left[110\right]$ at 200 mK. $B_c^{\mathrm{AF}1}$ and $B_c^{\mathrm{AF}2}$ are perfectly resolved while the monotonic behavior beyond $B_c^{\mathrm{AF}2}$ does not indicate any further phase transition. (c-f) Examples for the sample temperature response $T_{}\left(t\right)$ at 500 mK and different magnetic fields, all applied along the [110] direction. A single exponential function describes $T_{}\left(t\right)$ very well for lower fields of e.g. (a) 3.2 T and (b) 7 T, respectively. At higher fields well beyond the phase transition $B_c^{\mathrm{AF}2}$, the single exponential fit obviously deviates from the data points at (c) 10 T and (d) 12.6 T, respectively. This is even more obvious in the logarithmic plot in the insets where two time constants are present, manifesting themselves in different slopes.