Spin Thermal Hall Conductivity of a Kagome Antiferromagnet

A clear thermal Hall signal (${\kappa }_{xy}$) was observed in the spin-liquid phase of the $S=1/2$ kagome antiferromagnet Ca kapellasite $[{\mathrm{CaCu}}_3(\mathrm{OH}{)}_6{\mathrm{Cl}}_2·0.6{\mathrm{H}}_2\mathrm{O}]$. We found that ${\kappa }_{xy}$ is well reproduced, both qualitatively and quantitatively, using the Schwinger-boson mean-field theory with the Dzyaloshinskii-Moriya interaction of $D/J\sim 0.1$. In particular, ${\kappa }_{xy}$ values of Ca kapellasite and those of another kagome antiferromagnet, volborthite, converge to one single curve in simulations modeled using Schwinger bosons, indicating a common temperature dependence of ${\kappa }_{xy}$ for the spins of a kagome antiferromagnet.

Figure 1

(a),(b) Crystal structure of Ca kapellasite viewed along the $c$ axis (a) and the $a$ axis (b). The direction of the Dzyaloshinskii-Moriya interaction ($D_{ij}$) considered in Eq. (1) is shown by the $\odot$ symbols. The black arrows next to the $\odot$ symbols indicate the direction in the definition of $D_{ij}$ (see the SM [30]). $J_1$, $J_2$, and $J_d$ (the solid red lines) represent the nearest-neighbor, next-nearest-neighbor, and diagonal magnetic interactions, respectively. ${\mathrm{Ca}}^{2+}$ ions are randomly located in either of two Wyckoff positions that are slightly different along the $c$ axis [the solid and dotted circles in (b) [23] ]. (c) An illustration of our experimental setup. Three thermometers ($T_{\mathrm{High}}$, $T_{L1}$, $T_{L2}$) and a heater were attached to the sample. A heat current $Q\parallel x$ was applied within the $ab$ plane and a magnetic field was applied along the $c\parallel z$ axis. See the SM [30] for details. (d),(e) Temperature dependence of ${\kappa }_{xx}$ for samples No. 1 (the blue diamonds) and No. 2 (the green triangles) at high temperatures at 0 and 15 T (d) and that of ${\kappa }_{xx}/T$ for samples No. 1 and No. 3 (the pink circles) below 0.5 K at 0 and 14 T (e). The slope of ${\kappa }_{xx}(T)$ is decreased below $T^{*}=7.4\mathrm{K}$. The zero-temperature extrapolation of ${\kappa }_{xx}/T$ (the solid and dashed lines for 0 and 14 T, respectively) shows zero or negligibly small residual.

Figure 2

The field dependence (a) of the transverse temperature difference $\mathrm{\Delta }{T}_y(B)$ and (b) of ${\kappa }_{xy}(B)$. Solid lines in (b) represent linear fits. The field dependence of ${\kappa }_{xy}(B)$ at other temperatures is shown in the SM [30].

Figure 3

The temperature dependence of ${\kappa }_{xy}/TB$ of Ca kapellasite (samples No. 1 and No. 2) and that of volborthite [41]. The filled (open) symbols represent data estimated by the linear fit of ${\kappa }_{xy}$ (data at 15 T). The data of volborthite are taken from Ref. [41]. The error bars correspond to one standard deviation and are of the same order as the size of the symbol or smaller for data of Ca kapellasite.

Figure 4

Dimensionless thermal Hall conductivity ${\tilde{f}}_{\exp }(k_{B}T/J)$ and ${\tilde{f}}_{\mathrm{SBMF}}(k_{B}T/J)$. The solid line represents numerical results obtained using the Schwinger-boson mean-field theory of $D/J=0.1$ (see the SM [30] for more results). The values of parameters ($J/k_B$, $D/J$) used for the experimental data are (66, 0.12), (66, 0.06), and (60, $-0.07$) for Ca kapellasite No. 1, No. 2, and volborthite, respectively.