High-resolution Measurements of Thermal Conductivity Matrix and Search for Thermal Hall Effect in La$_2$CuO$_4$

We investigated the longitudinal thermal conductivity $κ_{xx}$ and thermal hall conductivity $κ_{xy}$ in La$_2$CuO$_4$ at temperatures $T$ between 2 and 20 K in magnetic fields $H$ up to 10 T. Within the temperature and field intervals studied, we do not resolve any thermal Hall signal with a conservative upper bound of $|κ_{xy}/T| <1\times10^{-4}$ ${\rm Wm^{-1}K^{-2}}$. The longitudinal thermal conductivity $κ_{xx}/T$ agrees well with previous studies, in both magnitude and $T$ dependence. In both channels, we performed measurements using the field-sweep protocol. To achieve high resolution, we carefully took into account relaxation effects after each step-increase in $H$. At low $T$, we find a linear decrease in $κ/T$ vs. $H$, as well as weak hysteresis near the meta-magnetic transition of the spin degrees.