Phonon heat transport and anisotropic tuning of quantum fluctuations in a frustrated honeycomb magnet

Honeycomb cobalt oxides containing $3d \mathrm{C}{\mathrm{o}}^{2+}$ ions might realize frustrated magnetism and novel quantum phases. Among candidate materials, $\mathrm{N}{\mathrm{a}}_{3}\mathrm{C}{\mathrm{o}}_{2}\mathrm{Sb}{\mathrm{O}}_{6}$ stands out for its distorted honeycomb lattice and significant in-plane anisotropy, motivating vector-field tuning inside the honeycomb plane. Here we use thermal transport down to the millikelvin regime to study twin-free crystals of $\mathrm{N}{\mathrm{a}}_{3}\mathrm{C}{\mathrm{o}}_{2}\mathrm{Sb}{\mathrm{O}}_{6}$ subject to in-plane vector fields. We find that the thermal conductivity $\ensuremath{\kappa}$ never exceeds the heat-transport capability of phonons, rendering its suppression primarily due to phonon scattering off magnetic excitations and/or domain boundaries. The system's field-driven quantum criticality manifests itself as an abundance of magnetic fluctuations hindering the heat transport, which further depends on the field direction in an intriguing manner. Our work establishes a powerful strategy of tuning magnetic quantum criticality with rotating fields and monitoring it with phonon heat transport.