Phase-coherent thermal transport in Josephson junctions based on altermagnets

We study the effects of $d$-wave altermagnetism on the thermal transport properties in Josephson junctions based on altermagnets. Owing to the anisotropic spin splitting in the altermagnet, the phase coherence of the thermal transport can be effectively modulated by the crystal orientation of the altermagnet. For the junction with pure ${d}_{xy}$-wave altermagnetism, despite the spin splitting of altermagnet, the spin-up and -down bands contribute equally to the thermal transport, and the total thermal conductance is insensitive to the altermagnetic exchange field strength. On the contrary, in the presence of ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave altermagnetism, the spin-up and -down bands manifest themselves distinctly in the thermal transport, and the total thermal conductance possesses pronounced oscillating profiles by varying the altermagnetic exchange field strength. These findings suggest that the altermagnet-based Josephson junction can serve a promising platform for realizing controllable phase-coherent thermal transport.