Spin-polarized Andreev molecules and anomalous nonlocal Josephson effects in altermagnetic junctions

Altermagnetism has emerged as a promising ingredient for realizing nontrivial Josephson phases, but so far explored in single Josephson junctions. In this work, we consider the coherent coupling of two Josephson junctions with spin-singlet $s$-wave superconductivity and demonstrate that $d$-wave altermagnetism gives rise to spin-polarized Andreev molecules due to the hybridization of Andreev bound states of each junction when the coupling is weak. Interestingly, these spin-polarized Andreev molecules induce an anomalous nonlocal Josephson effect, where the current flow across one Josephson junction due to phase changes across the other junction develops $0-\pi$ and ${\phi }_0$ transitions originating from altermagnetism. Furthermore, the nonlocal Josephson current carried by spin-polarized Andreev molecules exhibits nonreciprocal critical currents, enabling a nonlocal Josephson diode effect whose polarity is tunable by the altermagnetic strength and right phase. Our findings put forward altermagnetism as a promising arena for designing nonlocal spin Josephson phenomena.