Josephson effect and odd-frequency pairing in superconducting junctions with unconventional magnets

We consider Josephson junctions formed by coupling two conventional superconductors via an unconventional magnet and investigate the formation of Andreev bound states, their impact on the Josephson effect, and the emergent superconducting correlations. In particular, we focus on unconventional magnets known as $d$-wave altermagnets and $p$-wave magnets. We find that the Andreev bound states in $d$-wave altermagnet and ${p}_{y}$-wave magnet Josephson junctions strongly depend on the transverse momentum, with a spin splitting and low-energy minima as a function of the superconducting phase difference $\ensuremath{\varphi}$. In contrast, the Andreev bound states for ${p}_{x}$-wave magnets are insensitive to the transverse momentum. We then show that the Andreev bound states can be probed by the local density of states in the middle of the junction, which also reveals that ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$- and $p$-wave magnet junctions are prone to host zero energy peaks. While the zero-energy peak in ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave altermagnet junctions tends to oscillate with the magnetic order, it remains robust in $p$-wave magnet junctions. We then discover that the Josephson current in $d$-wave altermagnet junctions is composed of higher harmonics of $\ensuremath{\varphi}$, which originate a $\ensuremath{\phi}$-Josephson junction behavior entirely controlled by the magnetic order in ${d}_{xy}$-wave altermagnets. In contrast, the Josephson current in Josephson junctions with $p$-wave magnets exhibits a conventional sinelike profile with a fast sign change at $\ensuremath{\varphi}=\ensuremath{\pi}$ due to zero-energy Andreev bound states. We also demonstrate that the critical currents in $d$-wave altermagnet Josephson junctions exhibit an oscillatory decay with the increase of the magnetic order, while the oscillations are absent in $p$-wave magnet junctions albeit the currents exhibit a slow decay. Furthermore, we also demonstrate that the interplay of the Josephson effect and unconventional magnetic order of $d$-wave altermagnets and $p$-wave magnets originates from odd-frequency spin-triplet $s$-wave superconducting correlations that are otherwise absent. Our results can serve as a guide to pursue the new functionality of Josephson junctions based on unconventional magnets.