Equilibrium spin currents in altermagnet junctions: Josephson-like and anomalous transport

Altermagnets (AMs) offer a compelling platform for exploring novel spin-dependent phenomena in materials with zero net macroscopic magnetization. In this work, we theoretically investigate the emergence of equilibrium spin currents (ESCs) in two-dimensional AM heterostructures using a tight-binding lattice model. We first study an AM-normal metal-AM (AM-NM-AM) junction and demonstrate that the $\sigma_y$-polarized ESC exhibits a characteristic Josephson-like behavior, fundamentally governed by the relative angle ($\theta$) between the N\'eel vectors of the two AMs pointing in $xz$-plane. Crucially, we show that replacing the central normal metal with a $p$-wave magnet (PM) induces an anomalous ESC. Analogous to the anomalous Josephson effect, the breaking of spatial inversion symmetry by the PM allows a finite, dissipationless spin current to flow even when the N\'eel vectors are perfectly aligned ($\theta=0$). We establish that this anomalous transport is driven by an asymmetry in the quantum phases accumulated by right- and left-moving electrons undergoing spin-flip reflections. Finally, we show that the critical ESC exhibits pronounced fluctuations as a function of band filling, which we attribute to mesoscopic quantum size effects, including transverse subband quantization and longitudinal Fabry-P\'erot resonances. Our findings highlight the potential of altermagnet junctions for designing dissipationless, phase-tunable spintronic devices.