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We investigate theoretically the Josephson junction of semiconductor nanowirewith strong spin-orbit (SO) interaction in the presence of magnetic field. Byusing a tight-binding model, the energy levels $E_n$ of Andreev bound statesare numerically calculated as a function of phase difference $\varphi$ betweentwo superconductors in the case of short junctions. The DC Josephson current isevaluated from the Andreev levels. In the absence of SO interaction, a$0$-$\pi$ transition due to the magnetic field is clearly observed. In thepresence of SO interaction, the coexistence of SO interaction and Zeeman effectresults in $E_n (-\varphi) \ne E_n (\varphi)$, where the anomalous Josephsoncurrent flows even at $\varphi =0$. In addition, the direction-dependence ofcritical current is observed, in accordance with experimental results.

Anomalous Josephson effect induced by spin-orbit interaction and Zeeman effect in semiconductor nanowires