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The direct-current (DC) Josephson, or superconducting diode effect, i.e., the nonreciprocal behavior of critical current in a superconductor, has been observed in various systems exhibiting both time-reversal and parity symmetry breakings. However, we show that breaking these two types of symmetries is only a necessary but not sufficient condition for the diode effect, for which certain additional symmetries, including particle-hole symmetry and many others, must also be broken. The dependencies of the free energy on the gauge-independent phase difference across the junction and the magnetic field are classified, exhibiting the current-reversion (JR), field-reversion, and field-current reversion conditions, respectively. All symmetries satisfying the JR condition need to be broken for the DC Josephson diode effect. The relations of critical currents with respect to the magnetic field are classified into five classes, and three of them exhibit the diode effect. These symmetry considerations are applied to concrete examples. Our work reveals that the DC Josephson diode effect is a natural consequence of the JR symmetry breaking, and hence provides a guiding principle to understand or design a DC Josephson diode.
Wang et al. (2025) studied this question.