Superconductivity with Pairs in a Relative $p$ Wave

In previous treatments of superconducting systems with attractive interactions acting in odd-angular-momentum partial waves, the correlated electron pairs were formed in only two components of a spin triplet. This oversight is corrected here by a more general variational treatment, allowing all three components. In the case of a $p$-wave interaction, the present state is proved to give the absolute minimum of the free energy. Its rotational degeneracy is discussed. The energy spectrum is found to be isotropic (provided the normal phase is also) with the usual gap, and so to be completely equivalent thermodynamically to the BCS state. The charge-density autocorrelation is also isotropic, and the charge-current correlation vanishes. The state exhibits the conventional Meissner effect, and cannot be experimentally distinguished from the BCS state by means of electromagnetic or tunneling measurements, acoustic attenuation, or nuclear magnetic resonance (NMR) relaxation times. The paramagnetic spin susceptibility decreases with temperature from its value in the normal phase to a limiting ratio of $\frac{2}{3}$, in good agreement with results deduced from Knight shift measurements on mercury and tin, and in contrast to the BCS prediction. However, the addition of impurities is found to reduce the critical temperature sharply (again in contrast to the BCS case). Thus, the experimental observation of the $p$-wave pair state is expected to be difficult, and the agreement with Knight shift data is probably fortuitous. Finally, it is suggested that a similar effect in ${\mathrm{He}}^3$ might explain why the predicted superfluid phase has not been observed.