Josephson effect for SU(4) carbon-nanotube quantum dots

We present the theory of the Josephson effect in nanotube dots where an SU(4) symmetry can be realized. We find a remarkably rich phase diagram that significantly differs from the SU(2) case. In particular, $\ensuremath{\pi}$-junction behavior is largely suppressed. We analytically obtain the Josephson current in various parameter regions: (i) in the Kondo regime covering the full crossover from SU(4) to SU(2), (ii) for weak tunnel couplings, and (iii) for a large BCS gap. The transition between these regions is studied numerically.