Superconducting order parameters that change phase around the Fermi surface modify Josephson tunneling behavior, as in the phase-sensitive measurements that confirmed d order in the cuprates. This paper studies Josephson coupling when the individual grains break time-reversal symmetry; the specific cases considered are $p\pm \mathrm{ip}$ and $d\pm id,$ which may appear in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ and ${\mathrm{Na}}_x{\mathrm{CoO}}_2\cdot ({\mathrm{H}}_2\mathrm{O}{)}_y,$ respectively. T-breaking order parameters lead to frustrating phases when not all grains have the same sign of time-reversal symmetry breaking, and the effects of these frustrating phases depend sensitively on geometry for two-dimensional arrays of coupled grains. These systems can show perfect superconducting order with or without macroscopic T-breaking. The honeycomb lattice of superconducting grains has a superconducting phase with no spontaneous breaking of T but instead power-law correlations. The superconducting transition in this case is driven by binding of fractional vortices, and the zero-temperature criticality realizes a generalization of Baxter’s three-color model.

• Received 7 October 2003

DOI:https://doi.org/10.1103/PhysRevB.69.104511