Persistent Current for a genus g=2 structure

We report the first calculation of persistent current in two coupled rings which form a character ``8'' genus g=2 structure. We obtain an exact solution for the persistent current and investigated the exact solution numerically. For two large coupled rings with equal fluxes, we find that the persistent current in the two coupled rings is equal to that in a single ring. For opposite fluxes the energy has a chaotic structure. For both cases the periodicity is $h/e$. This results are obtained within an extension of Dirac's second class method to fermionic constraints. This theory can be tested in the ballistic regime.