Quantum mechanics for genus g=2-persistent current in coupled rings

We solve the quantum mechanical problem for electrons confined to two coupled metallic rings which form an '8' structure. The solution is achieved by gluing the wavefunction at the interface of the two rings using a modification of Dirac's constrained method. The Heisenberg equation of motion for the wavefunction is a fermionic and bosonic mixture, suggesting that it is impossible to solve the problem for two coupled rings by standard boundary conditions. As an explicit demonstration of our method we present an exact solution for the persistent current on coupled rings in the presence of two external magnetic fluxes. For 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. In order to compare our theory to the experimental situations we consider two rings with a finite width in the ballistic regime in the presence of 2K(F) impurity scattering.