For a large class of networks made of connected loops, in the presence of an external magnetic field of half flux quantum per loop, we show the existence of a large local symmetry group, generated by simultaneous flips of the electronic current in al
l the loops adjacent to a given node. Using an ultra-localized single particle basis adapted to this local Z_2 symmetry, we show that it is preserved by a large class of interaction potentials. As a main physical consequence, the only allowed tunneling processes in such networks are induced by electron-electron interactions and involve a simultaneous hop of two electrons. Using a mean-field picture and then a more systematic renormalization-group treatment, we show that these pair hopping processes do not generate a superconducting instability, but they destroy the Luttinger liquid behavior in the links, giving rise at low energy to a strongly correlated spin-density-wave state.
We consider arrays of Luttinger liquids, where each node is described by a unitary scattering matrix. In the limit of small electron-electron interaction, we study the evolution of these scattering matrices as the high-energy single particle states a
re gradually integrated out. Interestingly, we obtain the same renormalization group equations as those derived by Lal, Rao, and Sen, for a system composed of a single node coupled to several semi-infinite 1D wires. The main difference between the single node geometry and a regular lattice is that in the latter case, the single particle spectrum is organized into periodic energy bands, so that the renormalization procedure has to stop when the last totally occupied band has been eliminated. We therefore predict a strongly renormalized Luttinger liquid behavior for generic filling factors, which should exhibit power-law suppression of the conductivity at low temperatures E_{F}/(k_{F}a) << k_{B}T << E_{F}, where a is the lattice spacing and k_{F}a >> 1. Some fully insulating ground-states are expected only for a discrete set of integer filling factors for the electronic system. A detailed discussion of the scattering matrix flow and its implication for the low energy band structure is given on the example of a square lattice.
