Magnetic multipole analysis of kagome and artificial spin-ice dipolar arrays

We analyze an array of linearly extended monodomain dipoles forming square and kagome lattices. We find that its phase diagram contains two (distinct) finite-entropy kagome ice regimes---one disordered, one algebraic---as well as a low-temperature ordered phase. In the limit of the islands almost touching, we find a staircase of corresponding entropy plateaux, which is analytically captured by a theory based on magnetic charges. For the case of a modified square ice array, we show that the charges (``monopoles'') are excitations experiencing two distinct Coulomb interactions: a magnetic ``three-dimensional'' one as well as a logarithmic ``two-dimensional'' one of entropic origin.