We study the interplay between magnetic frustration and itinerant electrons. For example, how does the coupling to mobile charges modify the properties of a spin liquid, and does the underlying frustration favor insulating or conducting states? Supported by Monte Carlo simulations, our goal is in particular to provide an analytical picture of the mechanisms involved. The models under considerations exhibit Coulomb phases in two and three dimensions, where the itinerant electrons are coupled to the localized spins via double exchange interactions. Because of the Hund coupling, magnetic loops naturally emerge from the Coulomb phase and serve as conducting channels for the mobile electrons, leading to doping-dependent rearrangements of the loop ensemble in order to minimize the electronic kinetic energy. At low electron density rho, the double exchange coupling mainly tends to segment the very long loops winding around the system into smaller ones while it gradually lifts the extensive degeneracy of the Coulomb phase with increasing rho. For higher doping, the results are strongly lattice dependent, displaying loop crystals with a given loop length for some specific values of rho, which can melt into another loop crystal by varying rho. Finally, we contrast this to the qualitatively different behavior of analogous models on kagome or triangular lattices.
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