Ordering due to disorder in dipolar magnets on two-dimensional lattices

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Abstract
We calculate the effects of thermal fluctuations, dilution, and a uniform external field on the continuously degenerate ground state of dipolar magnets on the square and honeycomb lattices. In all cases, these perturbations make terms in the free energy that favor particular states, reducing the continuous degeneracy to a discrete symmetry. Dilution and thermal selection compete; i.e., they select different states. For the cases of thermal fluctuations and dilution, the discrete symmetry is fourfold on the square lattice and sixfold on the honeycomb lattice; for a uniform field it is twofold and threefold, respectively. In addition, dilution generates effective random axes, which disorder all of the selected phases in each system, except for the dilution-selected phase of the square lattice. We present a proposed phase diagram including speculations on the finite-temperature transition.