First-order transitions in anisotropic magnets with random fields and random uniaxial anisotropies

Long-range magnetic order is known to be replaced by a spin-glass phase for rotationally invariant N-component spin systems with random fields or random uniaxial anisotropies. Using exact calculations in the limit N, we show how long-range order is restored when a uniform anisotropy is added. For a uniaxial anisotropy g, long-range order is achieved above a critical value of g, at which a phase with an infinite susceptibility occurs. For higher-order (e.g., cubic or hexagonal) anisotropies long-range order is reached via first-order transitions. Scaling arguments and explicit calculations are used to obtain detailed predictions on the shape of the various phase boundaries.