We examine, principally via large $n$ methods ($n$ refers to the number of components of the classical field or the spin), correlation functions in systems that harbor competing long- and short-ranged interactions. We report, in detail, on various characteristics of the multiple correlation and modulation lengths that these systems generally exhibit. We compute the ground state stripe width of an Ising ferromagnet that is frustrated by long-range interactions. We investigate related systems with vectorial order parameters. The bulk of our results concerns the evolution of modulation lengths in such systems with temperature. We find that crossover temperatures $T^{*}$ mark the onset of modulations. For asymptotically weak frustration, the crossover temperature $T^{*}$ tends towards the critical temperature of the unfrustrated ferromagnetic system (in which the frustrating long-range interaction is absent). We illustrate that apart from certain special crossover points, the total number of correlation and modulation lengths remains conserved as temperature is varied. We illustrate that the correlation functions associated with the exact dipolar interactions differ substantially from those in which a simple scalar product form between the dipoles is assumed.

• Received 1 June 2011

DOI:https://doi.org/10.1103/PhysRevB.84.144402