Balancing long-range interactions and quantum pressure: solitons in the HMF model

The Hamiltonian Mean Field (HMF) model describes particles on a ring interacting via a cosine interaction, or equivalently, rotors coupled by infinite-range XY interactions. Conceived as a generic statistical mechanical model for long-range interactions such as gravity (of which the cosine is the first Fourier component), it has recently been used to account for self-organization in experiments on cold atoms with long-range optically mediated interactions. The significance of the HMF model lies in its ability to capture the universal effects of long-range interactions and yet be exactly solvable in the canonical ensemble. In this work we consider the quantum version of the HMF model in 1D and provide a classification of all possible stationary solutions of its generalized Gross-Pitaevskii equation (GGPE) which is both nonlinear and nonlocal. The exact solutions are Mathieu functions that obey a nonlinear relation between the wavefunction and the depth of the meanfield potential, and we identify them as bright solitons. Using a Galilean transformation these solutions can be boosted to finite velocity and are increasingly localized as the meanfield potential becomes deeper. In contrast to the usual local GPE, the HMF case features a tower of solitons, each with a different number of nodes. Our results suggest that long-range interactions support solitary waves in a novel manner relative to the short-range case.