Criticality and phase diagram of quantum long-range O(N) models

Several recent experiments in atomic, molecular, and optical systems motivated a huge interest in the study of quantum long-range systems. Our goal in this paper is to present a general description of their critical behavior and phases, devising a treatment valid in d dimensions, with an exponent d+? for the power-law decay of the couplings in the presence of an O(N) symmetry. By introducing a convenient ansatz for the effective action, we determine the phase diagram for the N-component quantum rotor model with long-range interactions, with N=1 corresponding to the Ising model. The phase diagram in the ?-d plane shows a nontrivial dependence on ?. As a consequence of the fact that the model is quantum, the correlation functions are anisotropic in the spatial and time coordinates for ? smaller than a critical value, and in this region the isotropy is not restored even at criticality. Results for the correlation length exponent ?, the dynamical critical exponent z, and a comparison with numerical findings for them are presented.