Quantum spherical model with competing interactions

We analyse the phase diagram of a quantum mean spherical model in terms of the temperature T , a quantum parameter g , and the ratio p = − J 2 / J 1 , where J 1 > 0 refers to ferromagnetic interactions between first-neighbour sites along the d directions of a hypercubic lattice, and J 2 < 0 is associated with competing antiferromagnetic interactions between second neighbours along m ≤ d directions. We regain a number of known results for the classical version of this model, including the topology of the critical line in the g = 0 space, with a Lifshitz point at p = 1 / 4 , for d > 2 , and closed-form expressions for the decay of the pair correlations in one dimension. In the T = 0 phase diagram, there is a critical border, g c = g c ( p ) for d ≥ 2 , with a singularity at the Lifshitz point if d < ( m + 4 ) / 2 . We also establish upper and lower critical dimensions, and analyse the quantum critical behavior in the neighborhood of p = 1 / 4 .