Field-induced quantum critical point in planar Heisenberg ferromagnets with long-range interactions: Two-time Green's function framework

The two-time Green's function method is used to study the critical properties and crossover phenomena near the field-induced quantum critical point (QCP) of a $d$-dimensional spin-$S$ planar Heisenberg ferromagnet with long-range interactions decaying as ${r}^{\ensuremath{-}\ensuremath{\alpha}}$ (with $\ensuremath{\alpha}gd$) with the distance $r$ between spins. We adopt the Callen scheme for spin $S$ and the Tyablikov decoupling procedure which is expected to provide suitable results at low temperatures. Different quantum critical regimes are found in the $(\ensuremath{\alpha},d)$ plane and the global structure of the phase diagram is determined showing the typical V-shaped region close to the QCP. Depending on the values of $\ensuremath{\alpha}$, we find that also for dimensionalities $d\ensuremath{\leqslant}2$ a finite-temperature critical line, ending in the QCP, exists with asymptotic behaviors and crossovers which can be employed as a useful guide for experimental studies. Moreover, these crossovers are shown to be suitably described in terms of $(\ensuremath{\alpha},d)$-dependent scaling functions and effective critical exponents.