Finite size scaling investigations in the quantum $\phi^4$-model with long-range interaction

In this paper, we study in details the critical behavior of the ${\cal O}(n)$ quantum $\phi^4$ model with long-range interaction decaying with the distances r by a power law as $r^{-d-\sigma}$ in the large n-limit. The zero-temperature critical behavior is discussed. Its alteration by the finite temperature and/or finite sizes in the space is studied. The scaling behaviours are studied in different regimes depending upon whether the finite temperature or the finite sizes of the system is leading. A number of results for the correlation length, critical amplitudes and the finite size shift, for different dimensionalities between the lower $d_<=\sigma/2$ and the upper $d_>=3\sigma/2$ critical dimensions, are calculated.