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The quantum-rotors model can be regarded as an effective model for the low-temperature behavior of the quantum Heisenberg antiferromagnets. Here, we consider a d-dimensional model in the spherical approximation confined to a general geometry of the form L dϪdЈ ϫϱ dЈ ϫL z (L-linear space size and L -temporal size͒ and subjected to periodic boundary conditions. Due to the remarkable opportunity it offers for rigorous study of finite-size effects at arbitrary dimensionality this model may play the same role in quantum critical phenomena as the popular Berlin-Kac spherical model in classical critical phenomena. Close to the zerotemperature quantum critical point, the ideas of finite-size scaling are utilized to the fullest extent for studying the critical behavior of the model. For different dimensions 1ϽdϽ3 and 0рdЈрd a detailed analysis, in terms of the special functions of classical mathematics, for the susceptibility and the equation of state is given. Particular attention is paid to the two-dimensional case.

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A detailed investigation of the scaling properties of the fully finite O(n) systems, under periodic boundary conditions, with long-range interaction, decaying algebraically with the interparticle distance r like r(-d-sigma), below their upper critical dimension, is presented. The computation of the scaling functions is done to one loop order in the nonzero modes. The results are obtained in an expansion of powers of sqrt[epsilon], where epsilon=2sigma-d up to O(epsilon(3/2)). The thermodynamic functions are found to depend upon the scaling variable z=RU(-1/2)L(2-eta-epsilon/2), where R and U are the coupling constants of the constructed effective theory, and L is the linear size of the system. Some simple universal results are obtained.