FINITE-SIZE SCALING FOR THE CORRELATION-FUNCTION OF THE SPHERICAL MODEL WITH LONG-RANGE INTERACTIONS

Finite-size scaling for the effective correlation length and the pair correlation function of the mean spherical model is studied in the case of general geometry L(d-d') x infinity d', periodic boundary conditions, and long-range interactions decaying like R(-d-sigma) at large distances R, with arbitrary real parameters 0 < sigma less-than-or-equal-to 2, sigma less-than-or-equal-to d less-than-or-equal-to 2-sigma, d' greater-than-or-equal-to 0. The analytical technique used is based on integral transformations with kernels of Mittag-Leffler type. It makes it possible to easily generalize a number of results available only for short-range interactions or in special cases of the L(d-d') x infinity d' geometry. The effective correlation length is identified at arbitrary temperatures from the finite-size large-distance asymptotic behavior of the pair correlation function. It is explicitly shown that the finite-size scaling functions are not singular in the epsilon-expansion when epsilon --> 0+ neither at d = sigma + epsilon, nor at d = 2-sigma - epsilon. Moreover, the applicability of the epsilon-expansion is tested in the specific finite-size case of d' = sigma +/- epsilon. The definition of the scaled field variable and the finite-size scaling in the regime of the first-order phase transition are also considered.