Triviality problem and high-temperature expansions of higher susceptibilities for the Ising and scalar-field models in four-, five-, and six-dimensional lattices

Abstract

High-temperature expansions are presently the only viable approach to the numerical calculation of the higher susceptibilities for the spin and the scalar-field models on high-dimensional lattices. The critical amplitudes of these quantities enter into a sequence of universal amplitude ratios that determine the critical equation of state. We have obtained a substantial extension, through order 24, of the high-temperature expansions of the free energy (in presence of a magnetic field) for the Ising models with spin s≥1/2 and for the lattice scalar-field theory with quartic self-interaction on the simple-cubic and the body-centered-cubic lattices in four, five, and six spatial dimensions. A numerical analysis of the higher susceptibilities obtained from these expansions yields results consistent with the widely accepted ideas, based on the renormalization group and the constructive approach to Euclidean quantum field theory, concerning the no-interaction ("triviality") property of the continuum (scaling) limit of spin-s Ising and lattice scalar-field models at and above the upper critical dimensionality.

High-temperature expansions of the higher susceptibilities for the Ising model in general dimensiond

“…We shall also estimate the critical amplitudes of the five lowest-order susceptibilities and a few universal ratios of these. In our analysis, we have simply assumed that all (higher) susceptibilities show MF exponents of divergence, as also our recent work [25,26] has contributed to confirm.…”

“…[44] and were studied in detail for d = 4,5,6 in Refs. [25,26]. For the first few values of r = 1,2,3, we have checked that as expected, also for d > 6, they take the MF values: I +MF = 154 within a relative accuracy generally higher than 10 −3 , although the single amplitudes entering into the ratios reach their MF value only in the d → ∞ limit.…”

“…Our estimates of the critical inverse temperatures K c (d), obtained from the ordinary susceptibility expansions, for several hsc lattices of dimensionality d > 4. We have marked by an asterisk the estimates in the cases in which expansions extend beyond the 20th order [26]. In particular for d = 5 our series extend to the 22nd order, and for d = 6 to the 21st order.…”

“…We can only briefly outline the now standard numerical approximation techniques that we have used for these analyses, since a more detailed discussion was already given in Refs. [25,26,29]. We have mainly employed the differential approximant (DA) method, that generalizes [39] the elementary well known Padé approximant method, to resum the HT expansions up to the border of their convergence disks.…”

“…The expressions presented here are obtained from recently derived [25][26][27] HT and low-field series expansions of the magnetization in presence of an external magnetic field for the spin-1/2 Ising model with nearest-neighbor interactions. Actually, we have produced a wider ranging set of data including as well other spin systems in the Ising model universality class, such as the general spin-s Ising model and the lattice scalar-field theories with polynomial self-interaction and thus, also for these models we are able to write exact expressions valid for general d-dimensional hsc lattices.…”