Critical specific heats of the N-vector spin models on the simple cubic and bcc lattices

We have computed through order β²¹ the high-temperature expansions for the nearest neighbor spin correlation function G(N,β) of the classical N-vector model, with general N, on the simple cubic and on the body centered cubic lattices. For this model, also known in quantum field theory as the lattice O(N) nonlinear σ model, we have presented in previous papers extended expansions of the susceptibility, of its second field derivative, and of the second moment of the correlation function. Here we study the internal specific energy and the specific heat C(N,β), obtaining updated estimates of the critical parameters and therefore a more accurate direct test of the hyperscaling relation dν(N)=2-α(N) on a range of values of the spin dimensionality N, including N=0 (the self-avoiding walk model), N=1 (the Ising spin 1/2 model), N=2 (the XY model), N=3 (the classical Heisenberg model). By the newly extended series we also compute the universal combination of critical amplitudes usually denoted by R_ξ⁺(N), in fair agreement with renormalization group estimates.