  We examine the Ising chain in a transverse field at zero temperature from the
point of view of a family of moment formalisms based upon the cumulant
generating function, where we find exact solutions for the generating functions
and cumulants at arbitrary couplings and hence for both the ordered and
disordered phases of the model. In a t-expansion analysis, the exact
Horn-Weinstein function E(t) has cuts along an infinite set of curves in the
complex Jt-plane which are confined to the left-hand half-plane Im Jt < -1/4
for the phase containing the trial state (disordered), but are not so for the
other phase (ordered). For finite couplings the expansion has a finite radius
of convergence. Asymptotic forms for this function exhibit a crossover at the
critical point, giving the excited state gap in the ground state sector for the
disordered phase, and the first excited state gap in the ordered phase.
Convergence of the t-expansion with respect to truncation order is found in the
disordered phase right up to the critical point, for both the ground state
energy and the excited state gap. However convergence is found in only one of
the Connected Moments Expansions (CMX), the CMX-LT, and the ground state energy
shows convergence right to the critical point again, although to a limited
accuracy.
