  In the path integral formulation of the partition function of quantum spin
models, most current treatments employ the so-called static approximation to
simplify the process of summing over all possible paths. Although sufficient
for studying the thermodynamic aspects of the system, static approximation
ignores the contributions made by time-dependent, or non-static, fluctuations
in the paths of the path integral. This non-static component is very small
relative to the static part, and its careful treatment is necessary for the
calculation of small non-extensive quantities such as the energy gap within the
path integral framework. We propose a formalism for incorporating non-static
effects into the path integral calculation of a class of spin models whose
partition functions are reducible to the trace of a single spin (often known as
the effective Hamiltonian). We first show that the time-dependent behavior of
the single spin trace is governed by the Pauli equation. Time-dependent
perturbation theory is used to obtain a perturbative expansion of the solution
of the Pauli equation, and then for the single spin trace. This gives us a
perturbative expansion of the path integral which can be integrated
systematically using standard techniques. In this paper, we develop the
theoretical framework outlined above in detail and apply it to a simple ordered
spin model, the infinite-range ferromagnetic Ising model in a transverse field.
We calculated two non-extensive quantites with this non-static approach: the
$N^0$ and $N^{-1}$ terms of the ground-state energy ($N$=number of spins) and
the energy gap between the ground and first-excited states. We checked our
results by comparing with those of Holstein-Primakoff transform and numerical
diagonalization of the Hamiltonian. The application of the method to quantum
spin-glasses is briefly discussed.
