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In the path integral formulation of the partition function of quantum spinmodels, most current treatments employ the so-called static approximation tosimplify the process of summing over all possible paths. Although sufficientfor studying the thermodynamic aspects of the system, static approximationignores the contributions made by time-dependent, or non-static, fluctuationsin the paths of the path integral. This non-static component is very smallrelative to the static part, and its careful treatment is necessary for thecalculation of small non-extensive quantities such as the energy gap within thepath integral framework. We propose a formalism for incorporating non-staticeffects into the path integral calculation of a class of spin models whosepartition functions are reducible to the trace of a single spin (often known asthe effective Hamiltonian). We first show that the time-dependent behavior ofthe single spin trace is governed by the Pauli equation. Time-dependentperturbation theory is used to obtain a perturbative expansion of the solutionof the Pauli equation, and then for the single spin trace. This gives us aperturbative expansion of the path integral which can be integratedsystematically using standard techniques. In this paper, we develop thetheoretical framework outlined above in detail and apply it to a simple orderedspin 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) andthe energy gap between the ground and first-excited states. We checked ourresults by comparing with those of Holstein-Primakoff transform and numericaldiagonalization of the Hamiltonian. The application of the method to quantumspin-glasses is briefly discussed.

Non-static effects in ordered and disordered quantum spin systems I:  theoretical formulation, energy gap and non-extensive terms of ground-state  energy of the ferromagnetic Ising model in a transverse field