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

Abstract

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.