Improved Taylor Expansion method in Ising model

We apply improved Taylor expansion method which is one of the variational schemes to Ising model in two-dimensions. It enables us to evaluate free energy and magnetization at strong coupling regions from weak coupling expansion even in the presence of the phase transition. We determine approximated transition point in this scheme. In the presence of external magnetic field we can see not only stable physical states but metastable one. Introduction. More often than not do we encounter a situation in which it is difficult to evaluate physical quantities by means of the standard perturbation methods because, for example, the theory under consideration does not have any small parameters to be expanded by. As non-perturbative methods, variational schemes have been applied to such circumstances with great success, in which one or more auxiliary parameters are introduced in the model. Optimized perturbation theory [1] is a systematic improvement of variational methods, formulated on the basis of the " principle of minimal sensitivity ". In zero and one dimensions it is proved that the optimized series converges [2]. This method has been applied to, among others, matrix models of superstring theory and their simplified toy models [3, 4, 5, 6, 7, 8, 9, 10]. It was first recognized in [9] that the minimal sensitivity is realized in auxiliary parameter space as plateau, a region on which physical quantities stay stable, and that the emergence of it might be regarded as a signal to judge whether the method works or not. It was also argued that the optimized perturbation theory can be concretely embodied in the form of an improvement of Taylor series which is obtained by the standard perturbation theory. In this article we would like to make our understandings of the method more keen through an application to Ising model in two-dimensions 1. Here we do not aim to calculate up to as high orders as possible in order to pursue more accurate estimates in a given scheme of approximation, but rather to show that the physical information can be extracted from relatively low order of perturbation by the method. In the actual problems such as QED, QCD, and so forth, no more than first three or four terms are available in perturbation theory, from which we are required to extract physical information 2. In the following we first briefly recall that the optimized perturbation method can be understood as improved Taylor …