Improved Taylor Expansion method in the Ising model

We apply an improved Taylor expansion method, which is a variational scheme to the Ising model in two dimensions. This method enables us to evaluate the free energy and magnetization in strong coupling regions from the weak coupling expansion, even in the case of a phase transition. We determine the approximate value of the transition point using this scheme. In the presence of an external magnetic field, we find both stable and metastable physical states. Introduction. More often than not, we encounter situations in which it is difficult to evaluate physical quantities by means of standard perturbation methods because, for example, the theory under consideration does not have any small parameters in which we can expand. As non-perturbative methods, variational schemes have been applied to such circumstances with great success. In such methods one or more auxiliary parameters are introduced into the model. Optimized perturbation theory [1, 2] is a systematic improvement of variational methods, formulated on the basis of the " principle of minimal sensitivity ". [2] In zero and one dimensions, it has been proved that the optimized series converges. [3] This method has been applied to, among others, matrix models of superstring theory and their simplified toy models. [4–11] It was first reported in Ref. [10] that minimal sensitivity is realized in auxiliary parameter space as a plateau (i.e. a region in which physical quantities are stable), and that the appearance of a plateau can be regarded as a signal reflecting whether or not the method works. It has also been argued that the optimized perturbation theory can be explicitly formulated as an improved Taylor series, which is obtained in standard perturbation theory. In this article, we attempt to improve our understanding of this method through application to the Ising model in two dimensions. 1 It is generally believed that physics in an ordered phase cannot be analyzed using a nä ive perturbation theory formulated in a disordered phase, because, in general, a perturbation theory yields an ill-defined expansion series when it is applied to a different phase than that in which it was constructed. We utilize a mathematical technique to reorganize the series resulting from a perturbation theory in a non-trivial manner which is known to yield a good approximation even in the case that the expansion is carried out about a point outside the radius of convergence. [10] We show that our scheme actually enables …