Theory Of Fidelity And Its One-dimensional Model

The theory of Fidelity (also known as Loschmidt Echo) is an important concept in the field of statistical mechanics, quantum information science and quantum chaos. It has been adopted as a standard measure for quantum reversibility and stability of quantum motion with respect to changes in an external parameter. It can be considered as the overlap of an initial state with the state obtained after the forward unperturbed evolution followed by a backward perturbed evolution. During the last years the theory of Fidelity is a subject of intensive research activity. This interest was motivated by various areas of physics. In this paper, we investigate the fidelity and study metal-insulator transition (MIT) in some familiar one-dimensional models. We can get many meaningful results.In this paper, firstly, we introduce some basic concepts of metal-insulator transition (MIT) and fidelity. Secondly, we numerically study the fidelity in the one-dimensional Harper model and in the one-dimensional slowly varying potential model. The extended and localized eigenstates can be distinguished by the fidelity. There are drastic decreases in fidelity of the individual eigenstates at mobility edges. In the curve of the spectrum averaged fidelity as a function of potential parameterλ,, a sharp transition exists at the metal -insulator transition pointλ_c=2.0. These results are consistence with that obtained by other methods. So it provides that the fidelity may be a good quantity to distinguish the extended and localized eigenstates and can reflect metal -insulator transition.