An algebraic approach to revealing magnetic structures of ground states in many-electron systems

Abstract: Mathematical understanding of the origin of ferromagnetism is still incomplete and remains an important research topic in mathematical physics. In this paper, we give a model-independent mathematical framework describing the magnetic features of the ground states in many-electron systems. Within this framework, we also present a new approach to understanding magnetic orders in macroscopic systems. Based on these, we construct a general theory that explains the stability of magnetic orders in the ground states despite the interaction of electrons with the environment. Methodologically, the theory presented in this paper is formulated using von Neumann algebras and their associated standard forms. A benefit of working in such an algebraic setting is that we can define operator inequalities that preserve the ordered structures that naturally follow from the standard forms; by exploiting these operator inequalities, we can develop new descriptions of the magnetic structures of the ground states. As specific applications of the proposed theory, we first analyze the Marshall--Lieb--Mattis theorem, Lieb's theorem, and their stabilities under various perturbations. Next, the Nagaoka--Thouless theorem and its stability are addressed. In addition, we interpret various other examples from the new theory and give a unified perspective on existing results.