Theory of Ferromagnetism and the Ordering of Electronic Energy Levels

Abstract

Consider a system of $N$ electrons in one dimension subject to an arbitrary symmetric potential, $V({x}_{1}, \ensuremath{\cdots}, {x}_{N})$, and let $E(S)$ be the lowest energy belonging to the total spin value $S$. We have proved the following theorem: $E(S)<E({S}^{\ensuremath{'}})$ if $S<{S}^{\ensuremath{'}}$. Hence, the ground state is unmagnetized. The theorem also holds in two or three dimensions (although it is possible to have degeneracies) provided $V({x}_{1}, {y}_{1}, {z}_{1}; \ensuremath{\cdots}; {x}_{N}, {y}_{N}, {z}_{N})$ is separately symmetric in the ${x}_{i}$, ${y}_{i}$, and ${z}_{i}$. The potential need not be separable, however. Our theorem has strong implications in the theory of ferromagnetism because it is generally assumed that for certain repulsive potentials, the ground state is magnetized. If such be the case, it is a very delicate matter, for a plausible theory must not be so general as to give ferromagnetism in one dimension, nor in three dimensions with a separately symmetric potential.