Some properties of the eigenstates in the many-electron problem

Abstract

A general hamiltonian $H$ of electrons in finite concentration, interacting via any two-body coupling inside a crystal of arbitrary dimension, is considered. For simplicity and without loss of generality, a one-band model is used to account for the electron-crystal interaction. The electron motion is described in the Hilbert space $S_{\varphi }$, spanned by a basis of Slater determinants of one-electron Bloch wave-functions. Electron pairs of total momentum $K$ and projected spin $\zeta =0,\pm 1$ are considered in this work. The hamiltonian then reads $H=H_D+{\sum }_{K,\zeta }{H}_{K,\zeta }$, where $H_D$ consists of the diagonal part of $H$ in the Slater determinant basis. $H_{K,\zeta }$ describes the off-diagonal part of the two-electron scattering process which conserves $K$ and $\zeta$. This hamiltonian operates in a subspace of $S_{\varphi }$, where the Slater determinants consist of pairs characterised by the same $K$ and $\zeta$. It is shown that the whole set of eigensolutions $\psi ,\epsilon$ of the time-independent Schrödinger equation $(H-\epsilon )\psi =0$ divides in two classes, ${\psi }_1,{\epsilon }_1$ and ${\psi }_2,{\epsilon }_2$. The eigensolutions of class 1 are characterised by the property that for each solution ${\psi }_1,{\epsilon }_1$ there is a single $K$ and $\zeta$ such that $(H_D+H_{K,\zeta }-{\epsilon }_1){\psi }_{K,\zeta }=0$ where in general ${\psi }_1\ne {\psi }_{K,\zeta }$, whereas each solution ${\psi }_2,{\epsilon }_2$ of class 2 fulfils $(H_D-{\epsilon }_2){\psi }_2=0$. We prove also that the eigenvectors of class 1 have off-diagonal long-range order whereas those of class 2 do not. Finally our result shows that off-diagonal long-range order is not a sufficient condition for superconductivity.

6 pages, no figure