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 ${\mathit{S}}_{\mathrm{\ensuremath{\varphi}}}$, spanned by a basis of Slater determinants of one-electron Bloch wave functions. Electron pairs of total momentum K and projected spin \ensuremath{\zeta}=0,\ifmmode\pm\else\textpm\fi{}1 are considered in this work. The Hamiltonian then reads H=${\mathit{H}}_{\mathit{D}}$+${\ensuremath{\sum}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$${\mathit{H}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$, where ${\mathit{H}}_{\mathit{D}}$ consists of the diagonal part of H in the Slater determinant basis. ${\mathit{H}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$ describes the off-diagonal part of the two-electron scattering process which conserves K and \ensuremath{\zeta}. This Hamiltonian operates in a subspace of ${\mathit{S}}_{\mathrm{\ensuremath{\varphi}}}$, where the Slater determinants consist of pairs characterized by the same K and \ensuremath{\zeta}. It is shown that the whole set of eigensolutions \ensuremath{\psi},\ensuremath{\epsilon} of the time-independent Schr\"odinger equation (H-\ensuremath{\epsilon})\ensuremath{\psi}=0 divides into two classes, ${\mathrm{\ensuremath{\psi}}}_{1}$,${\mathrm{\ensuremath{\epsilon}}}_{1}$ and ${\mathrm{\ensuremath{\psi}}}_{2}$,${\mathrm{\ensuremath{\epsilon}}}_{2}$. The eigensolutions of class 1 are characterized by the property that for each solution ${\mathrm{\ensuremath{\psi}}}_{1}$,${\mathrm{\ensuremath{\epsilon}}}_{1}$ there is a single K and \ensuremath{\zeta} such that (${\mathit{H}}_{\mathit{D}}$+${\mathit{H}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$-${\mathrm{\ensuremath{\epsilon}}}_{1}$)${\mathrm{\ensuremath{\psi}}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$=0 where, in general, ${\mathrm{\ensuremath{\psi}}}_{1}$\ensuremath{\ne}${\mathrm{\ensuremath{\psi}}}_{\mathit{K},\mathrm{\ensuremath{\zeta}}}$, whereas each solution ${\mathrm{\ensuremath{\psi}}}_{2}$,${\mathrm{\ensuremath{\epsilon}}}_{2}$ of class 2 fulfills (${\mathit{H}}_{\mathit{D}}$-${\mathrm{\ensuremath{\epsilon}}}_{2}$)${\mathrm{\ensuremath{\psi}}}_{2}$=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. \textcopyright{} 1996 The American Physical Society.