Some properties of the eigenstates in the many-electron problem

J. Szeftel, A. Khater

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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 Hubert space Sφ, spanned by a basis of Slater determinants of one-electron Bloch wave functions. Electron pairs of total momentum K and projected spin ζ=0,±1 are considered in this work. The Hamiltonian then reads H=HDK,ζHK,ζ, where HD consists of the diagonal part of H in the Slater determinant basis. HK,ζ describes the off-diagonal part of the two-electron scattering process which conserves K and ζ. This Hamiltonian operates in a subspace of Sφ, where the Slater determinants consist of pairs characterized by the same K and ζ. It is shown that the whole set of eigensolutions ψ,ε of the time-independent Schrödinger equation (H-ε)ψ=0 divides into two classes, ψ11 and ψ12,ε. The eigensolutions of class 1 are characterized by the property that for each solution ψ11, there is a single K and ζ such that (HD+HK,ζ1K,ζ=0 where, in general, ψ1ψ,ζ, whereas each solution ψ22 of class 2 fulfills (HD22=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.

Original languageEnglish
Pages (from-to)13581-13586
Number of pages6
JournalPhysical Review B - Condensed Matter and Materials Physics
Issue number19
Publication statusPublished - 15 Nov 1996
Externally publishedYes


ASJC Scopus subject areas

  • Condensed Matter Physics

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