### 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 Hubert space S_{φ}, 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=H_{D}+Σ_{K,ζ}H_{K,ζ}, where H_{D} consists of the diagonal part of H in the Slater determinant basis. H_{K,ζ} describes the off-diagonal part of the two-electron scattering process which conserves K and ζ. This Hamiltonian operates in a subspace of S_{φ}, 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, ψ_{1},ε_{1} and ψ_{1},ψ_{2},ε. The eigensolutions of class 1 are characterized by the property that for each solution ψ_{1},ε_{1}, there is a single K and ζ such that (H_{D}+H_{K,ζ}-ε_{1})ψ _{K,ζ}=0 where, in general, ψ_{1}≠_{ψ,ζ}, whereas each solution ψ_{2},ε_{2} of class 2 fulfills (H_{D}-ε_{2})ψ_{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.

Original language | English |
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Pages (from-to) | 13581-13586 |

Number of pages | 6 |

Journal | Physical Review B - Condensed Matter and Materials Physics |

Volume | 54 |

Issue number | 19 |

Publication status | Published - 15 Nov 1996 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Condensed Matter Physics

### Cite this

*Physical Review B - Condensed Matter and Materials Physics*,

*54*(19), 13581-13586.