### 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.

**Some properties of the eigenstates in the many-electron problem.** / Szeftel, J.; Khater, A.

Research output: Contribution to journal › Article

*Physical Review B - Condensed Matter and Materials Physics*, vol. 54, no. 19, pp. 13581-13586.

}

TY - JOUR

T1 - Some properties of the eigenstates in the many-electron problem

AU - Szeftel, J.

AU - Khater, A.

PY - 1996/11/15

Y1 - 1996/11/15

N2 - 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=HD+ΣK,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, ψ1,ε1 and ψ1,ψ2,ε. The eigensolutions of class 1 are characterized by the property that for each solution ψ1,ε1, there is a single K and ζ such that (HD+HK,ζ-ε1)ψ K,ζ=0 where, in general, ψ1≠ψ,ζ, whereas each solution ψ2,ε2 of class 2 fulfills (HD-ε2)ψ2=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.

AB - 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=HD+ΣK,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, ψ1,ε1 and ψ1,ψ2,ε. The eigensolutions of class 1 are characterized by the property that for each solution ψ1,ε1, there is a single K and ζ such that (HD+HK,ζ-ε1)ψ K,ζ=0 where, in general, ψ1≠ψ,ζ, whereas each solution ψ2,ε2 of class 2 fulfills (HD-ε2)ψ2=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.

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M3 - Article

VL - 54

SP - 13581

EP - 13586

JO - Physical Review B-Condensed Matter

JF - Physical Review B-Condensed Matter

SN - 0163-1829

IS - 19

ER -