论文标题
本电子问题中特征状态的某些特性
Some properties of the eigenstates in the many-electron problem
论文作者
论文摘要
考虑到有限浓度的一般性汉密尔顿$ h $，通过任何两体耦合在任意维度的晶体内进行相互作用。为了简单性并且没有一般性的损失，使用一个波段模型来说明电子 - 晶体相互作用。电子运动是在希尔伯特空间$ s_ϕ $中描述的，这是由单电子bloch波函数的基本决定因素的基础跨越的。总动量$ k $的电子对和预计旋转$ζ= 0，\ pm1 $在这项工作中考虑了。然后，哈密顿式读取$ h = h_d+\ sum_ {k，ζ} h_ {k，ζ} $，其中$ h_d $由$ h $的对角线组成，以$ h $的基础为基础。 $ h_ {k，ζ} $描述了保存$ k $和$ζ$的两电子散射过程的偏离部分。该哈密顿量在$ s_ϕ $的子空间中运行，其中slater的决定因素由以相同$ k $和$ζ$为特征的对组成。结果表明，独立于时间独立的schrödinger方程$（h-ε）ψ= 0 $的全部eigensolutions $ψ，ε$分为两个类别，$ψ_1，ε_1$和$ψ_2，ε_2$。第1类的特征是特征在于每个解决方案$ψ_1，ε_1$的属性，有一个$ k $和$ k $和$ζ$，因此$（h_d+h_ {k，ζ}-ε_1）ψ_{k，ζ} = 0 $ψ_2，2类的ε_2$满足$（H_D-ε_2）ψ_2= 0 $。我们还证明，1级的特征向量具有偏置长期顺序，而第二类则没有。最后，我们的结果表明，非对角线的长距离顺序不是超导性的足够条件。
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 $S_ϕ$, spanned by a basis of Slater determinants of one-electron Bloch wave-functions. Electron pairs of total momentum $K$ and projected spin $ζ=0,\pm1$ are considered in this work. The hamiltonian then reads $H=H_D+\sum_{K,ζ}H_{K,ζ}$, 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 characterised 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 in two classes, $ψ_1,ε_1$ and $ψ_2,ε_2$. The eigensolutions of class 1 are characterised 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 $ψ_1 \ne ψ_{K,ζ}$, whereas each solution $ψ_2,ε_2$ of class 2 fulfils $(H_D-ε_2)ψ_2=0$. 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.
