多体微扰论中的部分自洽性和解析性：粒子数守恒和一个广义求和规则
Partial self-consistency and analyticity in many-body perturbation theory: particle number conservation and a generalized sum rule
摘要：
我们考虑了一类保证多体微扰论中粒子数守恒的近似。要做到这一点，我们扩展的概念$Φ$-可导性的自能$Σ$到一个更大的类的图形条款，其中只有一些格林函数线包含完全打扮格林函数$G$。我们将$Σ$的相应近似称为部分$Φ$可导出的。这种近似的一个特殊子类，这是规范不变的，通过在$Φ$的图解展开中与$G$一致地修饰环而得到。这些近似是数量守恒的，但不必满足其他守恒定律，如能量守恒和动量守恒。从我们的形式主义，我们可以很容易地推断，如果常用的近似将满足连续性方程，这意味着粒子数守恒。我们进一步展示了部分$Φ$-可导性的概念如何在推导粒子数的广义求和规则中起着重要作用，该规则在均匀电子气的情况下简化为Luttinger-Ward定理，在Anderson模型的情况下简化为Friedel求和规则。要做到这一点，我们需要确保格林函数具有一定的复解析性质，如果谱函数是半正定的，则可以保证这一点。后一个性质可以为自能的部分$Φ$可导近似的子集，即那些可以从所谓的半图的平方中构造的子集。如果分析的要求没有得到满足，我们强调了一些微妙的问题有关的分支切割，极点结构和多值。我们还表明，各种计划的计算粒子数是一致的粒子数守恒近似。
英文摘要：
We consider a general class of approximations which guarantees the conservation of particle number in many-body perturbation theory. To do this we extend the concept of $Φ$-derivability for the self-energy $Σ$ to a larger class of diagrammatic terms in which only some of the Green's function lines contain the fully dressed Green's function $G$. We call the corresponding approximations for $Σ$ partially $Φ$-derivable. A special subclass of such approximations, which are gauge-invariant, is obtained by dressing loops in the diagrammatic expansion of $Φ$ consistently with $G$. These approximations are number conserving but do not have to fulfill other conservation laws, such as the conservation of energy and momentum. From our formalism we can easily deduce if commonly used approximations will fulfill the continuity equation, which implies particle number conservation. We further show how the concept of partial $Φ$-derivability plays an important role in the derivation of a generalized sum rule for the particle number, which reduces to the Luttinger-Ward theorem in the case of a homogeneous electron gas, and the Friedel sum rule in the case of the Anderson model. To do this we need to ensure that the Green's function has certain complex analytic properties, which can be guaranteed if the spectral function is positive semi-definite.The latter property can be ensured for a subset of partially $Φ$-derivable approximations for the self-energy, namely those that can be constructed from squares of so-called half-diagrams. In case the analytic requirements are not fulfilled we highlight a number of subtle issues related to branch cuts, pole structure and multi-valuedness. We also show that various schemes of computing the particle number are consistent for particle number conserving approximations.
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