Partial self-consistency and analyticity in many-body perturbation theory: Particle number conservation and a generalized sum rule

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 $\mathrm{\Phi }$ derivability for the self-energy $\mathrm{\Sigma }$ 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 $\mathrm{\Sigma }$ partially $\mathrm{\Phi }$ derivable. A special subclass of such approximations, which are gauge invariant, is obtained by dressing loops in the diagrammatic expansion of $\mathrm{\Phi }$ 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 whether commonly used approximations will fulfill the continuity equation, which implies particle number conservation. We further show how the concept of partial $\mathrm{\Phi }$ 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-semidefinite. The latter property can be ensured for a subset of partially $\mathrm{\Phi }$-derivable approximations for the self-energy, namely those that can be constructed from squares of so-called half diagrams. For the case in which the analytic requirements are not fulfilled we highlight a number of subtle issues related to branch cuts, pole structure, and multivaluedness. We also show that various schemes of computing the particle number are consistent for particle number conserving approximations.

Figure 1

Examples of partially dressed ring $\mathrm{\Phi }$ functionals (left) and the corresponding $\mathrm{\Sigma }=\frac{\delta \mathrm{\Phi }}{\delta G}$ (right). (a) A single partially dressed loop; thus $\mathrm{\Phi }$ is not gauge invariant and the approximation is not number conserving. This diagram is part of one-shot 2nd Born, as well as $G_0{W}_0$. (b) A single, fully dressed loop, which gives a gauge-invariant and thus particle-conserving approximation. This diagram is part of the $G{W}_0$ approximation. (c) Partially dressed loops, not gauge invariant. Several types of self-energy diagrams are obtained, where the first is part of the $G_{0}W$ approximation. (d) Fully dressed, thus conserving. Part of 2nd Born, and $GW$.

Figure 2

Examples of partially dressed 2nd-order exchange (2OE) $\mathrm{\Phi }$ functionals (left) and the corresponding $\mathrm{\Sigma }=\frac{\delta \mathrm{\Phi }}{\delta G}$ (right). (a) Partially dressed second-order exchange, not gauge invariant. Constitutes one-shot 2nd Born, together with Fig. 1. (b) Fully dressed second-order exchange, thus conserving. Constitutes 2nd Born, together with Fig. 1. The only way to obtain a number-conserving approximation for these types of diagrams is to fully dress all $G$ lines.

Figure 3

Examples of partially dressed, gauge-invariant, $\mathrm{\Phi }$ functionals (left) which fulfill particle number conservation, and their corresponding $\mathrm{\Sigma }=\frac{\delta \mathrm{\Phi }}{\delta G}$ (right). (a) A 4th-order $G{W}_0$ diagram. (b) Dressing two loops yields two nonequivalent classes of diagrams. (c) A 4th-order diagram in the particle-particle $T$-matrix approximation. Note that we would get the same diagram if we dressed only the inner loop.

Figure 4

Left: $-\frac{1}{\pi }Im{G}^M\left(\zeta \right)$ for a Gaussian spectral function, Eq. (36), with $a=1$. $G^M\left(\zeta \right)\to 1/\zeta$ for large $\zeta$. Right: Analytically continued $-\frac{1}{\pi }Im{G}^R\left(\zeta \right)$ for a Gaussian spectral function, Eq. (37), with $a=1$. $G^R\left(\zeta \right)$ is analytic everywhere and is unbounded in the lower half plane. High and low values have been cut off for the sake of clarity.

Figure 5

Upper: The imaginary part of the two poles ${\omega }_{\pm }$ of $G^R$ as function of $\alpha$. The branching occurs at $4\alpha =-b^2$. Lower: The spectral function $A\left(\omega \right)$ for $\alpha >0$, where $A\left(\omega \right)>0$, and $-4b^2<\alpha <0$, where $A\left(\omega \right)$ can become negative. For $\alpha <-4b^2$, $A\left(\omega \right)=0$, the delta function weights are zero. The parameters are $b=1$, with a finite broadening.

Figure 6

Spectral functions for the Anderson model (single impurity coupled to a featureless lead). 2OE0: Single-shot 2nd-order exchange diagram, Fig. 2. 2OE: Self-consistent 2nd-order exchange, Fig. 2. 2BA0: Single-shot 2nd Born, Fig. 1 $+$ Fig. 2. 2OE0 and 2OE are non-PSD. 2nd Born, however, is PSD. Compare with Fig. 7. The parameters are $U=6.5,\epsilon =-7$, and $\mathrm{\Gamma }=1$.

Figure 7

$-G^R\left(\omega \right)$ parametrized with $\omega \in (-\infty ,\mu ]$, starting from $\omega =-\infty$ where $G^R(-\infty )=0$, and ending at $\mu =0$, for the same situation as in Fig. 6. The 2nd-order exchange approximations are not PSD, and thus $-G^R\left(\omega \right)$ can cross the real axis. Single-shot 2OE breaks analyticity as well as the validity of the sum rules. Self-consistent 2OE does not. The single-shot 2nd Born approximation is PSD, and thus never crosses the real axis. For illustrative purposes, we have divided $G^R$ from 2OE0 by 3 and multiplied by 2 for 2BA0.

Figure 8

Number of particles $N_C$ in a 3-site system, for different dressings of ring approximations and single-shot 2nd-order exchange. We show the self-consistent ring approximation [Fig. 1], partially dressed ring diagram [Fig. 1], single-shot ring diagram [Fig. 1], and single-shot 2nd-order exchange [Fig. 2]. $I_{2,emb}$ and $I_{2,MB}$ are the embedding and many-body contributions to the Luttinger integral, Eq. (68), respectively. Note the different scale for $I_2$ in the non-self-consistent calculations.

Figure 9

The three eigenvalues of $-{\widehat{G}}^R\left(\omega \right)$ parametrized by $\omega$ for the system shown in Fig. 8, for the fully self-consistent ring approximation, Fig. 1. $\omega$ runs from $-\infty$ to $\omega =\mu =0$. The parameters are $U=10$, $\epsilon =-U/2$. Being a PSD approximation, no eigenvalue can cross the real axis. Numerically we find $N_C-I_1-I_2=0$.

Figure 10

The three eigenvalues of $-{\widehat{G}}^R\left(\omega \right)$ parametrized by $\omega$ for the system shown in Fig. 8, for the single-shot 2nd-order exchange, Fig. 2. $\omega$ runs from $-\infty$ to $\omega =\mu =0$. The parameters are $U=10$, $\epsilon =-U/2$. Being a non-PSD approximation, eigenvalues can, and do in this case, cross the real axis. Numerically, we find $N_C-I_1-I_2=-6$, since each eigenvalue yields a factor of 2 to $I_1$.

Figure 11

The eigenvalues ${\lambda }_k\left(\omega \right)$ of $-{\widehat{G}}^R\left(\omega \right)$ for the 3-site system in Sec. 4d, with $U=\epsilon =0$ and using the wide-band limit for simplicity. The curves start at $\omega =-\infty$, for which ${\lambda }_i(-\infty )=0$, and end at $\mu =0$. Neither of the eigenvalues cross the real axis, while their multiplication does.