Diagrammatic expansion for positive spectral functions beyond $GW$: Application to vertex corrections in the electron gas

We present a diagrammatic approach to construct self-energy approximations within many-body perturbation theory with positive spectral properties. The method cures the problem of negative spectral functions which arises from a straightforward inclusion of vertex diagrams beyond the $GW$ approximation. Our approach consists of a two-step procedure: We first express the approximate many-body self-energy as a product of half-diagrams and then identify the minimal number of half-diagrams to add in order to form a perfect square. The resulting self-energy is an unconventional sum of self-energy diagrams in which the internal lines of half a diagram are time-ordered Green's functions, whereas those of the other half are anti-time-ordered Green's functions, and the lines joining the two halves are either lesser or greater Green's functions. The theory is developed using noninteracting Green's functions and subsequently extended to self-consistent Green's functions. Issues related to the conserving properties of diagrammatic approximations with positive spectral functions are also addressed. As a major application of the formalism we derive the minimal set of additional diagrams to make positive the spectral function of the $GW$ approximation with lowest-order vertex corrections and screened interactions. The method is then applied to vertex corrections in the three-dimensional homogeneous electron gas by using a combination of analytical frequency integrations and numerical Monte Carlo momentum integrations to evaluate the diagrams.

Figure 1

The closed time-loop contour $\mathcal{C}$. The (forward) minus branch is denoted with a “$-$” label while the (backward) plus branch is denoted by a “$+$” label.

Figure 2

Diagrammatic structure of the functions $S^{*}\left(1\right)$ and $S\left(2\right)$ for the lesser self-energy. The external vertex points 1 and 2 have times on the minus and plus branch, respectively. Green's functions are denoted by the lines with arrows, while wavy lines correspond to the bare interparticle interaction.

Figure 3

Minimal example of irreducible $S^{*}$ diagram with self-energy insertions. When glued with some irreducible $S$ diagram it yields an irreducible self-energy diagram. Notice that this diagram is reducible in the “correlation function” sense because cutting a single Green's function line produces disconnected pieces.

Figure 4

An example of the distribution of $+$ and $-$ labels over the internal vertices of a lesser fourth-order self-energy diagram. Divisions with lines $v^{+-}$ or $v^{-+}$ vanish due to the time locality of the interaction and therefore are not shown. The third term after the equality sign must be discarded because it contains an isolated island of plus signs upon cutting the $+/-$ $g$ lines (see explanation in the text).

Figure 5

An approximate self-energy of many-body perturbation theory.

Figure 6

Partitions of the self-energy diagrams of Fig. 5 with five $+/-$ $g$ lines and decomposition in terms of $D-D^{*}$ diagrams.

Figure 7

(a) Partition of the first bubble diagram and the resulting half-diagrams. (b) Partition of the second-order exchange diagram and the resulting half-diagrams. (c) In order to form a a complete square with the second-order exchange diagram we need to add the right half-diagram with permuted $p_1$ and $p_2$ labels. This yields the 2B approximation.

Figure 8

(a) Partitions of the first and second bubble diagrams and the resulting half-diagrams. (b) The minimal completion of the square which yields a PSD spectrum.

Figure 9

(a) Bubble half-diagrams of the $GW$ approximation. (b) Ladder half-diagrams of the $T$-matrix approximation.

Figure 10

Next to leading order self-energy in the screened interaction $W$. Since $W$ is nonlocal in time, the thick wavy lines denoting the screened interaction can connect points on different branches of the Keldysh contour.

Figure 11

(a) Partitions of the first diagram on the right-hand side of Fig. 10. (b) Definition of the ${\tilde{D}}^{\left(j\right)}$ diagrams.

Figure 12

Partitions of the third diagram on the right-hand side of Fig. 10.

Figure 13

(a) Partitions of the fourth diagram on the right-hand side of Fig. 10. (b) Definition of the $D^{(i,j)}$ diagrams.

Figure 14

There are four extra diagrams that correct the self-energy with three $+/-$ $g$ lines (${\Sigma }_3$). They have (a) ${\tilde{D}}^{*}\left(1\right)\tilde{D}\left(2\right)$, (b) $D_P^{*}\left(1\right)\tilde{D}\left(2\right)$, ${\tilde{D}}_P^{*}\left(1\right)D\left(2\right)$ [similar to diagram (b) and thus not shown], and (c) ${\tilde{D}}_P^{*}\left(1\right)\tilde{D}\left(2\right)$ structures. Subscript $P$ denotes a diagram with permuted $p$ indices.

Figure 15

Leading order beyond-$GW$ self-energy with the PSD property. Thick wavy lines denote the screened Coulomb interaction in the random phase approximation.

Figure 16

The sum of the four self-energy partitions (a) + (b) + (c) + (d) yields a positive spectral function. Lines with arrows denote the electron propagator $G_0(k,\omega )$, whereas wavy lines stand for the screened Coulomb interaction $W_0(k,\omega )$. Vertices are labeled with $+$ ($-$) if the time lies on the minus (plus) branch of the Keldysh contour. Diagrams are translated into analytic expressions according to the definitions (47) and (48) and notations below Eq. (49). For instance $g_5(z-y_3)$, connecting vertices with “$-$” labels as in (c) and (d), corresponds to the time-ordered Green's function [see also Eq. (2)]: $\frac{\mathcal{B}\left(k_5\right)}{z-\Omega \left(y_3\right)-{\epsilon }_{k_5}-i\eta }+\frac{\mathcal{A}\left(k_5\right)}{z-\Omega \left(y_3\right)-{\epsilon }_{k_5}+i\eta }$.

Figure 17

The rate operator $\frac{1}{2}\Gamma (k,\omega )=-\mathrm{Im}{\Sigma }_c^{\mathrm{R}}(k,\omega )$ of the homogeneous electron gas at the density of $r_s=4$ and $k=k_F$ (the energy $\omega$ is measured with respect to $\mu$). Different line styles denote contributions of different orders: full, dotted, and dashed lines stand for first, second, and third order, respectively. Thick solid line denotes the sum of all contributions.

Figure 18

The rate operator $\frac{1}{2}\Gamma (k,\omega )=-\mathrm{Im}{\Sigma }_c^{\mathrm{R}}(k,\omega )$ as in Fig. 17 for a different momentum $k=1.2k_F$ (the energy $\omega$ is measured with respect to $\mu$). The inset magnifies the region of the logarithmic singularity. Notice the almost complete cancellation of ${\Sigma }_c^{>}(k,\omega )$ for $\omega >{\epsilon }_k+\Omega \left(0\right)$.

Figure 19

The real part of the retarded self-energy $\mathrm{Re}{\Sigma }^{\mathrm{R}}(k,\omega )$, see Eq. (52), and the rate operator $\frac{1}{2}\Gamma (k,\omega )=-\mathrm{Im}{\Sigma }_c^{\mathrm{R}}(k,\omega )$ of the homogeneous electron gas at the density of $r_s=4$ and $k=k_F$ (the energy $\omega$ is measured with respect to $\mu$). Shaded dashed curves denote the first-order calculation. Full lines denote contribution from diagrams shown in Fig. 16 plus the first-order contribution from the particle-hole excitations. The horizontal line bounding the shaded area in the graph of $\mathrm{Re}{\Sigma }^{\mathrm{R}}$ crosses the $y$ axis at the value of the exchange self-energy ${\Sigma }_x\left(k_F\right)/{\epsilon }_F=-\frac{2\alpha }{\pi }{r}_s\approx -1.327$.