Second-order perturbation theory

Second-order perturbation theory in the bare interaction is the simplest nontrivial approximation for both the two-particle vertex and the self-energy. It is worth writing out completely, for three reasons: it is the standard starting point for iterating the parquet equations, it is the natural benchmark against which any implementation of those equations should be validated, and it is the order at which all the prefactors can still be tracked by hand, namely the prefactors of the bubbles in the three two-particle channels, the spin sums, and the $\frac{1}{2}$ of the Schwinger-Dyson equation.

The purpose of this page is to derive explicit formulas for the second-order vertex and the second-order self-energy, with every prefactor, spin sum, frequency argument and Keldysh index in place.

We specialize throughout to a system with SU(2) spin symmetry and a local and instantaneous bare interaction, i.e. the Hubbard interaction of the Hubbard model example, whose spin components are

\begin{align} F_{0,\uparrow\downarrow} &= -U \, , & F_{0,\overline{\uparrow\downarrow}} &= +U \, , & F_{0,\uparrow\uparrow} &= 0 \, , \end{align}

(1)

and hence

\begin{align} F_{0,d} &= -U \, , & F_{0,m} &= +U \, , & F_{0,s} &= -2U \, , & F_{0,t} &= 0 \, . \end{align}

(2)

Second-order vertex¶

As derived in the section on two-particle channels, the second-order vertex is the sum of the bare vertex and one contribution per channel,

\begin{align} F = F_0 + \sum_{r \in \{\overline{ph},\, pp,\, ph\}} \Phi^r_{(2)} + \mathcal{O}(F_0^3) \, , \qquad\qquad \Phi^r_{(2)} \equiv F_0 \circ [\chi_0^{0}]^r \circ F_0 \, , \end{align}

(3)

where $[\chi_0^{0}]^r$ is the bare bubble in channel $r$ . Written out with the channel connectors, these are

\begin{align} [\Phi^{\overline{ph}}_{(2)}]_{1234} &= F_{0,1256}\, [\chi_0^{0}]^{\overline{ph}}_{6587}\, F_{0,7834} \, , \\ [\Phi^{pp}_{(2)}]_{1234} &= F_{0,1536}\, [\chi_0^{0}]^{pp}_{6857}\, F_{0,7284} \, , \\ [\Phi^{ph}_{(2)}]_{1234} &= F_{0,5236}\, [\chi_0^{0}]^{ph}_{8567}\, F_{0,1874} \, , \end{align}

(4)

with summation over the repeated internal indices $5,6,7,8$ implied. Recall that the channel prefactors are carried by the bubbles themselves, $[\chi_0^{0}]^{\overline{ph}}_{4321} = [\chi_0^{0}]_{4321}$ , $[\chi_0^{0}]^{pp}_{4321} = \frac{1}{2}[\chi_0^{0}]_{4321}$ and $[\chi_0^{0}]^{ph}_{4321} = \zeta [\chi_0^{0}]_{2341}$ , so that no further prefactors appear.

Spin structure¶

Because the propagator is diagonal in spin, the spin sums of the channel contractions collapse to a double sum, and the spin structure of the bubble differs between the channels. For the $ph$ and the $pp$ channel these contractions are derived in the section on spin parametrizations; the $\overline{ph}$ channel follows in exactly the same way. Collecting all three,

\begin{align} [A \circ \chi_0^{\overline{ph}} \circ B]_{\sigma_1 \sigma_2 \sigma_3 \sigma_4} &= \sum_{\sigma_5, \sigma_6} A_{\sigma_1 \sigma_2 \sigma_5 \sigma_6} \bullet \chi_0^{\overline{ph}} \bullet B_{\sigma_6 \sigma_5 \sigma_3 \sigma_4} \, , \\ [A \circ \chi_0^{pp} \circ B]_{\sigma_1 \sigma_2 \sigma_3 \sigma_4} &= \sum_{\sigma_5, \sigma_6} A_{\sigma_1 \sigma_5 \sigma_3 \sigma_6} \bullet \chi_0^{pp} \bullet B_{\sigma_6 \sigma_2 \sigma_5 \sigma_4} \, , \\ [A \circ \chi_0^{ph} \circ B]_{\sigma_1 \sigma_2 \sigma_3 \sigma_4} &= \sum_{\sigma_5, \sigma_6} A_{\sigma_5 \sigma_2 \sigma_3 \sigma_6} \bullet \chi_0^{ph} \bullet B_{\sigma_1 \sigma_5 \sigma_6 \sigma_4} \, , \end{align}

(5)

where $\bullet$ denotes the contraction over all variables except the spin indices. Evaluating these on the three independent spin components gives the following table, valid for arbitrary four-point objects $A$ and $B$ :

channel	$\uparrow\uparrow$	$\uparrow\downarrow$	$\overline{\uparrow\downarrow}$
$\overline{ph}$	$A_{\uparrow\uparrow} B_{\uparrow\uparrow} + A_{\overline{\uparrow\downarrow}} B_{\overline{\uparrow\downarrow}}$	$A_{\uparrow\downarrow} B_{\uparrow\downarrow}$	$A_{\uparrow\uparrow} B_{\overline{\uparrow\downarrow}} + A_{\overline{\uparrow\downarrow}} B_{\uparrow\uparrow}$
$pp$	$A_{\uparrow\uparrow} B_{\uparrow\uparrow}$	$A_{\uparrow\downarrow} B_{\uparrow\downarrow} + A_{\overline{\uparrow\downarrow}} B_{\overline{\uparrow\downarrow}}$	$A_{\uparrow\downarrow} B_{\overline{\uparrow\downarrow}} + A_{\overline{\uparrow\downarrow}} B_{\uparrow\downarrow}$
$ph$	$A_{\uparrow\uparrow} B_{\uparrow\uparrow} + A_{\uparrow\downarrow} B_{\uparrow\downarrow}$	$A_{\uparrow\downarrow} B_{\uparrow\uparrow} + A_{\uparrow\uparrow} B_{\uparrow\downarrow}$	$A_{\overline{\uparrow\downarrow}} B_{\overline{\uparrow\downarrow}}$

In this table the $\bullet \chi_0^r \bullet$ between the two factors is suppressed for readability.

The $\uparrow\uparrow$ and $\uparrow\downarrow$ entries of the $ph$ row and the whole $pp$ row are derived in the section on spin parametrizations. The remaining entries follow from the same two ingredients: spin conservation, $A_{\sigma_1\sigma_2\sigma_3\sigma_4} \sim \delta_{\sigma_1 + \sigma_3, \sigma_2 + \sigma_4}$ , and the definitions $A_{\sigma\sigma\sigma\sigma} = A_{\uparrow\uparrow}$ , $A_{\sigma\overline{\sigma}\overline{\sigma}\sigma} = A_{\uparrow\downarrow}$ , $A_{\sigma\sigma\overline{\sigma}\overline{\sigma}} = A_{\overline{\uparrow\downarrow}}$ .

$ph$ , $\overline{\uparrow\downarrow}$ component. Setting $(\sigma_1 \sigma_2 \sigma_3 \sigma_4) = (\sigma \sigma \overline{\sigma} \overline{\sigma})$ ,

\begin{align} [A \circ \chi_0^{ph} \circ B]_{\overline{\uparrow\downarrow}} &= \sum_{\sigma_5, \sigma_6} A_{\sigma_5 \sigma \overline{\sigma} \sigma_6} \bullet \chi_0^{ph} \bullet B_{\sigma \sigma_5 \sigma_6 \overline{\sigma}} \, . \end{align}

(6)

The first factor requires $\sigma_5 + \overline{\sigma} = \sigma + \sigma_6$ , which of the four choices of $(\sigma_5, \sigma_6)$ only $(\sigma, \overline{\sigma})$ satisfies. Hence

\begin{align} [A \circ \chi_0^{ph} \circ B]_{\overline{\uparrow\downarrow}} &= A_{\sigma \sigma \overline{\sigma} \overline{\sigma}} \bullet \chi_0^{ph} \bullet B_{\sigma \sigma \overline{\sigma} \overline{\sigma}} = A_{\overline{\uparrow\downarrow}} \bullet \chi_0^{ph} \bullet B_{\overline{\uparrow\downarrow}} \, . \end{align}

(7)

$\overline{ph}$ channel. Here $[\chi_0]^{\overline{ph}}_{\sigma_4 \sigma_3 \sigma_2 \sigma_1} \sim \delta_{\sigma_4,\sigma_1} \delta_{\sigma_2,\sigma_3}$ , and with the connector $[A \circ B]^{\overline{ph}}_{1234} = A_{1256} B_{6534}$ the four internal spin sums reduce to the double sum quoted above. For the $\uparrow\uparrow$ component, $A_{\sigma \sigma \sigma_5 \sigma_6}$ requires $\sigma_5 = \sigma_6$ , leaving $(\sigma,\sigma)$ and $(\overline{\sigma},\overline{\sigma})$ ,

\begin{align} [A \circ \chi_0^{\overline{ph}} \circ B]_{\uparrow\uparrow} &= A_{\sigma\sigma\sigma\sigma} \bullet \chi_0^{\overline{ph}} \bullet B_{\sigma\sigma\sigma\sigma} + A_{\sigma\sigma\overline{\sigma}\overline{\sigma}} \bullet \chi_0^{\overline{ph}} \bullet B_{\overline{\sigma}\overline{\sigma}\sigma\sigma} = A_{\uparrow\uparrow} B_{\uparrow\uparrow} + A_{\overline{\uparrow\downarrow}} B_{\overline{\uparrow\downarrow}} \, . \end{align}

(8)

For the $\uparrow\downarrow$ component, $A_{\sigma \overline{\sigma} \sigma_5 \sigma_6}$ requires $\sigma + \sigma_5 = \overline{\sigma} + \sigma_6$ , i.e. $(\sigma_5, \sigma_6) = (\overline{\sigma}, \sigma)$ , and

\begin{align} [A \circ \chi_0^{\overline{ph}} \circ B]_{\uparrow\downarrow} &= A_{\sigma \overline{\sigma} \overline{\sigma} \sigma} \bullet \chi_0^{\overline{ph}} \bullet B_{\sigma \overline{\sigma} \overline{\sigma} \sigma} = A_{\uparrow\downarrow} B_{\uparrow\downarrow} \, . \end{align}

(9)

For the $\overline{\uparrow\downarrow}$ component, $A_{\sigma \sigma \sigma_5 \sigma_6}$ again requires $\sigma_5 = \sigma_6$ , and

\begin{align} [A \circ \chi_0^{\overline{ph}} \circ B]_{\overline{\uparrow\downarrow}} &= A_{\sigma\sigma\sigma\sigma} \bullet \chi_0^{\overline{ph}} \bullet B_{\sigma\sigma\overline{\sigma}\overline{\sigma}} + A_{\sigma\sigma\overline{\sigma}\overline{\sigma}} \bullet \chi_0^{\overline{ph}} \bullet B_{\overline{\sigma}\overline{\sigma}\overline{\sigma}\overline{\sigma}} = A_{\uparrow\uparrow} B_{\overline{\uparrow\downarrow}} + A_{\overline{\uparrow\downarrow}} B_{\uparrow\uparrow} \, . \end{align}

(10)

Each row of the table satisfies the SU(2) identity $[\ldots]_{\uparrow\uparrow} = [\ldots]_{\uparrow\downarrow} + [\ldots]_{\overline{\uparrow\downarrow}}$ , as it must, upon inserting $A_{\overline{\uparrow\downarrow}} = A_{\uparrow\uparrow} - A_{\uparrow\downarrow}$ . $\checkmark$

Inserting $A = B = F_0$ , and using $F_{0,\uparrow\uparrow} = 0$ , the three channel contributions to the second-order vertex have the spin components

channel	$\uparrow\uparrow$	$\uparrow\downarrow$	$\overline{\uparrow\downarrow}$
$\overline{ph}$	$U^2$	$U^2$	0
$pp$	0	$2U^2$	$-2U^2$
$ph$	$U^2$	0	$U^2$

where the entries are the coefficients $c^r_x$ in

\begin{align} [\Phi^r_{(2)}]_x &= c^r_x\, [\chi_0^{0}]^r(\omega, \mathbf{q}) \, , & [\chi_0^{0}]^r(\omega, \mathbf{q}) &\equiv \int_{\nu, \mathbf{k}} [\chi_0^{0}]^r(\nu, \omega; \mathbf{k}, \mathbf{q}) \, . \end{align}

(11)

Here and below, a bubble written with bosonic arguments only is understood to be integrated over its fermionic ones. The result takes this form of a number times a bubble because the bare vertex of a local, instantaneous interaction is independent of frequency and momentum and can be pulled out of the contraction, as spelled out in the next subsection.

The $ph$ and $pp$ channels are naturally read in the spin bases in which their Bethe-Salpeter equations decouple, the density/magnetic basis and the singlet/triplet basis respectively, with $X_{d/m} = X_{\uparrow\uparrow} \pm X_{\uparrow\downarrow}$ (upper sign for $d$ ), $X_s = X_{\uparrow\downarrow} - X_{\overline{\uparrow\downarrow}}$ and $X_t = X_{\uparrow\uparrow}$ . In those bases the table reads

\begin{align} \Phi^{ph}_{(2),d} &= \Phi^{ph}_{(2),m} = U^2\, [\chi_0^{0}]^{ph}(\omega, \mathbf{q}) \, , & \Phi^{pp}_{(2),s} &= 4U^2\, [\chi_0^{0}]^{pp}(\omega, \mathbf{q}) \, , & \Phi^{pp}_{(2),t} &= 0 \, . \end{align}

(12)

Frequency and momentum parametrization¶

In the channel-native parametrizations, the bare bubbles read

\begin{align} [\chi_0^{0}]^{\overline{ph}}_{4321}(\nu, \omega) &= G_{0,41}(\nu)\, G_{0,23}(\nu + \omega) \, , \\ [\chi_0^{0}]^{pp}_{4321}(\nu, \omega) &= \tfrac{1}{2}\, G_{0,41}(\nu)\, G_{0,23}(-\nu - \omega) \, , \\ [\chi_0^{0}]^{ph}_{4321}(\nu, \omega) &= \zeta\, G_{0,21}(\nu)\, G_{0,43}(\nu + \omega) \, , \end{align}

(13)

and the channel contractions couple only the fermionic variables,

\begin{align} [A \circ \chi_0^{r} \circ B](\nu, \nu', \omega) &= \int_{\nu''} A(\nu, \nu'', \omega) \bullet \chi_0^{r}(\nu'', \omega) \bullet B(\nu'', \nu', \omega) \, . \end{align}

(14)

Because the bare vertex of a local, instantaneous interaction is independent of frequency and momentum, it drops out of the loop, and every $\Phi^r_{(2)}$ depends on the transfer variables of its own channel only,

\begin{align} \Phi^r_{(2)}(\omega, \mathbf{q}) &= F_0 \bullet [\chi_0^{0}]^{r}(\omega, \mathbf{q}) \bullet F_0 \, , \end{align}

(15)

with the integrated bubble defined above.

Result in the Keldysh formalism¶

For an instantaneous interaction the bare vertex takes the simple form derived in the section on the Keldysh formalism,

\begin{align} F_0^{k_1 k_2 k_3 k_4} &= \begin{cases} F_0 / 2 & \text{if } k_1 + k_2 + k_3 + k_4 \text{ odd} \\ 0 & \text{otherwise}\, , \end{cases} \end{align}

(16)

with $F_0$ the spin component under consideration, and the bubble inherits the Keldysh structure of the propagators, $\chi_0^{k_4 k_3 k_2 k_1} = G^{k_4 k_1} G^{k_2 k_3}$ . Collecting everything, the three channel contributions to the second-order vertex are

\begin{align} [\chi_0^{0,\overline{ph}}]^{k_4 k_3 k_2 k_1}(\nu, \omega; \mathbf{k}, \mathbf{q}) &= G_0^{k_4 k_1}(\nu, \mathbf{k})\, G_0^{k_2 k_3}(\nu + \omega, \mathbf{k} + \mathbf{q}) \, , \\ [\chi_0^{0,pp}]^{k_4 k_3 k_2 k_1}(\nu, \omega; \mathbf{k}, \mathbf{q}) &= \tfrac{1}{2}\, G_0^{k_4 k_1}(\nu, \mathbf{k})\, G_0^{k_2 k_3}(-\nu - \omega, -\mathbf{k} - \mathbf{q}) \, , \\ [\chi_0^{0,ph}]^{k_4 k_3 k_2 k_1}(\nu, \omega; \mathbf{k}, \mathbf{q}) &= \zeta\, G_0^{k_2 k_1}(\nu, \mathbf{k})\, G_0^{k_4 k_3}(\nu + \omega, \mathbf{k} + \mathbf{q}) \, , \end{align}

(17)

and, with the integrated bubbles $[\chi_0^{0,r}](\omega, \mathbf{q})$ defined as above,

\begin{align} [\Phi^{\overline{ph}}_{(2)}]^{k_1 k_2 k_3 k_4}(\omega, \mathbf{q}) &= F_0^{k_1 k_2 k_5 k_6}\ [\chi_0^{0,\overline{ph}}]^{k_6 k_5 k_8 k_7}(\omega, \mathbf{q})\ F_0^{k_7 k_8 k_3 k_4} \, , \\ [\Phi^{pp}_{(2)}]^{k_1 k_2 k_3 k_4}(\omega, \mathbf{q}) &= F_0^{k_1 k_5 k_3 k_6}\ [\chi_0^{0,pp}]^{k_6 k_8 k_5 k_7}(\omega, \mathbf{q})\ F_0^{k_7 k_2 k_8 k_4} \, , \\ [\Phi^{ph}_{(2)}]^{k_1 k_2 k_3 k_4}(\omega, \mathbf{q}) &= F_0^{k_5 k_2 k_3 k_6}\ [\chi_0^{0,ph}]^{k_8 k_5 k_6 k_7}(\omega, \mathbf{q})\ F_0^{k_1 k_8 k_7 k_4} \, , \end{align}

(18)

with the loop measures

\begin{align} \int_\nu &= \zeta i \int \frac{d\nu}{2\pi} = \int \frac{d\nu}{2\pi i} \quad \text{(fermions, Keldysh)} \, , & \int_{\mathbf{k}} &= \int_{\mathrm{BZ}} \frac{d^d k}{(2\pi)^d} \, . \end{align}

(19)

Second-order self-energy¶

The lowest nontrivial order of the Schwinger-Dyson equation already yields the complete second-order self-energy. Starting from its third form and replacing the full vertex and the full propagators by their bare counterparts, $F \rightarrow F_0$ and $G \rightarrow G_0$ , the dynamic term becomes

\begin{align} \Sigma^{(2)} &= \frac{1}{2}\, G_0 \cdot \Phi^{ph}_{(2)} \, , \end{align}

(20)

i.e. the self-energy closes exactly the $ph$ contribution to the second-order vertex derived above. Writing out the loop product,

\begin{align} \Sigma^{(2)}_{12} &= \frac{1}{2}\, [\Phi^{ph}_{(2)}]_{1 2 \tilde{1} \tilde{2}}\, G_{0,\tilde{2}\tilde{1}} \, . \end{align}

(21)

Spin structure of the closing loop¶

As shown in the section on the spin projection of the loop products, the loop product $[G \cdot X]$ projects the four-point object $X$ onto

\begin{align} [G \cdot X] \ \longrightarrow \ X_{\uparrow\uparrow} + X_{\overline{\uparrow\downarrow}} = \tfrac{1}{2}\left( X_d + 3 X_m \right) \, , \end{align}

(22)

which is a sum over the internal spin of the loop and hence contributes a factor 2 relative to a single spin component. Combined with the $\frac{1}{2}$ of the SDE,

\begin{align} \Sigma^{(2)}_{\sigma\sigma} &= \frac{1}{2}\left( [\Phi^{ph}_{(2)}]_{\uparrow\uparrow} + [\Phi^{ph}_{(2)}]_{\overline{\uparrow\downarrow}} \right) \bullet G_0 = \frac{1}{4}\left( [\Phi^{ph}_{(2)}]_{d} + 3\, [\Phi^{ph}_{(2)}]_{m} \right) \bullet G_0 \, , \end{align}

(23)

the familiar $\frac{1}{4}(d + 3m)$ weighting. Reading the $ph$ row of the table above, $[\Phi^{ph}_{(2)}]_{\uparrow\uparrow} = [\Phi^{ph}_{(2)}]_{\overline{\uparrow\downarrow}}$ and $[\Phi^{ph}_{(2)}]_{d} = [\Phi^{ph}_{(2)}]_{m}$ , so the two terms are equal and

\begin{align} \boxed{\ \Sigma^{(2)}_{\sigma\sigma} = F_{0,\uparrow\downarrow} \bullet [\chi_0^{0}]^{ph} \bullet F_{0,\uparrow\downarrow} \bullet G_0 \ } \, . \end{align}

(24)

Each $\bullet$ here stands for the contraction over the Keldysh indices together with the integration over the internal frequency and momentum, one integration inside the bubble and one for the closing loop; the spin sums have already been carried out. The $\frac{1}{2}$ of the SDE is compensated by the spin sum, so the dynamic second-order self-energy carries net prefactor 1, and it is obtained by inserting the single spin component $F_{0,\uparrow\downarrow} = -U$ at both ends of the bubble.

Important

This is the point at which a factor 2 is easily lost. The two surviving spin components of $\Phi^{ph}_{(2)}$ ,

\begin{align} [\Phi^{ph}_{(2)}]_{\uparrow\uparrow} + [\Phi^{ph}_{(2)}]_{\overline{\uparrow\downarrow}} &= F_{0,\uparrow\downarrow} \bullet [\chi_0^{0}]^{ph} \bullet F_{0,\uparrow\downarrow} + F_{0,\overline{\uparrow\downarrow}} \bullet [\chi_0^{0}]^{ph} \bullet F_{0,\overline{\uparrow\downarrow}} \, , \end{align}

(25)

are equal, so contracting the single scalar $F_0 = F_{0,\uparrow\downarrow} = -U$ accounts for exactly one half of the spin sum.

Frequency parametrization of the closing loop¶

It remains to parametrize the closing loop. It is the second index of $\Sigma$ that carries its arguments, opposite to $G_{21} = G(\nu_2)\delta(\nu_2 + \nu_1)$ , i.e.

\begin{align} \Sigma_{12} &= \Sigma(\nu_2)\, \delta(\nu_1 + \nu_2) \, . \end{align}

(30)

With this convention one finds for a general four-point object $A$ parametrized in the $ph$ channel

\begin{align} [G \cdot A](\nu) &= \int_{\nu'} A^{ph}(\nu', \nu', \nu - \nu')\, G(\nu') \, . \end{align}

(31)

The vertex thus enters with both of its fermionic arguments set to the loop variable, and with transfer variable $\omega = \nu - \nu'$ ; the momenta follow the same structure, with $\mathbf{q} = \mathbf{k} - \mathbf{k}'$ . This is the same result as the frequency parametrization of the loop in the section on the $GW$ approximation; we repeat the derivation here with the index convention for $\Sigma$ made explicit.

That the arguments of $\Sigma$ sit on its second index follows from the Dyson equation, which in multi-index notation reads $G_{21} = G_{0,21} + G_{0,2\tilde{1}} \Sigma_{\tilde{1}\tilde{2}} G_{\tilde{2}1}$ . The two propagators fix $\nu_{\tilde{1}} = -\nu_2$ and $\nu_{\tilde{2}} = -\nu_1 = \nu_2$ , so that $G(\nu_2) = G_0(\nu_2) + G_0(\nu_2) \Sigma(\nu_2) G(\nu_2)$ requires $\Sigma_{\tilde{1}\tilde{2}} = \Sigma(\nu_{\tilde{2}}) \delta(\nu_{\tilde{1}} + \nu_{\tilde{2}})$ .

Using $G_{\tilde{2}\tilde{1}} = G(\nu_{\tilde{2}}) \delta(\nu_{\tilde{2}} + \nu_{\tilde{1}})$ together with $A_{1 2 \tilde{1} \tilde{2}} = A^{ph}(-\nu_{\tilde{1}}, \nu_{\tilde{2}}, \nu_{\tilde{1}} + \nu_2)\, \delta(\nu_1 + \nu_2 + \nu_{\tilde{1}} + \nu_{\tilde{2}})$ , we have

\begin{align} [G \cdot A]_{12} &= A_{1 2 \tilde{1} \tilde{2}}\, G_{\tilde{2}\tilde{1}} \\ &= \int_{\nu_{\tilde{1}}, \nu_{\tilde{2}}} A^{ph}(-\nu_{\tilde{1}}, \nu_{\tilde{2}}, \nu_{\tilde{1}} + \nu_2)\, \delta(\nu_1 + \nu_2 + \nu_{\tilde{1}} + \nu_{\tilde{2}})\, G(\nu_{\tilde{2}})\, \delta(\nu_{\tilde{2}} + \nu_{\tilde{1}}) \\ &= \int_{\nu_{\tilde{2}}} A^{ph}(\nu_{\tilde{2}}, \nu_{\tilde{2}}, \nu_2 - \nu_{\tilde{2}})\, G(\nu_{\tilde{2}})\, \delta(\nu_1 + \nu_2) \, , \end{align}

(32)

where the $\nu_{\tilde{1}}$ integration was performed with $\delta(\nu_{\tilde{2}} + \nu_{\tilde{1}})$ , setting $\nu_{\tilde{1}} = -\nu_{\tilde{2}}$ and reducing the second delta function to $\delta(\nu_1 + \nu_2)$ . Identifying $\nu = \nu_2$ and renaming $\nu_{\tilde{2}} \rightarrow \nu'$ gives the expression quoted above. $\checkmark$

Result in the Keldysh formalism¶

Collecting the spin result, the parametrization, and the Keldysh indices, the dynamic second-order self-energy of the Hubbard interaction is

\begin{align} [\chi_0^{0,ph}]^{k_4 k_3 k_2 k_1}(\nu, \omega; \mathbf{k}, \mathbf{q}) &= \zeta\, G_0^{k_2 k_1}(\nu, \mathbf{k})\, G_0^{k_4 k_3}(\nu + \omega, \mathbf{k} + \mathbf{q}) \, , \\ [\Phi^{ph}_{(2)}]^{k_1 k_2 k_3 k_4}(\omega, \mathbf{q}) &= F_0^{k_5 k_2 k_3 k_6} \left( \int_{\nu'', \mathbf{k}''} [\chi_0^{0,ph}]^{k_8 k_5 k_6 k_7}(\nu'', \omega; \mathbf{k}'', \mathbf{q}) \right) F_0^{k_1 k_8 k_7 k_4} \, , \\ \Sigma^{(2), k_1 k_2}(\nu, \mathbf{k}) &= \int_{\nu', \mathbf{k}'} [\Phi^{ph}_{(2)}]^{k_1 k_2 k_3 k_4}(\nu - \nu', \mathbf{k} - \mathbf{k}')\, G_0^{k_4 k_3}(\nu', \mathbf{k}') \, , \end{align}

(33)

with $F_0^{k_1 k_2 k_3 k_4} = -U/2$ for odd index sum and zero otherwise, and summation over repeated Keldysh indices implied. There are exactly two loops, and hence two factors of $\frac{1}{2\pi i}$ and two factors of $(2\pi)^{-d}$ , and no further numerical prefactor.

Note

Self-consistency at second order means iterating the propagator alone: one replaces $G_0 \rightarrow G$ in the bubble and in the closing loop, and determines $G$ from the Dyson equation, while the vertex stays at second order in $F_0$ . The boxed result above survives this unchanged, because $\Phi^{ph}_{(2),d} = \Phi^{ph}_{(2),m}$ rests only on $F_{0,m} = -F_{0,d}$ and on the two vertices entering quadratically, neither of which involves the propagators.

The SDE becomes exact only when one of the two bare vertices is promoted to the full vertex $F$ , as in the parquet or $GW$ schemes. Then the density and magnetic components no longer coincide and the spin-resolved form

\begin{align} \Sigma^{k_1 k_2}(\nu, \mathbf{k}) &= \frac{1}{4} \int_{\nu', \mathbf{k}'} \left( [\Phi^{ph}_{d}]^{k_1 k_2 k_3 k_4} + 3\, [\Phi^{ph}_{m}]^{k_1 k_2 k_3 k_4} \right)\!(\nu - \nu', \mathbf{k} - \mathbf{k}')\, G^{k_4 k_3}(\nu', \mathbf{k}') \end{align}

(34)

must be used, with $\Phi^{ph}_{d/m} = F_{d/m} \bullet \chi_0^{ph} \bullet F_{0,d/m}$ (the spin table above holds for $A \neq B$ as well). The vertex is then no longer independent of the fermionic variables, and both of its fermionic arguments are set to the loop variables $\nu'$ and $\mathbf{k}'$ , as derived above.

Fourier convolution¶

Both loops evaluated above are convolutions: the bubble ties the two propagators together at a fixed transfer $(\omega, \mathbf{q})$ , and the closing loop of the self-energy probes $\Phi^{ph}_{(2)}$ at $\omega = \nu - \nu'$ and $\mathbf{q} = \mathbf{k} - \mathbf{k}'$ . Each therefore becomes a pointwise product after transforming to time and space, which is what makes the second-order expressions cheap to evaluate numerically: two nested integrations are replaced by a handful of transforms and one multiplication.

Transform conventions¶

We use the Fourier transform of the section on frequency parametrizations, i.e. $e^{-i\nu t}$ from time to frequency and the opposite sign for the momenta,

\begin{align} \mathcal{F}\{f\}(\nu, \mathbf{k}) &\equiv \int dt \sum_{\mathbf{x}} e^{-i \nu t}\, e^{i \mathbf{k}\cdot\mathbf{x}}\, f(t, \mathbf{x}) \, , \\ \mathcal{F}^{-1}\{f\}(t, \mathbf{x}) &\equiv \int \frac{d\nu}{2\pi} \int_{\mathrm{BZ}} \frac{d^d k}{(2\pi)^d}\, e^{i \nu t}\, e^{-i \mathbf{k}\cdot\mathbf{x}}\, f(\nu, \mathbf{k}) \, , \end{align}

(35)

and we write $f(t, \mathbf{x})$ for $\mathcal{F}^{-1}\{f\}(t, \mathbf{x})$ throughout. The two relations that do all the work below are

\begin{align} \int d\nu\, e^{-i\nu(t_1 + t_2)} &= 2\pi\, \delta(t_1 + t_2) \, , & \int_{\mathrm{BZ}} d^d k\, e^{i \mathbf{k}\cdot(\mathbf{x}_1 + \mathbf{x}_2)} &= (2\pi)^d\, \delta_{\mathbf{x}_1, -\mathbf{x}_2} \, . \end{align}

(36)

Bubble¶

Transforming both propagators of the integrated bare $ph$ bubble gives

\begin{align} \boxed{\ [\chi_0^{0,ph}]^{k_4 k_3 k_2 k_1}(\omega, \mathbf{q}) = \frac{\zeta}{i}\, \mathcal{F}\Big\{ G_0^{k_2 k_1}(-t, -\mathbf{x})\, G_0^{k_4 k_3}(t, \mathbf{x}) \Big\}\ } \, , \end{align}

(37)

where, as before, $[\chi_0^{0,ph}](\omega,\mathbf{q})$ denotes the bubble already integrated over its fermionic variables. For fermions $\zeta/i = i$ .

Self-energy¶

The closing loop works the same way. Since it pairs $\Phi^{ph}_{(2)}(\nu - \nu', \mathbf{k} - \mathbf{k}')$ with $G_0(\nu', \mathbf{k}')$ rather than shifting a common variable, the two time arguments come out equal instead of opposite,

\begin{align} \boxed{\ \Sigma^{(2), k_1 k_2}(\nu, \mathbf{k}) = \frac{1}{i}\, \mathcal{F}\Big\{ [\Phi^{ph}_{(2)}]^{k_1 k_2 k_3 k_4}(t, \mathbf{x})\, G_0^{k_4 k_3}(t, \mathbf{x}) \Big\}\ } \, . \end{align}

(40)

Evaluated in this order, one first builds $[\chi_0^{0,ph}](\omega, \mathbf{q})$ , contracts it with the two bare vertices to obtain $\Phi^{ph}_{(2)}(\omega, \mathbf{q})$ , and only then transforms again for the closing loop. That is two transform pairs in total.

Self-energy in one step¶

The second transform pair is avoidable. Since $\mathcal{F}$ and $\mathcal{F}^{-1}$ are inverse to each other and the bare vertices are constants in $(\omega, \mathbf{q})$ , the time-domain vertex follows from the bubble result without ever returning to the transfer variables,

\begin{align} [\Phi^{ph}_{(2)}]^{k_1 k_2 k_3 k_4}(t, \mathbf{x}) &= \frac{\zeta}{i}\, F_0^{k_5 k_2 k_3 k_6}\, G_0^{k_6 k_7}(-t, -\mathbf{x})\, G_0^{k_8 k_5}(t, \mathbf{x})\, F_0^{k_1 k_8 k_7 k_4} \, , \end{align}

(43)

where the Keldysh indices are contracted pointwise in $(t, \mathbf{x})$ . Inserting this into the self-energy and using $\frac{1}{i}\frac{\zeta}{i} = -\zeta$ gives the second-order self-energy as a single transform of a product of three propagators and two bare vertices,

\begin{align} \boxed{\ \Sigma^{(2), k_1 k_2}(\nu, \mathbf{k}) = -\zeta\, \mathcal{F}\Big\{ F_0^{k_5 k_2 k_3 k_6}\, G_0^{k_6 k_7}(-t, -\mathbf{x})\, G_0^{k_8 k_5}(t, \mathbf{x})\, F_0^{k_1 k_8 k_7 k_4}\, G_0^{k_4 k_3}(t, \mathbf{x}) \Big\}\ } \, , \end{align}

(44)

with $-\zeta = +1$ for fermions. Only the three nonzero Keldysh components of $G_0(t, \mathbf{x})$ and their reflections $G_0(-t, -\mathbf{x})$ are needed, so one transform of the propagator and one of the result suffice.

References¶

Wentzell, N., Li, G., Tagliavini, A., Taranto, C., Rohringer, G., Held, K., Toschi, A., & Andergassen, S. (2020). High-frequency asymptotics of the vertex function: Diagrammatic parametrization and algorithmic implementation. Physical Review B, 102(8). 10.1103/physrevb.102.085106