The GW approximation

The $GW$ approximation is one of the oldest and most widely used approximations for the self-energy of an interacting many-body system. Its name refers to its defining structure: the self-energy is given by the product of a propagator $G$ and a screened interaction $W$. In the language of the single boson exchange (SBE) decomposition, it arises very naturally as the approximation in which the Hedin vertices are replaced by their lowest-order contribution, $\gamma^r \simeq \mathbf{1}^r$ and $\overline{\gamma}^r \simeq \mathbf{1}^r$.

The purpose of this page is to spell out what this means concretely, in the conventions used throughout this documentation. This is worth doing carefully, because “$GW$” is used for a whole family of related but inequivalent approximations in the literature. Two choices in particular have to be made explicit, and neither of them is visible at second order in the bare interaction:

1. Which two-particle channel is resummed. The Schwinger-Dyson equation (SDE) can be written in three ways, one per channel, which are equivalent for the exact vertex $F$ but not for a truncated one. As shown below, the two particle-hole forms turn out to yield the same $GW$ self-energy, whereas the particle-particle form yields a genuinely different approximation.

2. How the spin structure is handled. For SU(2)-symmetric systems, the density and magnetic ladders in the $ph$ channel differ already at third order in $F_0$, and the self-energy picks them up with different weights.

We work in the Matsubara formalism throughout, and we specialize 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. This is the case in which the SBE machinery simplifies most drastically, and it is the case relevant for most applications.

Spin structure of the bare interaction¶

We first record the spin components of the antisymmetrized bare vertex of the Hubbard model, which were derived in the Hubbard model example of the section on the starting point,

where we suppressed the (trivial) frequency and momentum dependence. Reading off the three independent spin components defined in the section on spin parametrizations gives

and hence, in the density/magnetic and singlet/triplet bases,

We will also need the spin components of the $ph$ channel unit. Two related objects appear in what follows: the channel identity operator $\mathbb{1}^{ph}_{1234} = \delta_{1,2}\delta_{3,4}$, which is the unit with respect to the full connector ∘, and the unit vertex $\mathbf{1}^{ph}$, which is the unit with respect to $\bullet$, i.e. with respect to all degrees of freedom except frequency and momentum. Both carry the same spin part, $\delta_{\sigma_1, \sigma_2}\delta_{\sigma_3, \sigma_4}$, so that

From here on we consistently use $\mathbf{1}^{ph}$, since that is the object appearing in the SBE equations.

Spin projection of the loop products¶

The self-energy is obtained from a four-point object by closing two of its legs with a propagator. As discussed in the section on the Schwinger-Dyson equation, there are two such loop products,

Because the propagator is diagonal in spin, $G_{\tilde{2}\tilde{1}} \sim \delta_{\tilde{\sigma}_2, \tilde{\sigma}_1}$, and because the self-energy is diagonal in spin as well, each loop product projects the four-point object $X$ onto one specific spin combination. These two combinations are different:

The two projections look inequivalent, but for a crossing-symmetric $X$ they are not: the identity $[X\cdot G] = \zeta [G \cdot X]$ quoted in the section on the Schwinger-Dyson equation follows from $X_{1 \tilde{2} \tilde{1} 2} = \zeta X_{1 2 \tilde{1} \tilde{2}}$. For the bare vertex, which is crossing symmetric and frequency independent, we can check this on the spin components directly:

Polarization and screened interaction in the $ph$ channel¶

We now specialize to the $ph$ channel, which is the channel in which the density and magnetic fluctuations live, as is evident from its interpretation in terms of the density-density correlator discussed in the section on two-particle channels.

The $GW$ approximation consists of replacing the Hedin vertices in the SBE equations by the unit vertex,

Everything else follows. The polarization then reduces to the bare bubble,

Its spin structure is particularly simple: inserting the unit vertices into the $ph$ contraction gives $P^{ph}_{1234} = \delta_{\sigma_1, \sigma_2}\delta_{\sigma_3, \sigma_4} P^{ph}$, i.e. the polarization is proportional to the $ph$ unit vertex in spin space, with the spin-independent scalar

With the polarization at hand, the screened interaction follows from the bosonic Dyson equation,

Since $P^{ph}$ is proportional to the $ph$ unit vertex in spin space, the $\bullet$ contraction carries the same spin algebra as the ∘ contraction. That algebra is worked out explicitly in the section on spin parametrizations — the spin sums there never touch the frequency arguments, which is precisely why the result is already stated with $\bullet$ between the two factors — and it decouples in the density/magnetic basis. The corresponding spin table for all three channels is collected in the section on second-order perturbation theory. The bosonic Dyson equation therefore reduces to two independent scalar equations,

with the closed-form solutions

It is often convenient to split off the bare interaction and work with the decaying part of the screened interaction, which we denote by a tilde,

where we identified the susceptibilities in the two channels,

The $GW$ self-energy¶

We start from the $ph$ form of the Schwinger-Dyson equation,

and use the exact SBE identity $F \circ \chi_0^{r} \circ F_0 = \overline{\gamma}^{r} \bullet W^{r} - F_0$ derived in the section on the SDE in SBE form. Setting $\overline{\gamma}^{ph} \simeq \mathbf{1}^{ph}$ gives $F \circ \chi_0^{ph} \circ F_0 \simeq W^{ph} - F_0$, and therefore

which is precisely the third line of the SBE form of the SDE with the Hedin vertex set to unity. Projecting onto spin components with $[G \cdot X] \rightarrow \tfrac{1}{2}(X_d + 3X_m)$, the factor $\tfrac{1}{2}$ combines with the $\tfrac{1}{2}$ of the SDE into the weights $\tfrac{1}{4}$ and $\tfrac{3}{4}$:

where the spin sum has been carried out, so that the remaining $\cdot$ denotes only the frequency and momentum contraction of the loop.

Frequency parametrization¶

The loop closes the two legs of the screened interaction that carry the bosonic transfer frequency of the $ph$ channel. Writing out the frequency structure gives

Finally, splitting off the static contribution and using $\widetilde{W}_{d/m} = U^2 \chi_{d/m}$ yields the form in which the $GW$ self-energy is most easily recognized,

with the density per spin $n = \int_\nu G(\nu) e^{i\nu 0^+}$.

Limits and checks¶

Reduction to second-order perturbation theory¶

Expanding the screened interactions to leading order, $\widetilde{W}_{d/m} = U^2 P^{ph} + \mathcal{O}(U^3)$, both spin channels contribute identically, and the weights add up to one. For the dynamic part of the self-energy we thus obtain

Written out, and using $\zeta = -1$,

which is the familiar second-order (dynamic) self-energy of the Hubbard model, with the static Hartree term $Un$ accounted for separately.

Stoner criterion¶

The magnetic screened interaction $W_m = U / (1 - U P^{ph})$ diverges when $U P^{ph}(\omega) = 1$, while the density one, $W_d = -U/(1 + U P^{ph})$, stays finite for $P^{ph} > 0$. At $\omega = 0$ and for fermions,

which, if evaluated with the bare propagator, equals $\partial n / \partial \mu$ and is positive. The divergence condition then becomes $U \, \partial n / \partial\mu = 1$, which is the Stoner criterion for a magnetic instability. That the instability appears in the magnetic and not in the density channel is a direct consequence of the relative sign $F_{0,m} = -F_{0,d}$, and is a useful sanity check on the signs above.

Comparison with the literature¶

The result agrees with the $GW$ equations for the Hubbard atom given in Sec. 4.2 of Kiese et al., SciPost Phys. Codebases 24 (2024), who write

Their polarization does not carry the closed-loop sign, so $P = -P^{ph}$, and consequently their screened interactions are related to ours by an overall sign,

With $\tfrac{1}{4}F_{0,d} + \tfrac{3}{4}F_{0,m} = U/2$ accounting for their explicit $Un/2$, the two expressions are identical term by term.

Closed-form ladder resummation with additional indices¶

Everything above was derived for objects that are scalars in every degree of freedom other than spin and the bosonic transfer variable, as announced in the scope admonition at the top of this page. Under that assumption the bosonic Dyson equation decoupled into two scalar equations and could be solved by ordinary division. This section removes the restriction. It keeps a finite set of additional indices on the two-particle objects (Keldysh indices in the application we have in mind, although the algebra only uses that they are finite in number, so orbital indices are covered by the same argument), and shows that the density and magnetic ladders still sum in closed form, with the scalar division replaced by a matrix inversion at every $(\omega, \mathbf{q})$.

Two properties of the local Hubbard interaction survive the addition of these indices, and they are what makes the closed form possible:

1. The bare vertex $F_0$ is independent of frequency and momentum.

2. Every rung of the $ph$ ladder carries the same bosonic transfer variable $(\omega, \mathbf{q})$. Inserting one more bubble is therefore an elementwise product in $(\omega, \mathbf{q})$, not a new integration.

Together these mean that at each fixed $(\omega, \mathbf{q})$ the ladder recursion is a linear map on a finite-dimensional space, and the ladder itself is a geometric series in that map.

The rung as a linear map¶

Let $k_1 k_2 k_3 k_4$ collect the additional indices, each running over a finite range, and let $P^{ph}_{k_1 k_2 k_3 k_4}(\omega,\mathbf{q})$ denote the integrated $ph$ bubble carrying them, the index-resolved generalization of the polarization $P^{ph}(\omega)$ of the section above, to which it reduces when the indices are dropped. Write the $ph$-reducible vertex with $\ell$ bubble insertions in spin channel $c \in \{d, m\}$ as $\Phi^{(\ell)}_{c; k_1 k_2 k_3 k_4}(\omega, \mathbf{q})$; the parenthesized superscript counts bubble insertions and is not the channel label of the $\Phi^r$ used elsewhere in this documentation.

The additional indices carry no spin label, because the spin projection has already been carried out: the bare vertex entering here is one of the two components $F_{0,d} = -U$ and $F_{0,m} = +U$. Inserting one further bubble between a bare vertex on the left and $\Phi_c^{(\ell)}$ on the right therefore defines one linear map per spin channel,

where every factor is evaluated at the same $(\omega, \mathbf{q})$, which is therefore suppressed. Since $F_{0,m} = -F_{0,d}$ for a local interaction, the two maps differ only by an overall sign,

so a single map suffices, and we abbreviate $M \equiv M_d$ from here on. The ladder recursion is then simply

with $\Phi_c^{(1)} = F_{0,c} \bullet P^{ph} \bullet F_{0,c}$ the second-order vertex. Only the square of the bare vertex enters at first order, so $\Phi_m^{(1)} = \Phi_d^{(1)}$, and we write $\Phi^{(1)}$ for their common value. Note that applying $M_c$ is itself one more $\bullet$ contraction, $M_c X = F_{0,c} \bullet P^{ph} \bullet X$, and that this contraction is associative: chains of it need no bracketing, and the resolvents below may be manipulated exactly like matrix inverses.

The geometric series¶

Because $\Phi_c^{(\ell)} = M_c^{\ell-1}\Phi^{(1)}$ with $M_m = -M$, the magnetic ladder is the alternating counterpart of the density one,

which is where the relative sign between the two channels enters. The two channel sums defined above therefore become geometric series in the single map $M$,

and the $GW$ vertex is the weighted combination

This is the announced replacement of the scalar division by a matrix inversion. It is a pointwise statement: at each $(\omega, \mathbf{q})$ one inverts a single finite matrix, and no integration couples different transfer variables. The cost is therefore independent of how many ladder orders would have been needed to reach the same accuracy by direct summation.

The static offsets¶

The objects above are the dynamic parts, marked by the tilde exactly as in the scalar case: the static contributions $F_{0,d} = -U$ and $F_{0,m} = +U$ are deliberately not included. They combine into the Hartree term, which is conventionally absorbed into the propagator, and leaving them out keeps the resummed objects decaying at large transfer frequency. Restoring them, if wanted, is the substitution $W_{d/m} = \widetilde{W}_{d/m} + F_{0,d/m}$.

Convergence and the matrix Stoner criterion¶

The geometric series converges if and only if the spectral radius of the rung, meaning the largest modulus among its eigenvalues $\lambda_i$,

stays below one at every $(\omega, \mathbf{q})$. The closed forms, by contrast, are defined wherever $\mathbb{1} \mp M$ is invertible, which is a strictly weaker condition. This is the same relationship as between the scalar geometric series in $U P^{ph}$ and the closed-form $W_{d/m}$ noted in the section on $G_0 W_0$ and self-consistent $GW$ at the end of this page. The condition $\rho(M) < 1$ is the matrix form of the Stoner criterion: stability of the ladder is a statement about the leading eigenvalue of $M$, not about a scalar exceeding one.

Susceptibilities¶

The susceptibility in channel $c$ is the same $ph$ ladder that produced $\widetilde{W}_c$, except that it begins and ends with a bubble instead of with a bare vertex,

Each further order prepends one more factor $P^{ph} \bullet F_{0,c}$, so the term with $j$ interaction lines is $\big(P^{ph} \bullet F_{0,c}\big)^{j} P^{ph}$. This is a geometric series of exactly the kind summed above, and it closes the same way,

with the density and magnetic cases differing only through $F_{0,m} = -F_{0,d}$. Setting all additional indices to a single value gives $\chi_d \rightarrow P^{ph}/(1 + U P^{ph})$ and $\chi_m \rightarrow P^{ph}/(1 - U P^{ph})$, the two susceptibilities of the scalar case. $\checkmark$

The rung of this series carries the same two factors as $M_c$ but in the opposite order, a bubble followed by a vertex rather than a vertex followed by a bubble, simply because the ladder now starts at a bubble. Written out in the additional indices it is the formula for $M_c$ with the two factors exchanged,

again at fixed $(\omega, \mathbf{q})$. The indices $k_1$ and $k_4$ are spectators here too, so the block structure noted for $M_c$ carries over unchanged. Dropping the indices collapses both orders to the same product $F_{0,c} P^{ph}$, which is why a single map sufficed in the scalar case.

These are RPA susceptibilities in the strict sense only if $P^{ph}$ is built from bare propagators. The same caveat as in the scalar case applies here verbatim.

Which form of the Schwinger-Dyson equation?¶

The SDE can be written in three equivalent ways, and the SBE identities turn each of them into a $GW$-like expression once the Hedin vertices are set to unity,

For the exact vertex the three forms of the SDE are identical. The three expressions above, however, are not related by an exact identity: each of them involves the replacement $\gamma^r \simeq \mathbf{1}^r$ made in a specific channel, and discarding $T^r \circ \chi_0^r \circ \mathbf{1}^r$ is a different approximation for different $r$. So the question of which form to start from is a real one.

It has a partly reassuring answer:

• the $ph$ form resums density and magnetic fluctuations, and is the one derived above;

• the $\overline{ph}$ form resums the crossed particle-hole ladder. It decouples not in the density/magnetic basis but in the combinations $X_\pm = X_{\uparrow\uparrow} \pm X_{\overline{\uparrow\downarrow}}$, for which the bare interaction takes the values $F_{0,+} = +U$ and $F_{0,-} = -U$. Nevertheless it gives exactly the same self-energy as the $ph$ form, to all orders;

• the $pp$ form resums the particle-particle ladder, i.e. it is a $T$-matrix approximation. In the singlet/triplet basis the bare interaction is $F_{0,s} = -2U$ and $F_{0,t}=0$, so only the singlet channel is screened. It agrees with the other two at $\mathcal{O}(U^2)$, as it must, but differs from $\mathcal{O}(U^3)$ onwards.