Luttinger–Ward Functionals and Φ-Derivable Self-Energies

A Φ-derivable approximation begins from a functional of the full propagator whose derivative is the self-energy. When the selected skeleton functional respects a continuous symmetry and Dyson’s equation is solved self-consistently, the approximation obeys the associated macroscopic conservation laws. This construction is powerful, but it does not by itself guarantee crossing symmetry, positivity, uniqueness, or small quantitative error. The theorem and its stationarity hypotheses are stated in Baym 1962, pp. 1393–1397.

Required background. Dyson Equations and Self-Energy defines the dressed propagator, Current Vertices and Ward-Consistent Response defines the compatible response kernel, and Connected Correlators and Cumulants supplies the generating-functional relations.

The Luttinger–Ward functional

For a specified interaction and regularization, let $\Phi[G]$ be a sum of closed, two-particle-irreducible skeleton diagrams built with full propagators and bare interaction vertices. Choose the derivative convention

$$\Sigma(1,2)=\frac{\delta\Phi[G]}{\delta G(2,1)}.$$

A corresponding grand-potential functional can be written schematically as

$$\Omega[G] =\Omega_0-\operatorname{Tr}\ln(-G^{-1}) -\operatorname{Tr}\bigl[(G_0^{-1}-G^{-1})G\bigr] +\Phi[G],$$

with statistics-dependent signs and additive constants fixed by convention. Its stationary condition $\delta\Omega/\delta G=0$ gives Dyson’s equation. The stationary property suppresses first-order changes in $\Omega$ under a small propagator error, but not errors caused by omitting skeletons from $\Phi$.

A Hartree example

For two fermion species with contact interaction $U n_\uparrow n_\downarrow$, retain the first closed skeleton,

$$\Phi_H=U\int_x G_\uparrow(x,x^+)G_\downarrow(x,x^+).$$

Functional differentiation gives

$$\Sigma_\uparrow^H=U n_\downarrow, \qquad \Sigma_\downarrow^H=U n_\uparrow.$$

Solving Dyson’s equation with these densities is self-consistent Hartree theory. Evaluating the same formula once with an unrelated reference density is not the stationary Φ-derivable solution.

At higher order, retaining a selected skeleton set produces self-consistent Born, $T$-matrix, or GW-like approximations depending on the interaction organization. The names do not replace an explicit list of skeletons and fields.

Why conservation follows

Under a local phase transformation of $G$, an invariant $\Phi$ has zero variation. Inserting $\Sigma=\delta\Phi/\delta G$ turns that identity into the collision-term cancellation underlying the continuity equation. Space-time translations similarly yield energy and momentum conservation when the regulator and interactions preserve them. The argument requires:

1. a symmetry-invariant selected functional;
2. full self-consistency on a physical solution branch; and
3. a response vertex generated from the same kernel $\delta\Sigma/\delta G$.

Breaking any of these conditions can break the corresponding identity.

Current Vertices and Ward-Consistent Response develops the corresponding response construction in detail.

Scope of the guarantee

Φ derivability guarantees the conservation consequences of the retained symmetry, not all exact properties. A truncation can violate crossing symmetry because it treats two-particle channels unequally. Some self-consistent approximations can produce unphysical spectral features or multiple solutions. Formal functionals may also be non-single-valued in strongly correlated regimes, so branch selection and comparison with direct observables remain necessary.

Renormalized contact theories add another condition: counterterms and functional differentiation must be implemented in the same regulator. A formally conserved but cutoff-dependent result is not predictive.

Common pitfalls

Using a skeleton self-energy without self-consistency. The conservation proof applies at the stationary solution, not to an arbitrary one-shot evaluation.

Computing response with a bare vertex. The compatible kernel is $\delta\Sigma/\delta G$; omitting it discards part of the construction.

Treating conservation as an accuracy estimate. Exact particle number can coexist with a poor spectrum or equation of state.

Exercises

Differentiate the Hartree functional

Use the displayed $\Phi_H$ to obtain $\Sigma_\uparrow$.

Solution

Varying $G_\uparrow$ opens its closed line and leaves $U G_\downarrow(x,x^+)$. With $n_\downarrow=G_\downarrow(x,x^+)$ in the adopted imaginary-time convention, $\Sigma_\uparrow^H=Un_\downarrow$. Alternative sign conventions move a minus sign into the definition of equal-time $G$ but give the same physical Hartree shift.

Test stationarity

If $G_\star$ is stationary, how does $\Omega[G_\star+\delta G]-\Omega[G_\star]$ begin in $\delta G$?

Solution

The linear term is $\int(\delta\Omega/\delta G)_{G_\star}\delta G=0$. The leading change is quadratic, provided the functional is differentiable on that branch. This does not say that the stationary value is close to the exact one when diagrams are omitted.

Continue

Baym–Kadanoff Conservation and Validity turns the formal construction into a practical set of independent checks. Hedin Equations, Screened Interactions, and GW gives a central application. Spectral Moments and Many-Body Sum Rules tests exact constraints beyond stationarity.

References

• Baym, Gordon. “Self-Consistent Approximations in Many-Body Systems.” Physical Review 127 (1962): 1391–1401. DOI.

Further reading

• Baym, Gordon, and Leo P. Kadanoff. “Conservation Laws and Correlation Functions.” Physical Review 124 (1961): 287–299. DOI.
• Luttinger, J. M., and J. C. Ward. “Ground-State Energy of a Many-Fermion System. II.” Physical Review 118 (1960): 1417–1427. DOI.
• Potthoff, Michael. “Non-Perturbative Construction of the Luttinger–Ward Functional.” Condensed Matter Physics 9 (2006): 557–567. DOI.