Conservation, crossing symmetry, and completeness in diagrammatic theories

TL;DR

The paper analyzes a fundamental clash in diagrammatic many-body theory: conserving Φ-derivable approximations tend to break crossing symmetry, while parquet approaches aiming for crossing symmetry lose strict conservation when truncated. Using Kraichnan's stochastic Hamiltonian embedding, it separates pairwise stu channels from irreducible, non-pairwise contributions, reformulating the parquet equations within a conserving, Hamiltonian framework. It demonstrates that the exact Luttinger-Ward functional decomposes into a stu part Φ^stu and a complementary Φ^{cmp} plus a redressed term, and derives Kraichnan-parquet equations for the exact ground state, clarifying why simple truncations cannot satisfy both conservation and crossing symmetry. The work also discusses completeness, uniqueness, and two-body consistency, highlighting how averaging over collective indices preserves conservation but breaks crossing symmetry, explaining why construction of a simultaneously conserving and crossing-symmetric truncated theory is impossible.

Abstract

The diagrammatic analysis of interacting particle assemblies harbors a fundamental mismatch between two of its main implementations: Phi-derivable (conserving) approximations and parquet (crossing symmetric) models. No termwise expansion, short of the exact theory itself, can be both conserving and crossing symmetric. This work applies the Kraichnan embedded-Hamiltonian formalism for strongly coupled systems to investigate consistency of the interplay between purely pair-mediated correlations and pair-irreducible ones. The approach sheds a different light on the issue of crossing symmetry versus conservation. In the process, the parquet equations acquire a different formulation.

Figures (10)

• Figure 1: Construction of the Kraichnan Hamiltonian. (a) The exact Hamiltonian is embedded first in an arbitrarily large sum of ${\cal N}$ identical but distinguishable copies, indexed by $n = 1,2, ...{\cal N}$. A Fourier transform over the index generates a collective description. The interaction ${\langle k_1k_2|\overline V|k_3k_4 \rangle}$ is augmented with a parameter $\varphi_{\nu_1\nu_2|\nu_3\nu_4}$ transforming in its Fourier indices $\nu$ exactly as $V$ in its physical indices $k$. (b) The collective Hamiltonian is next embedded in an arbitrarily large sum of ${\cal M}$ topologically identical replicas, except that each now carries a unique set of factors $\varphi$. The extended ${\cal NM}$-sized Hamiltonian remains Hermitian. Setting $\varphi$ to unity recovers the exact physical expectations. When $\varphi$ is tailored to be randomly assigned over the ${\cal M}$-fold ensemble of collective Hamiltonians, a selected subset of correlation diagrams is distinguished by their total product of coupling factors factoring out to unity. On averaging over the stochastic distribution, random phasing suppresses everything else. All canonical relationships valid for the underlying Hamiltonian remain valid in the reduced model.
• Figure 3: (a) Definition of the primary all-order $s, t$ and $u$ interactions. Dark ovals: antisymmetrized potential $\overline V$; linking lines are one-body propagators. Kraichnan couplings from Eq, (\ref{['irr02']}), each selecting for its channel, are shown. In the $s$ channel to leading order, the full Hartree term appears with its Fock exchange; for $t$ and $u$ it is their superposition as an exchange pair that generates the full Hartree-Fock term. (b) Symbolic definition of $\Phi^{\rm stu}$, the LW correlation energy functional (combinatorial weightings pn are understood) following the Kraichnan average over all K couplings according to Eq, (\ref{['irr02']}). Subtraction of two second-order skeletons corrects for threefold overcounting in the $s,t$ and $u$ channels. While the skeleton graphs for $\Phi^{\rm stu}$ appear simple, their complexity lies in the selfconsistent nesting of self-energy insertions in the one-body propagators. The $stu$ topology is fully revealed only when the response to an external probe is extracted. Universality of the LW functional potthofflin1 means that the topology of its constitutive diagrams in (b) is unaltered in moving the interaction from $V$ to $\overline V$ when the Hamiltonian is itself invariant with respect to exchange. This does not affect the overall accounting.
• Figure 4: Index convention for a Kraichnan vertex, associating with it the nominal K coupling $\phi_{\nu_1\nu_2|\nu_3\nu_4}$. A response term results when $G$ lines are attached left and right and join at a perturbation node. The diagram contributes if and only if the internal sum of coupling phases cancels on connection to the effective K coupling $\phi_{\nu_1\nu_2|\nu_3\nu_4}\rightarrow \varphi_{\nu_1\nu_2|\nu_2\nu_1} \equiv 1$ as in Eq. (\ref{['irr02.1']}). This construct manifests the same closed topology previously implicit in the unitary structure of $\Phi$.
• Figure 5: Systematic removal of a propagator $G$ internal to the self-energy $\Sigma[\varphi\overline V; G] = \Lambda\!:\!G$ after Kadanoff and Baym kb1kb2. This generates the primary $stu$ scattering kernel $\Lambda' = \delta^2 \Phi^{\rm stu}/\delta G \delta G$. Removing $G(32)$, solid line, simply regenerates $\Lambda$. Removing any internal $G$ lines (dotted) other than $G(32)$ yields the additional vertices required by microscopic conservation. (a) Beyond the $s$-channel ladder $\Lambda_s$ the noncrossing symmetric $t$-like term $\Lambda_{s;t}$ and $u$-term $\Lambda_{s;u}$ are generated. (b) Generation of $\Lambda_t$ and the nonsymmetric $\Lambda_{t;s}$, $\Lambda_{t;u}$. (c) Generation of $\Lambda_u$ with $\Lambda_{u;t}$ and $\Lambda_{u;s}$. No diagrammatic structure emerges that is not already incorporated recursively in the propagators $G$ within $\Phi$. Note that $\Lambda' \!-\! \Lambda$ starts at third order in $\overline V$.
• Figure 6: Recursive construction of response kernel within $\Phi$ derivability. Dots: external perturbation nodes. (a) Two contributions A and B to the particle-hole response function combine into (b), a new contribution. Graphs (c) and (d) show a more complex combination with a third allowed contribution C. In the response description Eq. (\ref{['kII18.2']}), the system cannot tell a direct perturbation from one that is induced, so a perturbation node may be freely replaced with an induced perturbation. Fusion of the response terms produces a new contribution to the total. In the process the internal topology of the resultant response, virtual within the renormalization of $\Phi$, becomes manifest. All the kernel parts in Fig. \ref{['F5']} are recursively convolved in this way.