Do the Matsubara two-particle irreducible vertex divergences persist on the real-frequency axis?

In the Matsubara (imaginary-frequency) formalism, the two-particle irreducible (2PI) vertex of the half-filled Hubbard model - obtained by inverting the generalized susceptibility channel by channel - diverges along lines in the interaction-temperature plane. These divergences are tied to zero-crossings of susceptibility eigenvalues and to the multivaluedness of the Luttinger-Ward functional, and the zero-temperature Mott endpoint is an accumulation point of infinitely many divergence lines. Kugler, Lee and von Delft [Phys. Rev. X 11, 041006 (2021)] derived spectral representations that make multipoint correlation functions and vertices computable directly on the real-frequency axis, in the zero-temperature and Keldysh formalisms, and posed the question whether the Matsubara divergences persist for real frequencies - or whether they disappear or transform into a different singularity structure. Later work by the same groups resolved adjacent items of that paper's outlook (generalized fluctuation-dissipation relations for Keldysh multipoint correlators; symmetric estimators yielding all Keldysh components of the four-point vertex) and built real-frequency vertex-computation infrastructure; the perturbative fRG- and parquet-based computations reach only weak-to-moderate coupling, where no divergences occur, and the existing nonperturbative real-frequency four-point data have not been analyzed for divergences. No published study determines the fate of the divergences on the real axis. This problem asks for that determination - analytically for the Hubbard atom and/or numerically for the single-impurity Anderson model or a dynamical mean-field theory solution at parameters where the Matsubara vertex is known to diverge - together with the characterization of whatever singularity structure replaces or realizes the Matsubara divergences.

Background

Two-particle irreducible vertices and their divergences. In the Matsubara (imaginary-frequency) formulation of finite-temperature many-body theory, the four-point correlation function of fermions - the generalized susceptibility $\chi$ - is related to the two-particle irreducible (2PI) vertex $\gamma$ by a matrix inversion carried out separately in each channel (particle-hole, crossed particle-hole, particle-particle). The 2PI vertex acts as an effective interaction: it is the input to parquet and functional renormalization group (fRG) treatments and to the diagrammatic extensions of dynamical mean-field theory (DMFT, the self-consistent impurity description of local correlations in the Hubbard model, whose impurity problem is the single-impurity Anderson model, AIM). Since 2013 it is known that $\gamma$ of the half-filled Hubbard model diverges along lines in the interaction-temperature plane: the divergences appear as precursors of the Mott metal-insulator transition (Schäfer et al., Phys. Rev. Lett. 2013), form a landscape of divergence lines around the critical endpoint (Schäfer et al., Phys. Rev. B 2016), occur in the AIM and in the Hubbard atom, i.e. the zero-hybridization local limit of both models, where they can be analyzed fully analytically (Chalupa et al. 2018; Thunström et al. 2018), and multiply without bound as the temperature is lowered, making the zero-temperature Mott endpoint an accumulation point of infinitely many divergence lines (Pelz et al. 2023). Mathematically, each divergence occurs when an eigenvalue of $\chi$, viewed as a matrix in the discrete fermionic Matsubara frequencies, crosses zero, so that the inversion defining $\gamma$ becomes singular. The phenomenon has been tied to the nonexistence and multivaluedness of the Luttinger-Ward functional (the generating functional of skeleton diagrams, whose functional derivative gives the self-energy) and hence to the possible misleading convergence of self-consistent diagrammatic expansions (Kozik et al. 2015; Vučičević et al. 2018).

Real-frequency formalisms. Spectral and dynamical observables require the same objects on the real-frequency axis. The zero-temperature formalism (ZF) works directly with real frequencies. The Keldysh formalism (KF) treats real times on a two-branched contour; in the Keldysh basis every correlator becomes a matrix whose entries, labeled by Keldysh indices 1 or 2 on each external leg, encode response and fluctuation content. Kugler, Lee and von Delft [Phys. Rev. X 11, 041006 (2021)] derived general spectral (Lehmann-type) representations for $l$-point correlation and vertex functions in the Matsubara, ZF and Keldysh formalisms, built from formalism-independent partial spectral functions convolved with formalism-specific kernels; together with the companion paper on the multipoint numerical renormalization group (NRG, an impurity solver that yields real-frequency data without analytic continuation), this makes local four-point functions of quantum impurity models computable in all three formalisms. They showed that the fully retarded Keldysh components follow from Matsubara functions by a direct analytic continuation $i\omega \to \omega + i0^+$, whereas components carrying two or more Keldysh indices equal to 2 involve $\delta$-function terms and possess no region of analyticity in frequency space.

The open question. In the Matsubara formalism the divergences of $\gamma$ are singularities of a matrix inversion over discrete frequencies. On the real-frequency axis the same inversion acts on functions of continuous frequencies with an additional Keldysh matrix structure, and the question - raised in the Outlook of that paper, which asks whether the divergences of the 2PI vertex persist for real frequencies - is whether they persist, disappear, or turn into a different singularity structure (poles, branch points, $\delta$-function terms). The answer matters on two counts: it would clarify whether the vertex divergences and the associated Luttinger-Ward multivaluedness are physical singularities of the correlation structure or representation-specific artifacts of the imaginary-frequency description; and the real-frequency local vertex is the natural input for real-frequency diagrammatic extensions of DMFT (such as a Keldysh fRG or a real-frequency ladder dynamical vertex approximation), so knowing whether and where this input diverges determines the stability of such schemes.

Problem Statement

Determine the fate on the real-frequency axis of the two-particle irreducible (2PI) vertex divergences established in the Matsubara formalism. Using the spectral representations of multipoint correlation functions (or any equivalent route), compute the 2PI vertex directly in the real-frequency zero-temperature formalism and/or in the Keldysh formalism for a model and parameter values where the Matsubara 2PI vertex is known to diverge - the Hubbard atom at its analytically known divergence loci, and/or the single-impurity Anderson model or a dynamical mean-field theory (DMFT) solution of the half-filled Hubbard model at parameters on a documented Matsubara divergence line - and establish: (i) whether the real-frequency 2PI vertex diverges at the corresponding points, stated per Keldysh component where applicable, with the divergence criterion made explicit (pole location, or unbounded growth under frequency-grid refinement and broadening reduction); and (ii) if it does not diverge, what replaces the Matsubara divergences - finite values, branch-point singularities at continuum thresholds, or delta-function structures. The comparison to the Matsubara results must be made at matched model parameters, with the Matsubara divergence locus at those parameters documented by citation or recomputation.

References

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