Improved treatment of fermion-boson vertices and Bethe-Salpeter equations in non-local extensions of dynamical mean field theory

A. Katanin

REVIEW 3 major objections 4 minor 37 references

Improved treatment of fermion-boson vertices and Bethe-Salpeter equations in non-local extensions of dynamical mean field theory

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper derives analytic formulas that remove most of the finite-frequency-box error in non-local dynamical mean-field theory, improving the truncation error from O(1/ν_max) to O(1/ν_max^4).

desk verdict A genuinely useful, honestly-scoped methodological advance: analytic out-of-box corrections for non-local DMFT vertices, with the advertised O(1/ν^4) scaling resting on an unproven but plausible decay assumption. read the letter →

arxiv 1908.03198 v2 pith:RFYEWLBK submitted 2019-08-08 cond-mat.str-el

A. Katanin This is my paper · ORCID

classification cond-mat.str-el

keywords fermion-bosonvertexBethe-Salpeterequationdynamicalmean-fieldtheorynon-localcorrelationsfrequencyboxasymptoticstwo-particleirreducibleHubbardmodel

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that, in non-local extensions of dynamical mean-field theory, the error caused by truncating the Matsubara frequency grid to a finite box can be reduced from a slow $1/\nu_{\max}$ decay to a much faster $1/\nu_{\max}^4$ decay. The author derives closed formulas that analytically account for all contributions of frequencies outside the box, so only the numerically exact vertices inside a small box are needed. These formulas express the full fermion-boson vertices, susceptibilities, and the two-particle irreducible vertex in terms of two correction functions, $X_q$ and $Z_{\nu q}$, built from the known high-frequency tail of the vertex. The central identity, Eq. (25), relates the physical two-particle irreducible vertex to the vertex obtained by inverting the Bethe-Salpeter equation inside the box. This makes finite-box artifacts nearly negligible and allows much smaller frequency boxes in practice.

What carries the argument

The analytic tail correction $X_q$ and the vertex correction factor $Z_{\nu q}$ are the objects that carry the argument. $X_q$ is the sum over frequencies outside the box of the bare bubble plus one insertion of the asymptotic vertex tail; $Z_{\nu q}$ measures how much the three-leg vertex is renormalized by the tail outside the box. Both are computed from the asymptotic form (3)-(4), which uses only local charge, spin, and particle-particle susceptibilities, so they are cheap to evaluate for arbitrarily large frequencies. Eq. (25) is the load-bearing identity: $\Phi_{\nu\nu'q} = \Phi^{\mathrm{box}}_{\nu\nu'q} + U_q - Z_{\nu q}\tilde U_q Z_{\nu'q}$, with $\tilde U_q = U_q/(1-U_q X_q)$. This identity removes the $1/\nu_{\max}$ error and leaves the smaller $1/\nu_{\max}^4$ error.

What would settle it

Take a model in which the asymptotic form is known to fail, such as a non-ladder vertex with retarded non-local interactions, evaluate the full two-particle irreducible vertex on a very large frequency box, and compare the left and right sides of Eq. (25) as the box size grows. If the difference does not shrink like $1/\nu_{\max}^4$ — or if the correction pushes the vertex away from the exact large-box value — the assumed tail is the culprit.

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Extended reading notes

Core claim

The central claim is that the out-of-box frequency contributions, which previously required numerical treatment of the asymptotic tail in a larger box, can be evaluated analytically to leading order. Starting from the asymptotic form (3) of the two-particle irreducible vertex, the paper defines $X_q$ and $Z_{\nu q}$ and derives Eqs. (8), (11), (14), (18), and (25) for the full fermion-boson vertex, the non-local susceptibility, the reduced fermion-boson vertex, the irreducible susceptibility, and the two-particle irreducible vertex, respectively. The relation $\Phi_{\nu\nu'q} = \Phi^{\mathrm{box}}_{\nu\nu'q} + U_q - Z_{\nu q}\tilde U_q Z_{\nu'q}$ is the key identity: it converts a matrix inversion performed only inside the box into the physical vertex. In the companion numerical test on the two-dimensional Hubbard model, the corrected vertices change very little with box size, while the uncorrected ones extrapolate slowly; the corrected error is consistent with $1/\nu_{\max}^4$ scaling.

Load-bearing premise

The whole scheme depends on the high-frequency asymptotic form (3) of the two-particle irreducible vertex, together with the assumption that deviations from this form decay at least as $1/\max(|\nu|,|\nu'|)^3$; if that tail is wrong or decays too slowly, the size of the correction terms $X_q$ and $Z_{\nu q}$ would be misestimated.

Editorial extensions

If this is right

A single calculation with a small frequency box should suffice for ladder-type non-local DMFT calculations, instead of running several box sizes and extrapolating.
The method gives a direct relation between the local and non-local two-particle irreducible vertices, making it easier to feed local input into non-local Bethe-Salpeter equations.
The same formulas apply to both charge and spin channels, and they can be adapted to cases where the vertex depends on more than one momentum transfer.
The claimed accuracy means the main remaining error is set by the physical decay of the irreducible vertex toward its asymptotic form, not by the truncation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the same asymptotic form holds in the particle-particle channel, an analogous set of box corrections could be derived for pairing susceptibilities, where the tail is built from the same local susceptibilities.
The quartic improvement is conditional on the decay of the deviation from Eq. (3); in models with long-range or retarded interactions the effective exponent would be lower, and the size of $X_q$ and $Z_{\nu q}$ would reveal that.
The construction suggests a general recipe for any truncated Matsubara summation: add back the analytically known tail before inverting, rather than enlarging the box.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

This paper gives closed-form analytic corrections for finite-frequency-box truncation in non-local extensions of DMFT. What is actually new: instead of numerically treating vertex asymptotics in a large outer box, as in the box-splitting schemes of Kunes and Tagliavini et al., Katanin derives formulas involving X_q and Z_q so that only the small box needs numerical treatment. The core relation, Eq. (25), connecting the physical 2PI vertex to the box vertex via U_q − Zν Ũ_q Zν′, is the useful piece. The derivation is parameter-free, and the paper is honest that Eq. (25) can be obtained from Method 2 of Ref. [33]; the new contribution is making the correction analytic and cheap.

What it does well: the algebra is internally consistent, and the numerical tests on the 2D Hubbard model show corrected vertices approaching CT-QMC susceptibilities to 0.01–0.3% and convergence with box size that is much flatter than the uncorrected 1/ν_max behavior. That is credible evidence the machinery works in the tested regime. The citation pattern is fine; prior work is credited, and self-citation is not a problem here.

Soft spots: first, the central accuracy claim—O(1/ν_max^4) error—rests on the assumed 1/ν^3 decay of the irreducible vertex beyond the Eq. (3) asymptotic. That exponent is not derived; it is described as expected, and the numerics do not directly measure the decay of δΦ. This is a real limitation, but not a fatal one: the method remains useful even if the true decay is slower, it just weakens the advertised scaling. Second, the step from Eq. (24) to Eq. (25) is compressed as algebraic transformations. The author points to the alternative derivation, but a reader should not have to reconstruct Ref. [33] to verify the main result. Third, the numerical validation is one model, one parameter set, spin channel only, no error bars on the fits. For a method paper aimed at broad use, another channel or a second model would strengthen the claim. These are addressable, not disqualifying.

Who it is for: anyone doing ladder DΓA, dual fermion/boson, TRILEX, or 2PI-fRG who is currently brute-forcing frequency boxes. I would send it to a serious referee, with the request to expand the derivation of Eq. (25) and clarify the status of the decay assumption. It is not a breakthrough, but it is a practical and honest improvement.

Referee Report

3 major / 4 minor

Summary. The manuscript proposes an analytic treatment of fermionic frequency tails in Bethe-Salpeter equations and fermion-boson vertices for non-local extensions of DMFT. The key idea is to split all frequency summations into a small box, where vertices are known numerically, and the outside region, where the asymptotic form of the irreducible vertex, Eq. (3), is used. The outside contribution is expressed in closed form through the quantities X_q and Z_q, yielding explicit formulas for the full and reduced fermion-boson vertices (Eqs. (8), (14), (21)), the full and irreducible susceptibilities (Eqs. (11), (18)), and the two-particle irreducible vertex (Eq. (25)). The method is tested on spin-sector DMFT results for a two-dimensional Hubbard model, with reported agreement with CT-QMC local susceptibilities and a much weaker dependence on frequency-box size than the uncorrected calculation.

Significance. If the central claims hold, the work is a useful contribution: the correction formulas are explicit, parameter-free, and reduce the need for large frequency boxes and for separate numerical treatment of the large box in earlier schemes. The algebraic nature of the derivation and the concrete numerical demonstration of improved convergence with box size are strengths. However, the headline O(1/ν_max^4) error-scaling claim rests on an unproven decay assumption for the residual irreducible vertex, and the numerical support is limited to one channel and one model parameter set, which weakens the generality asserted in the abstract and conclusion.

major comments (3)

[Section IV, discussion following Eq. (25)] The central claim that the method achieves O(1/ν_max^4) accuracy is not established. The argument assumes that the deviation of the true irreducible vertex from the asymptotic form (3) decays as 1/max(|ν|,|ν'|)^3; this exponent is not derived, and the text describes it only as 'expected.' Because X_q and Z_q are built entirely from the U_q + \barΦ tail, any component of Φ outside that tail with a slower power law will enter the out-of-box sums at lower order. The numerical fits in Sec. V (Fig. 3) use an a + b/ν_max^4 + c/ν_max^5 form, which presupposes the claimed scaling and therefore cannot by itself confirm the exponent. To support the conclusion, the author should either derive (or cite a derivation of) the 1/ν^3 decay of the residual, or directly measure this residual as a function of frequency and demonstrate the absence of 1/ν^2 or 1/ν^3 contamination.
[Section IV, transition from Eq. (24) to Eq. (25)] The key relation (25) is obtained by 'algebraic transformations' that are not shown, and Eq. (21) similarly refers to 'algebraic manipulations' of Appendix C of Ref. [30]. Since Eq. (25) is the central new result and is used to convert the box-restricted vertex Φ^{box} into the physical Φ, the derivation should be reproduced in detail, for example in an appendix. Without this, the reader cannot verify the consistency of signs, the treatment of the out-of-box sums, or the order at which terms are neglected. The numerical checks in Sec. V are too limited to substitute for this derivation.
[Section V] The numerical validation covers only the spin channel of a single two-dimensional Hubbard model with U=10t, t'=0.15t, n=0.96, T=0.08t (and one higher-temperature half-filled run). The paper advertises the method for a broad class of non-local DMFT extensions (ladder DΓA, dual fermion/boson, TRILEX, DMF2RG), but the asymptotic form (3) and the assumed 1/ν^3 decay of its correction are model- and approximation-dependent. In addition, the comparison of the local susceptibility from Eq. (16) with the CT-QMC result is partly a consistency check, because the tail functions X_q and Z_q themselves use local susceptibilities obtained from the same CT-QMC solver. At minimum, the authors should state these limitations explicitly in the conclusions and provide evidence in at least one further channel, parameter set, or non-local context.

minor comments (4)

[Fig. 3] The axis label 'tX0, Z' in panel (b) is unclear; please spell out the plotted quantities (e.g., tX_0^{(1)}, tX_0^{(2)}, Z_{ν,0}-1) and ensure that all axis labels render correctly.
[Sec. III, after Eq. (8)] The statement that the second term in X_q and the difference Z_q-1 are 'verified numerically' should be qualified: the verification is performed only for the spin channel of the Hubbard model, not for the full range of claimed applications.
[Sec. V, Fig. 3] The extrapolations shown by dotted lines and the fits with a+b/ν_max^4+c/ν_max^5 are presented without error bars or fit parameters; reporting these numbers would make the claimed scaling easier to assess.
[Sec. V] The sentence 'we find the results of extrapolation consistent with those for vertices, obtained without account of finite frequency box effects' is not quantified; please provide the extrapolated vertex values and the corresponding fit parameters.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is algebraic and parameter-free, built on an externally sourced vertex asymptotics; self-citations are minor and non-load-bearing.

full rationale

The central derivation chain starts from the assumed asymptotic form of the two-particle irreducible vertex, Eq. (3), and the corresponding asymptotics of the reducible vertex, Eq. (5), both attributed to prior work by Tagliavini et al. (Ref. [33]), not to the present author. The correction functions X_q and Z_q are then obtained as explicit sums over the outside-box region using that asymptotic tail, with no fitted parameters and no quantity being renamed as a prediction. Equations (8), (11), (14), (18), (21), and (25) are algebraic rearrangements of the Bethe-Salpeter equation under the stated asymptotics; none of them reduces by construction to its own input. The self-citation to Ref. [30] is used only for 'algebraic manipulations, similar to those described in Appendix C' and does not carry the load-bearing argument. The numerical section does contain a consistency-check element: the local susceptibilities entering the asymptotic tail in Eq. (4) are also compared with the CT-QMC result in Sec. V, so that particular comparison is partly a self-consistency test rather than an independent benchmark for those susceptibilities. However, this overlap does not make the derivation circular, because the derived vertex correction formulas do not fit those susceptibilities to the final vertices. The main caveat is instead a correctness risk, not circularity: the claimed O(1/ν_max^4) error improvement relies on the statement that deviations from Eq. (3) 'is expected to scale as 1/max(|ν|, |ν'|)^3', which is asserted as an expectation rather than derived, and the numerical verification covers only one model and one parameter set. That is a limitation of the evidence, but not an equivalence between the output and the input by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The derivation uses no fitted parameters; model parameters and local susceptibilities are external inputs from the impurity solver. The main axioms are the high-frequency asymptotic behavior of vertices and the associated decay estimates, which are standard in ladder-based diagrammatic methods but are not re-derived here. No new physical entities are introduced.

assumptions (5)

domain assumption Two-particle irreducible vertex Φ_{νν'q} has the asymptotic form U_q^{c(s)} + Φ̄_{νν'ω} for large ν or ν' (Eq. 3).
Section II, Eq. (3). Taken from prior work (Ref. [33]); all out-of-box correction terms X_q and Z_q rely on this tail.
domain assumption The tail Φ̄_{νν'ω} is local and expressed through local charge, spin, particle-particle susceptibilities and local interaction v_c(ω), decaying as (|ν|-|ν'|)^{-2} for large frequency differences.
Section II, Eq. (4) and following. Needed to compute the auxiliary functions Z_q and X_q analytically.
standard math The bubble χ0_{νq} decays as 1/ν^2 for large fermionic frequency.
Used in Section III to estimate the size of X_q, Z_q - 1 and to justify truncating out-of-box sums; standard high-frequency behavior of G G products.
domain assumption The deviation of the actual vertex from the asymptotic form (3) is O(1/max(|ν|,|ν'|)^3), so neglected terms contribute O(1/ν_max^4).
Section IV and V. Not proven analytically; the scaling is inferred from numerical fits a + b/ν_max^4 + c/ν_max^5.
domain assumption In the ladder approximation the irreducible vertex Φ is local and identical for the local and non-local problems.
Section IV, after Eq. (25). Used to relate box vertices of the local and non-local problems.

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Pith. "Pith review of Improved treatment of fermion-boson vertices and Bethe-Salpeter equations in non-local extensions of dynamical mean field theory." pith.science (2026). https://pith.science/paper/RFYEWLBK

@misc{pith2026190803198,
  author       = {Pith},
  title        = {Pith review of: Improved treatment of fermion-boson vertices and Bethe-Salpeter equations in non-local extensions of dynamical mean field theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RFYEWLBK}},
  note         = {Machine review of arXiv:1908.03198}
}

read the original abstract

We reconsider the procedure of calculation of fermion-boson vertices and numerical solution of Bethe-Salpeter equations, used in non-local extensions of dynamical mean-field theory. Because of the frequency dependence of vertices, finite frequency box for matrix inversions is typically used, which often requires some treatment of asymptotic behaviour of vertices. Recently [Phys. Rev. B 83, 085102 (2011); 97, 235140 (2018)] it was proposed to split the considered frequency box into smaller and larger one; in the smaller frequency box the numerically exact vertices are used, while beyond this box asymptotics of vertices are applied. Yet, this method requires numerical treatment of vertex asymptotics (including corresponding matrix manipulations) in the larger frequency box and/or knowing fermion-boson vertices, which may be not convenient for numerical calculations. In the present paper we derive the formulae which treat analytically contribution of vertices beyond chosen frequency box, such that only numerical operations with vertices in the chosen small frequency box are required. The method is tested on the Hubbard model and can be used in a broad range of applications of non-local extensions of dynamical mean-field theory.

Figures

Figures reproduced from arXiv: 1908.03198 by the authors.

**Figure 2.** FIG. 2: The 2PI vertex Φ [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** FIG. 3: The dependence of (a,c) triangular vertex [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [33]

Tagliavini, S

A. Tagliavini, S. Hummel, N. Wentzell, S. Andergassen, A. Toschi, and G. Rohringer, Phys. Rev. B 97, 235140 (2018)

work page 2018

[2] [30]

A. A. Katanin, Phys. Rev. B 99, 115112 (2019)

work page 2019

[3] [1]

Fradkin, S

E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Rev. Mod. Phys. 87, 457 (2015)

2015

[4] [2]

Dagotto, Rev

E. Dagotto, Rev. Mod. Phys. 66, 763 (1994)

1994

[5] [3]

W. E. Pickett, Rev. Mod. Phys. 61, 433 (1989)

work page 1989

[6] [4]

Haule and G

K. Haule and G. Kotliar, New J. Phys. 11, 025021 (2009); A. Georges, L. de’ Medici, and J. Mravlje, Ann. Rev. Cond. Matt. Physics 4, 137 (2013); L. Fanfarillo and E. Bascones, Phys. Rev. B 92, 075136 (2015); A. S. Belozerov, A. A. Katanin, and V. I. Anis imov, Phys. Rev. B 97, 115141 (2018)

work page 2009

[7] [5]

Metzner and D

W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324 (1989)

1989

[8] [6]

Georges, G

A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg, Rev. Mod. Phys. 68, 13 (1996); G. Kotliar and D. Vollhardt, Physics Today 57, 53 (2004)

work page 1996