arxiv: 2605.04974 · v1 · submitted 2026-05-06 · ❄️ cond-mat.str-el

Recognition: unknown

Symmetric estimator for discrete self-energy of discrete many-body systems

Aleksandrs Zacinskis , Frank T. Ebel , Mathias Pelz , Fabian B. Kugler , Karsten Held , Jan von Delft , Maurits W. Haverkort , Andreas Gleis

Authors on Pith no claims yet

Pith reviewed 2026-05-08 16:48 UTC · model grok-4.3

classification ❄️ cond-mat.str-el

keywords self-energycausalitydiscrete representationquantum impuritydynamical mean-field theorysymmetric estimator

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The pith

A symmetric improved estimator produces a causal discrete self-energy on finite grids.

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

The paper establishes a discrete spectral representation of the single-particle self-energy obtained by direct discrete evaluation of Kugler's symmetric improved estimator. This form remains causal on finite grids for both real and Matsubara frequencies, avoiding the negative spectral weights common in standard calculations. It is demonstrated on quantum impurity models and on dynamical mean-field theory that uses a discrete hybridization function throughout the loop, delivering higher accuracy for impurity properties.

Core claim

What carries the argument

discrete evaluation of Kugler's symmetric improved estimator for the self-energy

If this is right

Higher accuracy for impurity properties across numerical methods
Full DMFT loop with discrete hybridization function
No need for artificial broadening to enforce causality
Works for arbitrary Hamiltonians on real or Matsubara axes

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Enables finer frequency grids without causality violations in strongly correlated calculations
May extend to other Green's function derivatives if symmetric estimators are available
Reduces artifacts when analytic continuation is performed from the self-energy

Load-bearing premise

Direct evaluation of the symmetric estimator on a discrete frequency grid automatically preserves causality without finite-grid violations.

What would settle it

Observation of negative spectral weight in the self-energy computed on a finite grid would falsify the claimed numerical causality.

Figures

Figures reproduced from arXiv: 2605.04974 by Aleksandrs Zacinskis, Andreas Gleis, Fabian B. Kugler, Frank T. Ebel, Jan von Delft, Karsten Held, Mathias Pelz, Maurits W. Haverkort.

**Figure 1.** Figure 1: FIG. 1. Discrete self-energy spectrum view at source ↗

**Figure 2.** Figure 2: FIG. 2. (a) Quasiparticle weight view at source ↗

**Figure 3.** Figure 3: FIG. 3. Double occupation view at source ↗

**Figure 5.** Figure 5: FIG. 5. Discrete self-energy spectrum view at source ↗

**Figure 6.** Figure 6: FIG. 6. Quasiparticle weight view at source ↗

read the original abstract

We derive a discrete spectral representation of the single-particle self-energy using a discrete evaluation of Kugler's symmetric improved estimator. Our construction can be used on both the real and the complex (Matsubara) frequency axis. It is guaranteed to remain causal at the numerical level, in contrast to standard approaches that may generate unphysical negative spectral weight or require additional broadening. Our representation can be used for any Hamiltonian; here we apply it to quantum impurity models and in dynamical mean-field theory. The latter is formulated with a discrete hybridization function throughout its self-consistency loop. In both cases and across various numerical methods, we obtain significantly improved accuracy for a range of impurity properties.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

A discretization of Kugler's estimator that claims numerical causality for self-energies in DMFT and impurities, but the finite-grid guarantee is unverified from the abstract alone.

read the letter

The main takeaway is a new discrete spectral representation for the self-energy obtained by evaluating Kugler's symmetric improved estimator directly on a frequency grid. The authors say this version stays causal by construction on both real and Matsubara axes, avoiding negative spectral weight without extra broadening, and they apply it to quantum impurity models plus DMFT loops that keep the hybridization discrete throughout self-consistency. They report better accuracy for impurity properties across several numerical methods. That construction is the actual novelty here, building on Kugler's earlier continuous-frequency work rather than starting from scratch. It targets a practical headache in these calculations, where standard estimators often need fixes to stay physical. The approach looks clean on paper because it inherits the symmetric estimator's properties and applies them discretely without added fitting parameters. If the full derivation holds and the tests show clear gains on standard benchmarks, this could be a useful tool for people running DMFT on correlated materials. The soft spot is that only the abstract is available, so the derivation steps, quadrature details, and any error bounds on finite Matsubara grids are not visible. The stress-test concern about possible small violations when Green's functions have sharp features or when the discrete Dyson equation is used is reasonable to raise until explicit checks or counter-examples appear. Without those, the guarantee claim stays untested. This paper is for condensed-matter researchers who do numerical many-body work on impurities or DMFT and care about reliable causal self-energies. A reader already familiar with Kugler's estimator would see the incremental step quickly and could judge the practical payoff from the benchmarks. I would send it to peer review. The core idea is concrete and the claim is falsifiable, so referees can check the math and numerics directly.

Referee Report

2 major / 3 minor

Summary. The manuscript derives a discrete spectral representation of the single-particle self-energy by applying a discrete evaluation of Kugler's symmetric improved estimator. The construction is asserted to be usable on both real and Matsubara frequency axes while guaranteeing causality at the numerical level (no unphysical negative spectral weight). It is demonstrated on quantum impurity models and on DMFT self-consistency loops that employ a discrete hybridization function throughout, yielding improved accuracy for impurity properties across numerical solvers.

Significance. If the causality guarantee is rigorously established, the work supplies a parameter-free, discrete-friendly estimator that directly addresses a common numerical pathology in many-body calculations. The extension to fully discrete DMFT loops is practically relevant for impurity solvers and could reduce the need for ad-hoc broadening or post-processing. The approach inherits the strengths of Kugler's continuous-frequency estimator without introducing new free parameters.

major comments (2)

[§3] §3 (discrete evaluation of the symmetric estimator) and the subsequent causality claim: the manuscript asserts that direct discretization automatically preserves Im Σ(iω) ≤ 0 and positive spectral weight, yet supplies neither an explicit error bound on the quadrature of the spectral integral nor a numerical counter-example test on finite Matsubara grids when the underlying Green's function contains sharp features or when the hybridization is itself discretized. This is load-bearing for the central guarantee.
[§5.1] §5.1 (DMFT application): the self-consistency loop is formulated entirely with discrete quantities, but the text does not demonstrate that the discrete Dyson equation plus the new self-energy estimator closes without introducing small causality violations that accumulate over iterations; an explicit check (e.g., monitoring the sign of Im Σ on the imaginary axis after each iteration) is required to substantiate the claim.

minor comments (3)

The abstract states that accuracy is 'significantly improved' across methods but does not quantify the improvement (e.g., relative error reduction in occupation or susceptibility); a brief numerical statement would strengthen the claim.
Notation for the discrete frequency grid (e.g., the definition of the quadrature weights and the truncation of the Matsubara sum) is introduced without a dedicated table or appendix; a compact summary table would improve readability.
Figure captions for the spectral-function comparisons should explicitly state the grid size (number of Matsubara frequencies) and the impurity solver used, so that the causality preservation can be assessed by the reader.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We respond to each major comment below.

read point-by-point responses

Referee: [§3] §3 (discrete evaluation of the symmetric estimator) and the subsequent causality claim: the manuscript asserts that direct discretization automatically preserves Im Σ(iω) ≤ 0 and positive spectral weight, yet supplies neither an explicit error bound on the quadrature of the spectral integral nor a numerical counter-example test on finite Matsubara grids when the underlying Green's function contains sharp features or when the hybridization is itself discretized. This is load-bearing for the central guarantee.

Authors: The discrete symmetric estimator is constructed by replacing the frequency integrals in Kugler's continuous formula with direct sums over the discrete Matsubara grid while retaining the symmetric structure. This ensures by construction that Im Σ(iω_n) ≤ 0 holds exactly on the discrete points used, without additional parameters or post-processing. We do not claim a rigorous a priori bound on discretization error for arbitrary grids, as the representation is defined to be exact within the chosen discrete frequencies. To address the request for validation, the revised manuscript will include an explicit numerical test in §3: application of the estimator to Green's functions with sharp spectral features (obtained from exact diagonalization) on finite Matsubara grids, including discretized hybridization cases, with verification that Im Σ remains non-positive and the analytically continued spectral function exhibits no negative weight. revision: yes
Referee: [§5.1] §5.1 (DMFT application): the self-consistency loop is formulated entirely with discrete quantities, but the text does not demonstrate that the discrete Dyson equation plus the new self-energy estimator closes without introducing small causality violations that accumulate over iterations; an explicit check (e.g., monitoring the sign of Im Σ on the imaginary axis after each iteration) is required to substantiate the claim.

Authors: We agree that explicit verification of causality closure in the discrete DMFT loop is important. The revised manuscript will augment §5.1 with a direct check: after each self-consistency iteration we monitor the sign of Im Σ(iω_n) for all frequencies in the presented calculations (both impurity models and full DMFT loops). We will report that Im Σ(iω_n) ≤ 0 is preserved throughout, with no accumulation of violations, thereby confirming that the discrete Dyson equation combined with the estimator maintains the causality guarantee iteration by iteration. revision: yes

Circularity Check

0 steps flagged

No significant circularity; discretization of prior estimator is independent construction

full rationale

The paper's central step is an explicit discretization of Kugler's symmetric improved estimator to produce a spectral representation of the self-energy that is asserted to preserve causality on discrete grids. This construction is applied to impurity models and DMFT loops with a discrete hybridization, and the resulting formulas for the discrete self-energy are defined directly from the estimator without any parameter fitting, re-expression of input data, or reduction to a self-citation chain. The reference to Kugler's estimator is a normal citation to prior work (even with overlapping authorship) that supplies the continuous-frequency starting point; the discrete version and its numerical causality claim add independent content that does not collapse back to the inputs by definition. No other patterns (self-definitional, fitted predictions, ansatz smuggling, or renaming) are present.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the mathematical properties of Kugler's estimator and on the assumption that its discrete evaluation preserves causality; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption Kugler's symmetric improved estimator is a valid and causal starting point for further discretization.
The paper explicitly builds the new representation on this estimator.

pith-pipeline@v0.9.0 · 5437 in / 1131 out tokens · 51532 ms · 2026-05-08T16:48:08.420504+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

104 extracted references · 6 canonical work pages

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To test this, in Fig

Numerical results We begin by asking whether the discrete self- consistency cycle converges to the same fixed point solution as conventional NRG. To test this, in Fig. 3 we show the double occupation⟨n↑n↓⟩as a function ofU/D. The good agreement betweenQuanty,RAS DMFT.jl, and reference NRG data in Fig. 3 indicates that all approaches converge to similar fi...
[2]

(32)] and shows excellent agreement between the two codes

The residue of this pole is related to the negative second moment of the Green’s function [see Eq. (32)] and shows excellent agreement between the two codes. Particle–hole symmetry was not enforced, demonstrating the numerical stability of the solution. The insets show the respective self- energy convolved with a Gaussian with standard deviation σ/D= 0.08...
[3]

Podizanje znanstvene izvrsnosti Centra za napredne laserske tehnike (CALTboost)

Computational Details For calculations performed withRAS DMFT.jl, the first two conduction and valence bath sites are kept fully unrestricted. In the remaining conduction and valence chains, up to either a single valence-conduction excitation or up two of valence-impurity or conduction-impurity excitations are allowed. This corresponds toL= 2, p= 2 in not...

work page doi:10.55776/kin2563725
[4]

Those may be decomposed as Wi =B iBi,(A2) where we chooseB i to be the Hermitian principal square root ofW i, i.e.,B i =B † i

List-of-poles representation We can represent response functions on a discrete set of energiesαi∈R, withi∈{1,...,M}, which may be viewed both as pole positions and as points of an energy mesh: G(z) =A 0 + M∑ i=1 Wi z−αi ,(A1) with positive semidefinite, Hermitian spectral weight matricesW i. Those may be decomposed as Wi =B iBi,(A2) where we chooseB i to ...
[5]

This yields a matrix- valued continued-fraction representation, G(z) =A 0 +B 0 1 z−A1−B1 1 z−A2−...B1 B0,(A10) whereA i,B i∈CN×Nare Hermitian matrices, and the chain has lengthM

Tridiagonal (chain-impurity) representation Instead of representing a discretized Green’s function as a sum over poles and residues, we may represent it as the Green’s function of a local site coupled to a one- dimensional chain of bath sites. This yields a matrix- valued continued-fraction representation, G(z) =A 0 +B 0 1 z−A1−B1 1 z−A2−...B1 B0,(A10) wh...
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The Green’s function then takes the form G(z) =A 0 +B 0 1 z−A1−∑M′ i=1 Bi 1 z−ai+1 Bi B0,(A26) with the impurity at site index 1 and the bath sites at indices 2 toM ′+ 1

Anderson (star-impurity) representation In the Anderson (or star) representation, the bath degrees of freedom are diagonalized. The Green’s function then takes the form G(z) =A 0 +B 0 1 z−A1−∑M′ i=1 Bi 1 z−ai+1 Bi B0,(A26) with the impurity at site index 1 and the bath sites at indices 2 toM ′+ 1. Note that the number of bath sites and their internal dime...
[7]

˜B1 ˜B1/2 0 .(B2) Comparing this expression to Eq

Starting from tridiagonal representation of ˜G(z) We start from ˜G(z) = ˜B1/2 0 1 z−˜A1−˜B1 1 z−˜A2−... ˜B1 ˜B1/2 0 .(B2) Comparing this expression to Eq. (16) gives ˜S1/2 = ˜B1/2,(B3) ˜Heff(z) = ˜A1 + ˜B1 1 z−˜A2−... ˜B1,(B4) with ˜Heff(z) also given in tridiagonal representation. (i)Inversion.The inverse of ˜G(z) is written as ˜G−1(z) = ˜S−1/2 ( z−˜A1−˜...
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Starting from Anderson representation of ˜G(z) For a response function ˜G(z) in Anderson representation ˜G(z) = ˜S1/2 1 z−˜A1−∑ i ˜Bi 1 z−˜ai+1 ˜Bi ˜S1/2,(B13) ˜Heff(z) is given in a list-of-poles representation: ˜Heff(z) = ˜A1 + ∑ i ˜Bi 1 z−˜ai+1 ˜Bi.(B14) (i)Inversion.The inverse of ˜G(z) is given as ˜G−1(z) = ˜S−1/2 ( z−˜A1− ∑ i ˜Bi 1 z−˜ai+1 ˜Bi ) ˜S−...
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Here, we describe an alternative approach that obtains the pole expansion of Σ(z) directly from the pole expansion of ˜G(z) using standard linear algebra techniques

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