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A mathematical analysis of the discretized IPT-DMFT equations

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Pith's one-line read The paper proves that the discretized IPT-DMFT equations admit physically admissible solutions for small hopping at any finite Matsubara cutoff, unique under a stronger smallness condition, with multiple admissible solutions possible for…

desk verdict Rigorous existence and uniqueness for the MaF-discretized IPT-DMFT equations, with a real but non-fatal gap: the algebraic reduction behind the Nω=1 solution counts is asserted, not proved. read the letter →

arxiv 2505.21287 v1 pith:37GHNUQ7 submitted 2025-05-27 math.NA cs.NA

E. Cancès , A. Kirsch , S. Perrin--Roussel This is my paper
classification math.NAcs.NA MSC 65H1065H2082B80
keywords dynamicalmean-fieldtheoryIPT-DMFTequationsMatsubarafrequencydiscretizationexistenceanduniquenessHubbarddimerPickcriterionparticle-holesymmetryalgebraicsolutioncounting
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 proves that the Matsubara-frequency discretization of the IPT-DMFT equations—the version of dynamical mean-field theory that solves the impurity model by second-order perturbation theory—is mathematically well posed. For each cutoff $N_\omega$, existence of a physically admissible solution (components in $-C_+$, i.e. with nonpositive imaginary part) is shown under the condition that the dimensionless hopping $t=\beta T$ is not too large, with the allowed range shrinking with the graph degree. Uniqueness, and linear convergence of the natural fixed-point iteration, holds when $t^2u^2\deg(G_H)$ is below a cutoff-dependent constant. For bipartite systems the equations reduce to a sparse real algebraic system; for the Hubbard dimer at $N_\omega=0$ the number of admissible solutions is fully characterized in the low-temperature limit, with a transition at $\alpha=U/T=3\pi/2$. Numerical simulations exhibit a conductor–insulator transition and show that, for large enough $U$ at small $N_\omega$, the converged discrete Green's function cannot be interpolated by a Pick function.

What carries the argument

The Matsubara-frequency (MaF) discretization: the unknown functions $\Delta$ and $\Sigma$, originally analytic on the upper half-plane, are represented by their values at the lowest $N_\omega+1$ imaginary Matsubara frequencies $i\omega_n$, $\omega_n=(2n+1)\pi/\beta$. The nonlocal IPT equation is replaced by the rational map $F_{N_\omega}$ defined by a convolution over triples of frequencies (equation (18)), and the full fixed-point map is $F^{\mathrm{DMFT}}_{N_\omega}(\Delta)_n = t^2 w^T (i(2n+1)\pi - t h^0_\perp - u^2[F_{N_\omega}(\Delta)]_n)^{-1} w$. Existence is proved by Brouwer's fixed-point theorem on the compact set $D_t^{N_\omega}=\{z: |z_n|\le t^2\deg(G_H)/((2n+1)\pi)\}$, while uniqueness and linear convergence follow from the contraction estimate of Lemma 3.6. For bipartite systems, the map preserves the imaginary axis, and in the variables $x_n=1/(1+y_n)$ the equations reduce to the sparse polynomial system (57), whose solution count is studied with symbolic and numerical algebraic geometry.

What would settle it

At a single parameter pair with $\alpha=U/T$ fixed just above $3\pi/2$ and $t$ large, solve the $N_\omega=0$ dimer equation (48) with arbitrary precision; Theorem 4.2 predicts exactly three admissible solutions, so any count other than three would falsify the characterization.

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Extended reading notes

Core claim

The central claim is Theorem 3.1: for every Matsubara cutoff $N_\omega$ there is a positive constant $t_{N_\omega}$ such that, whenever $0\le t \le t_{N_\omega}/\sqrt{\deg(G_H)}$, equations (19)–(20) have a solution $(\Delta,\Sigma)$ with all components in the physically admissible cone $-C_+$. The obstruction this overcomes is that the discretized IPT map $F_{N_\omega}$ does not preserve $-C_+$ for $N_\omega>0$ (the counterexample is equation (23)); the proof uses Brouwer's fixed-point theorem on the compact set $D_t^{N_\omega}$. Theorem 3.5 shows uniqueness of that solution—and linear convergence of the simple fixed-point iteration (33)–(34)—under the additional smallness condition $t^2u^2\deg(G_H)<\eta_{N_\omega}$. For bipartite graphs, particle-hole symmetry forces solutions onto the imaginary axis and the discretized equations become a sparse polynomial system in variables $x_n\in(0,1]$; the paper completely solves the $N_\omega=0$ Hubbard dimer in the low-temperature limit, locating a transition in the count of admissible solutions at $\alpha=3\pi/2$, and reports that for $N_\omega=2$ and $5$ the converged numerical solution can have a negative Pick-matrix eigenvalue, meaning no Pick function interpolates the discrete Green's function. The paper conjectures (Remark 5.1) that as $N_\omega\to\infty$ the Pick-criterion violations disappear and the discrete solutions converge to the continuous ones.

Load-bearing premise

The physical reading of the theorems rests on the unproved assumption that solutions of the discretized equations converge to solutions of the continuous IPT-DMFT equations as the Matsubara cutoff $N_\omega$ grows; the paper only conjectures this in Remark 5.1.

Editorial extensions

If this is right

  • For any finite Matsubara cutoff, the IPT-DMFT fixed-point loop has a guaranteed starting point in $D_t^{N_\omega}$ from which a physically admissible solution exists, as long as $t$ is below the graph-degree-adjusted bound.
  • In the uniqueness regime $t^2u^2\deg(G_H)<\eta_{N_\omega}$, the simple fixed-point iteration converges linearly, with a rate controlled by the same constants used in the existence proof.
  • For the Hubbard dimer at $N_\omega=0$ and low temperature, the number of admissible solutions changes at $\alpha=U/T=3\pi/2$: below this ratio only one solution exists (escaping to infinity with $t$), while above it two additional bounded solutions appear.
  • Numerical simulations on the Hubbard dimer at $\beta=1$ show the spectral density at zero frequency $\rho(0)$ vanishing between $U\approx 6$ and $8$, identifying a conductor–insulator transition in the IPT-DMFT approximation.
  • For $N_\omega=2$ and $5$, the converged discrete Green's function can have a negative Pick-matrix eigenvalue, proving that no Pick function interpolates the discrete data; the paper conjectures this artifact disappears as $N_\omega\to\infty$.
  • The physical reading of the theorems rests on the unproved assumption that solutions of the discretized equations converge to solutions of the continuous IPT-DMFT equations as the Matsubara cutoff grows; the paper only conjectures this in Remark 5.1.

Reading between the lines

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

  • Editorial inference: the sharp threshold $\alpha=3\pi/2$ at $N_\omega=0$, where the count of admissible solutions jumps from 1 to 3, is a plausible finite-cutoff precursor of the Mott transition; tracking how this threshold moves with $N_\omega$ would give a quantitative test of the paper's convergence conjecture.
  • Editorial inference: the lowest eigenvalue of the Pick matrix computed from a converged discrete Green's function could serve as a practical a posteriori error indicator, telling a user how many Matsubara frequencies are needed before the discrete data become analytically continuable.
  • Editorial inference: because the existence proof only covers small $t$, the strong-coupling regime where the numerics find insulator-like solutions may lie outside the range of guaranteed admissibility—an open question whether those solutions are physical or artifacts of the finite cutoff.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. This paper studies the Matsubara-frequency (MaF) discretization of the IPT-DMFT equations for translation-invariant Hubbard models. In the dimensionless formulation (19)–(20), the authors prove via Brouwer's fixed-point theorem that for each cutoff N_ω there exists t_{N_ω} > 0 such that for 0 ≤ t ≤ t_{N_ω}/sqrt(deg(G_H)) the discretized equations admit a solution (Δ,Σ) with nonpositive imaginary parts (Theorem 3.1). Under the additional smallness condition t^2 u^2 deg(G_H) < η_{N_ω}, they prove uniqueness and linear convergence of the fixed-point iteration (Theorem 3.5). For bipartite graphs, particle-hole symmetry reduces the problem to a polynomial system in the variables x_n = 1/(1+y_n); the dimer case N_ω = 0 is fully characterized as a function of α = U/T (Theorem 4.2), and for N_ω = 1 numerical algebraic geometry suggests 16 complex solutions and 1–5 admissible solutions. The final section reports TRIQS and Julia simulations showing a conductor-to-insulator transition and cases where the converged values violate the Pick criterion.

Significance. These appear to be the first existence and uniqueness results for the discretized IPT-DMFT equations, and they are obtained from explicit fixed-point estimates rather than by fitting parameters. The N_ω = 0 dimer theorem is a complete, rigorous characterization of the admissible solutions in the low-temperature limit. The algebraic-geometry viewpoint in Section 4 is promising and, if fully justified, would give a systematic way to count and classify solutions for larger N_ω. The numerical experiments are clearly labelled as illustrations and include a careful high-precision check of the Pick criterion, which is a useful caution for practitioners of the MaF discretization. The main limitations are that the algebraic reduction in Section 4.2 is not proved in the manuscript, and the convergence of discretized solutions to the continuous IPT-DMFT solutions is only conjectured (Remark 5.1).

major comments (3)
  1. [Section 4.2, Eqs. (46) and (57)] The passage from the rational equations (19)–(20) to the polynomial system is asserted with 'it follows that' and no proof is given. This equivalence is load-bearing for the N_ω = 1 results in Section 4.3.2, because all solution counts are computed on the polynomial system (58)–(59), not on the original equations. A rigorous derivation must justify the change of variables x_n = 1/(1+y_n), the spectral decomposition of h_perp^0, and the clearing of denominators, and it must rule out spurious roots introduced by that clearing and the possible loss of boundary solutions (e.g., x_n = 1, i.e., y_n = 0). Please provide the full derivation in the text or in an appendix.
  2. [Proof of Theorem 3.5] The Lipschitz constant L_{N_ω} is defined as a supremum over D_t^{N_ω}, which depends on the parameter t. The subsequent identification η_{N_ω} = π^2/L_{N_ω} therefore makes η_{N_ω} depend on t, contrary to the theorem's statement that η_{N_ω} depends only on N_ω. The argument can be repaired by taking the supremum over the largest admissible set, D_{t_{N_ω}/sqrt(deg(G_H))}^{N_ω}, which depends only on N_ω, but as written the proof is incomplete.
  3. [Section 4.3.2, Eqs. (58)–(59)] The assertions that (58)–(59) have 16 complex solutions and that the number of admissible solutions ranges from 1 to 5 are obtained from HomotopyContinuation.jl, not from a proved theorem. Similarly, the separation line b = 3(1 + a/10)^3 is stated without computation. If these are numerical observations, the text should label them as such; if they are intended as rigorous 'some results for N_ω = 1', the manuscript must supply proofs or certificates. This matters because the abstract promises results for N_ω = 1.
minor comments (6)
  1. [Eq. (34)] The Matsubara frequency in the definition of F_DMFT_Nω is written 'i2(n+1)π' rather than 'i(2n+1)π'; this is inconsistent with all other occurrences and appears to be a typo.
  2. [Lemma 3.4] The statement says F_Nω(D_t^{Nω}) ⊂ −C_+ for all 0 ≤ t ≤ t_Nω, but because D_t^{Nω} = t^2 deg(G_H) C_Nω, the proof actually requires t ≤ t_Nω / sqrt(deg(G_H)); the statement and proof should be aligned.
  3. [Figure 2 caption] The caption writes 'ω_n = 2(n+1)π/β'; the correct Matsubara frequency is ω_n = (2n+1)π/β.
  4. [Theorem 4.2, first bullet] The bullet states that there is a unique solution 'in C+' with iΔ∞_{α,t} → +∞; for the admissible set −C_+ this should read 'in −C_+' so that the stated limit is consistent.
  5. [Proof of Lemma 3.4] The quantity φ(n,Nω) = π^3 Im(F_Nω(0)_n) is described as a 'positive rational number', but it is negative (for example φ(0,0) = −3); the word 'positive' should be removed or the sign convention explained.
  6. [Section 1, last paragraph] The phrase 'it is highlited' should be 'it is highlighted'.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the discretized existence/uniqueness proofs are self-contained fixed-point arguments, and self-citations to the prior continuous IPT-DMFT paper are background, not load-bearing reductions.

full rationale

This paper's central claims — existence (Theorem 3.1), uniqueness and linear convergence (Theorem 3.5), and the Nω=0 characterization (Theorem 4.2) — are proved by Brouwer/Picard fixed-point arguments on explicit parameter-dependent sets D_t^{Nω} and by an explicit quartic analysis; no fitted parameters are renamed as predictions. The discretized equations (19)-(20) are formulated independently of the results they are used to prove. The main self-citations to [5] are background: [5] supplies the continuous IPT construction, the sign of F_IPT(0) used in Lemma 3.4, and standard Nevanlinna-Pick facts, but the monotonicity in Nω, the invariant-set estimates, and the contraction estimates are derived here. Lemma 3.4 appeals to [5, Prop. 2.18] only to evaluate a fixed limit at zero hybridization; this is an independent quantitative fact, not a restatement of the discretized existence theorem. Section 4.2's algebraic reduction is compressed, and the homotopy-continuation counts of (58)-(59) are conditional on that reduction, but this is a derivation gap or numerical verification issue, not circularity: the variable change x_n=1/(1+y_n) is a bijection on the admissible domain and the polynomial system is an algebraic reformulation, not a fitted target. Section 5 numerical results are explicitly presented as simulations, and Remark 5.1 labels the Nω→∞ convergence as a conjecture, so the paper does not disguise an unproved link as a theorem. Overall, no step reduces by construction to its own input, and no load-bearing conclusion is forced by a self-citation chain.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

This ledger covers the background the central theorems draw on. The main mathematical ingredients are standard fixed-point theorems and Pick function theory; no ad hoc entities or fitted constants are introduced. The physical setting is the IPT-DMFT approximation as defined in the authors' previous paper, which is assumed as the starting point.

assumptions (7)
  • standard math Brouwer fixed-point theorem
    Used in the proof of Theorem 3.1 to prove existence of a fixed point of the MaF-discretized IPT-DMFT map on a compact convex set.
  • standard math Picard fixed-point theorem (contraction mapping theorem)
    Used in the proof of Theorem 3.5 to prove uniqueness and linear convergence of the fixed-point iteration.
  • standard math Nevanlinna-Riesz representation and Pick function theory
    Provides the representation of Green's function and self-energy as negatives of Pick functions, used throughout to define the admissible solution set -C_+.
  • standard math Resolvent identities and Schur complement methods
    Used in Lemma 3.3 and Section 4.1 to bound the bath-update map and to show preservation of purely imaginary solutions for bipartite systems.
  • domain assumption IPT approximation is a valid impurity solver and the continuous IPT-DMFT equations from [5] are the correct starting point
    The paper analyzes a discretization of the continuous IPT-DMFT equations introduced in the authors' previous paper; no justification for the IPT approximation itself is given.
  • domain assumption Finite vertex-transitive Hubbard graph and paramagnetic single-site setting
    The DMFT construction is specialized to translation-invariant, paramagnetic, single-site impurities, stated in Section 1 and Definition 2.1.
  • domain assumption For bipartite systems, half-filling and particle-hole symmetry
    Section 4.1 assumes the Hubbard model is at half-filling so that the grand canonical Hamiltonian is invariant under the particle-hole transformation, leading to purely imaginary solutions.

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Pith. "Pith review of A mathematical analysis of the discretized IPT-DMFT equations." pith.science (2026). https://pith.science/paper/37GHNUQ7

@misc{pith2026250521287,
  author       = {Pith},
  title        = {Pith review of: A mathematical analysis of the discretized IPT-DMFT equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/37GHNUQ7}},
  note         = {Machine review of arXiv:2505.21287}
}
abstract

In a previous contribution (E. Canc\`es, A. Kirsch and S. Perrin--Roussel, arXiv:2406.03384), we have proven the existence of a solution to the Dynamical Mean-Field Theory (DMFT) equations under the Iterated Perturbation Theory (IPT-DMFT) approximation. In view of numerical simulations, these equations need to be discretized. In this article, we are interested in a discretization of the \acrshort{ipt}-\acrshort{dmft} functional equations, based on the restriction of the hybridization function and local self-energy to a finite number of points in the upper half-plane $\left(i\omega_n\right)_{n \in |[0,N_\omega]|}$, where $\omega_n=(2n+1)\pi / \beta$ is the $n$-th Matsubara frequency and $N_\omega \in \mathbb N$. We first prove the existence of solutions to the discretized equations in some parameter range depending on $N_\omega$. We then prove uniqueness for a smaller range of parameters. We also study more in depth the case of bipartite systems exhibiting particle-hole symmetry. In this case, the discretized IPT-DMFT equations have purely imaginary solutions, which can be obtained by solving a real algebraic system of $(N_\omega+1)$ equations with $(N_\omega+1)$ variables. We provide a complete characterization of the solutions for $N_\omega=0$ and some results for $N_\omega=1$ in the simple case of the Hubbard dimer. We finally present some numerical simulations on the Hubbard dimer.

Figures

Figures reproduced from arXiv: 2505.21287 by the authors.

Figure 1
Figure 1. Left: Pariser–Parr–Pople model of benzene C [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. The set (iωn)n∈[[0,Nω]] of points in C+ used for the discretization of ∆ and Σ with Nω = 4 where ωn = 2(n+1)π β is the n-th Matsubara frequency and F IPT β,Nω is defined by (9). Remark 2.2 (comment on the set of admissible solutions). Note that we restrict ourselves to solutions satisfying the condition ∆, Σ ∈ −C+ Nω+1. This is motivated by the fact that ∆ and Σ are negatives of Pick functions. Nevertheless, for Nω … view at source ↗
Figure 3
Figure 3. Number of admissible solutions (x0, x1) ∈ (0, 1]2 to (58)-(59) in the range of parameters (a, b) ∈ [0, 10] × [0, 25]. This gives the number of purely imaginary solutions to the dimensionless MaF￾discretized IPT-DMFT equations (19)–(20) for the Hubbard dimer in the range of parameters (t, u) ∈ [0, √ 10π] × [0, 5π 2 ]. to exemplify the results of Section 4.3, we then switch to a Julia implementation with arbitrary fin… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Density ρ of the spectral function A obtained by analytic continuation using Pade approximants, for different values of the on-site repulsion U. Other parameters are fixed as in (64). • As U increases, ρ(0) decreases to reach approximately 0 between U = 6 and 8: for U …
Figure 5
Figure 5. Figure 5: Linear convergence of the fixed-point algorithm. The residual [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]
Figure 6
Figure 6. Figure 6: Density ρ of the spectral function A, for different values of the on-site repulsion U (focus around 0 on the right). For U = 7 and U = 7.5, the density ρ takes negatives values around 0. 5.2 Pick criterion on the converged solution In this section, we are interested in…
Figure 7
Figure 7. Figure 7: Test of the Pick criterion (lowest eigenvalue of the Pick matrix) of a solution to the IPT-DMFT [PITH_FULL_IMAGE:figures/full_fig_p025_7.png]

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Works this paper leans on

41 extracted references · 38 canonical work pages

  1. [1]

    Complexity of quantum impurity problems.Communications in Mathematical Physics, 356(2):451–500, December 2017

    Sergey Bravyi and David Gosset. Complexity of quantum impurity problems.Communications in Mathematical Physics, 356(2):451–500, December 2017. arXiv:1609.00735 [cond-mat, physics:math- ph, physics:quant-ph]

    arXiv 2017

  2. [2]

    HomotopyContinuation.jl: A Package for Homotopy Continuation in Julia

    Paul Breiding and Sascha Timme. HomotopyContinuation.jl: A Package for Homotopy Continuation in Julia. InInternational Congress on Mathematical Software, pages 458–465. Springer, 2018

    work page 2018

  3. [3]

    Springer Science & Business Media, 2011

    Andries E Brouwer and Willem H Haemers.Spectra of graphs. Springer Science & Business Media, 2011

    2011

  4. [4]

    A mathematical analysis of the GW0 method for computing electronic excited energies of molecules.Reviews in Mathematical Physics, 28(04):1650008, 2016

    Eric Canc` es, David Gontier, and Gabriel Stoltz. A mathematical analysis of the GW0 method for computing electronic excited energies of molecules.Reviews in Mathematical Physics, 28(04):1650008, 2016

    work page 2016

  5. [5]

    A mathematical analysis of IPT-DMFT, June

    ´Eric Canc` es, Alfred Kirsch, and Solal Perrin-Roussel. A mathematical analysis of IPT-DMFT, June

    work page

  6. [6]

    Interpolation by pick functions.Monotone Matrix Functions and Analytic Continuation, pages 117–127, 1974

    William F Donoghue and William F Donoghue. Interpolation by pick functions.Monotone Matrix Functions and Analytic Continuation, pages 117–127, 1974

    work page 1974

  7. [7]

    Nevanlinna Analytical Continuation.Physical Review Letters, 126(5):056402, February 2021

    Jiani Fei, Chia-Nan Yeh, and Emanuel Gull. Nevanlinna Analytical Continuation.Physical Review Letters, 126(5):056402, February 2021

    work page 2021

  8. [8]

    Analytical continuation of matrix- valued functions: Carath´ eodory formalism.Physical Review B, 104(16):165111, 2021

    Jiani Fei, Chia-Nan Yeh, Dominika Zgid, and Emanuel Gull. Analytical continuation of matrix- valued functions: Carath´ eodory formalism.Physical Review B, 104(16):165111, 2021

    work page 2021

Show all 41 references
  1. [9]

    Hubbard model in infinite dimensions.Physical Review B, 45(12):6479–6483, March 1992

    Antoine Georges and Gabriel Kotliar. Hubbard model in infinite dimensions.Physical Review B, 45(12):6479–6483, March 1992

    1992

  2. [10]

    Rozenberg

    Antoine Georges, Gabriel Kotliar, Werner Krauth, and Marcelo J. Rozenberg. Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions.Reviews of Modern Physics, 68(1):13–125, January 1996

    1996

  3. [11]

    On Matrix–Valued Herglotz Functions.Mathematische Nachrichten, 218(1):61–138, 2000

    Fritz Gesztesy and Eduard Tsekanovskii. On Matrix–Valued Herglotz Functions.Mathematische Nachrichten, 218(1):61–138, 2000

    2000

  4. [12]

    Grayson and Michael E

    Daniel R. Grayson and Michael E. Stillman. Macaulay2, a software system for research in algebraic geometry. Available athttp://www2.macaulay2.com

  5. [13]

    J. E. Gubernatis, Mark Jarrell, R. N. Silver, and D. S. Sivia. Quantum Monte Carlo simulations and maximum entropy: Dynamics from imaginary-time data.Physical Review B, 44(12):6011–6029, 1991

    1991

  6. [14]

    Millis, Alexander I

    Emanuel Gull, Andrew J. Millis, Alexander I. Lichtenstein, Alexey N. Rubtsov, Matthias Troyer, and Philipp Werner. Continuous-time Monte Carlo methods for quantum impurity models.Reviews of Modern Physics, 83(2):349–404, 2011

    2011

  7. [15]

    Gutzwiller

    Martin C. Gutzwiller. Effect of Correlation on the Ferromagnetism of Transition Metals.Physical Review Letters, 10(5):159–162, March 1963

    1963

  8. [16]

    Mott-Hubbard Metal-Insulator Transition in Paramagnetic V$ 2$O$ 3$: An LDA+DMFT (QMC) Study.Physical review letters, 86(23):5345, 2001

    Karsten Held, Georg Keller, Volker Eyert, Dieter Vollhardt, and Vladimir I Anisimov. Mott-Hubbard Metal-Insulator Transition in Paramagnetic V$ 2$O$ 3$: An LDA+DMFT (QMC) Study.Physical review letters, 86(23):5345, 2001. 26

    2001

  9. [17]

    Robust analytic continuation of Green’s functions via projection, pole estimation, and semidefinite relaxation.Physical Review B, 107(7):075151, February

    Zhen Huang, Emanuel Gull, and Lin Lin. Robust analytic continuation of Green’s functions via projection, pole estimation, and semidefinite relaxation.Physical Review B, 107(7):075151, February

  10. [18]

    Electron correlations in narrow energy bands.Proceedings of the Royal Society of London

    John Hubbard. Electron correlations in narrow energy bands.Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 276(1365):238–257, 1963

    1963

  11. [19]

    Mark Jarrell and J. E. Gubernatis. Bayesian inference and the analytic continuation of imaginary- time quantum Monte Carlo data.Physics Reports, 269(3):133–195, May 1996

    1996

  12. [20]

    Electron Correlation and Ferromagnetism of Transition Metals.Progress of Theoretical Physics, 30(3):275–289, 1963

    Junjiro Kanamori. Electron Correlation and Ferromagnetism of Transition Metals.Progress of Theoretical Physics, 30(3):275–289, 1963

    1963

  13. [21]

    Interaction-expansion inchworm Monte Carlo solver for lattice and impurity models.Physical Review B, 105(16):165133, 2022

    Jia Li, Yang Yu, Emanuel Gull, and Guy Cohen. Interaction-expansion inchworm Monte Carlo solver for lattice and impurity models.Physical Review B, 105(16):165133, 2022

    2022

  14. [22]

    Elliott H. Lieb. The Hubbard model: Some Rigorous Results and Open Problems. InCondensed Matter Physics and Exactly Soluble Models: Selecta of Elliott H. Lieb. Berlin, Heidelberg, 2004

    2004

  15. [23]

    Sparsity Pattern of the Self-energy for Classical and Quantum Impurity Problems.Annales Henri Poincar´ e, 21(7):2219–2257, 2020

    Lin Lin and Michael Lindsey. Sparsity Pattern of the Self-energy for Classical and Quantum Impurity Problems.Annales Henri Poincar´ e, 21(7):2219–2257, 2020

    2020

  16. [24]

    Martin, Lucia Reining, and David M

    Richard M. Martin, Lucia Reining, and David M. Ceperley.Interacting Electrons: Theory and Computational Approaches. Cambridge University Press, Cambridge, 2016

    2016

  17. [25]

    Correlated Lattice Fermions in d =$\infty$Dimensions

    Walter Metzner and Dieter Vollhardt. Correlated Lattice Fermions in d =$\infty$Dimensions. Physical Review Letters, 62(3):324–327, January 1989

    1989

  18. [26]

    Uber beschrankte Funktionen, die in gegeben Punkten vorgeschrieben Werte an- nehmen.Ann

    Rolf Nevanlinna. Uber beschrankte Funktionen, die in gegeben Punkten vorgeschrieben Werte an- nehmen.Ann. Acad. Sci. Fenn. Ser. A 1 Mat. Dissertationes, 1919

    1919

  19. [27]

    Interpolating Blaschke products solving Pick-Nevanlinna problems.Journal d’Analyse Mathematique, 62(1):199–224, December 1994

    Artur Nicolau. Interpolating Blaschke products solving Pick-Nevanlinna problems.Journal d’Analyse Mathematique, 62(1):199–224, December 1994

    1994

  20. [28]

    The Nevanlinna-Pick Interpolation Problem

    Artur Nicolau. The Nevanlinna-Pick Interpolation Problem. 2015

    2015

  21. [29]

    TRIQS: A toolbox for research on interacting quantum systems.Computer Physics Communications, 196:398–415, 2015

    Olivier Parcollet, Michel Ferrero, Thomas Ayral, Hartmut Hafermann, Igor Krivenko, Laura Messio, and Priyanka Seth. TRIQS: A toolbox for research on interacting quantum systems.Computer Physics Communications, 196:398–415, 2015

    2015

  22. [30]

    Rudolph Pariser and Robert G. Parr. A Semi-Empirical Theory of the Electronic Spectra and Electronic Structure of Complex Unsaturated Molecules. II.The Journal of Chemical Physics, 21(5):767–776, 1953

    1953

  23. [31]

    ¨Uber die Beschr¨ ankungen analytischer Funktionen, welche durch vorgegebene Funk- tionswerte bewirkt werden.Mathematische Annalen, 77(1):7–23, March 1915

    Georg Pick. ¨Uber die Beschr¨ ankungen analytischer Funktionen, welche durch vorgegebene Funk- tionswerte bewirkt werden.Mathematische Annalen, 77(1):7–23, March 1915

    1915

  24. [32]

    John A. Pople. Electron interaction in unsaturated hydrocarbons.Transactions of the Faraday Society, 49(0):1375–1385, 1953

    1953

  25. [33]

    A. N. Rubtsov, V. V. Savkin, and A. I. Lichtenstein. Continuous-time quantum Monte Carlo method for fermions.Physical Review B, 72(3):035122, July 2005

    2005

  26. [34]

    TRIQS/CTHYB: A continuous- time quantum Monte Carlo hybridisation expansion solver for quantum impurity problems.Computer Physics Communications, 200:274–284, 2016

    Priyanka Seth, Igor Krivenko, Michel Ferrero, and Olivier Parcollet. TRIQS/CTHYB: A continuous- time quantum Monte Carlo hybridisation expansion solver for quantum impurity problems.Computer Physics Communications, 200:274–284, 2016

    2016

  27. [35]

    Shapiro.A Fixed-Point Farrago

    Joel H. Shapiro.A Fixed-Point Farrago. Universitext. Springer International Publishing, Cham, 2016

    2016

  28. [36]

    Extended dynamical mean-field theory and GW method.Physical Review B, 66(8):085120, 2002

    Ping Sun and Gabriel Kotliar. Extended dynamical mean-field theory and GW method.Physical Review B, 66(8):085120, 2002

    2002

  29. [37]

    Doping a Wigner- Mott insulator: Exotic charge orders in transition metal dichalcogenide moir´ e heterobilayers.Physical Review Research, 5(4):043190, 2023

    Yuting Tan, Pak Ki Henry Tsang, Vladimir Dobrosavljevi´ c, and Louk Rademaker. Doping a Wigner- Mott insulator: Exotic charge orders in transition metal dichalcogenide moir´ e heterobilayers.Physical Review Research, 5(4):043190, 2023. 27

    2023

  30. [38]

    Philipp Werner, Armin Comanac, Luca de’ Medici, Matthias Troyer, and Andrew J. Millis. Continuous-Time Solver for Quantum Impurity Models.Physical Review Letters, 97(7):076405, 2006

    2006

  31. [39]

    Philipp Werner and Andrew J. Millis. Hybridization expansion impurity solver: General formulation and application to Kondo lattice and two-orbital models.Physical Review B, 74(15):155107, 2006

    2006

  32. [40]

    X. Y. Zhang, M. J. Rozenberg, and G. Kotliar. Mott transition in the d =\infty Hubbard model at zero temperature.Physical Review Letters, 70(11):1666–1669, March 1993. 28

    1993

  33. [2023]

    arXiv:2210.04187 [cond-mat, physics:physics]

    arXiv

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