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REVIEW 4 major objections 4 minor 29 references

Real-frequency TPSC+DMFT investigation of the square-lattice Hubbard model

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read TPSC+DMFT at the one-shot level reproduces the pseudogap, the Mott insulator, and the doping-driven Fermi-pockets-to-arcs crossover in the square-lattice Hubbard model.

desk verdict A useful real-frequency TPSC+DMFT implementation with clear qualitative physics, but the acknowledged NCA solver bias and missing quantitative benchmark keep the central claim conditional. read the letter →

arxiv 2501.05346 v1 pith:OPZBTHUT submitted 2025-01-09 cond-mat.str-el

Lei Geng , Jiawei Yan , Philipp Werner This is my paper · ORCID
classification cond-mat.str-el PACS 71.10.Fd71.27.+a71.30.+h
keywords HubbardmodelTPSC+DMFTpseudogapFermiarcsMottinsulatorspinfluctuationsreal-frequencymethodssquarelattice
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 sets out to show that a one-shot hybrid of Dynamical Mean-Field Theory (DMFT), which supplies local correlations, and Two-Particle Self-Consistent theory (TPSC), which supplies nonlocal spin and charge fluctuations, can describe the two-dimensional Hubbard model in and beyond the moderately correlated regime. The central claim is that using DMFT's double occupation to fix TPSC's effective vertices, then adding TPSC's momentum-dependent self-energy to DMFT's local self-energy, reproduces the pseudogap (a partial suppression of low-energy spectral weight at the antinode), the Mott insulator (an insulating state driven by electron repulsion) at large interaction strength, and the doping-driven evolution of Fermi pockets into Fermi arcs. A reader should care because the calculation runs entirely on the real-frequency axis, avoiding analytic continuation, and is far cheaper than cluster methods that have previously produced these features. If the claim holds, TPSC+DMFT is a practical bridge between weak- and strong-coupling physics of a central model of correlated electrons.

What carries the argument

The load-bearing object is the TPSC nonlocal self-energy, built from renormalized spin and charge susceptibilities $\chi_{\rm sp}$ and $\chi_{\rm ch}$. TPSC replaces the bare interaction $U$ with two effective vertices $U_{\rm sp}$ and $U_{\rm ch}$, chosen so that the local susceptibilities satisfy the Pauli-principle sum rules; the double occupation entering those sum rules is supplied by DMFT rather than by the usual phenomenological Ansatz. The resulting self-energy averages longitudinal and transverse channels, $\Sigma^C_{i,j}(t,t') = -\frac{iU}{8} G^{(1)}_{i,j}(t,t')\,[U_{\rm ch}\chi^{\rm ch}_{j,i}(t',t)+3U_{\rm sp}\chi^{\rm sp}_{j,i}(t',t)]$, and is added once to the DMFT local self-energy inside the Dyson equation. The spin-channel part of this nonlocal self-energy produces a mirrored band, an image of the noninteracting dispersion that suppresses the antinodal region and creates the pseudogap; the charge-channel part generates high-energy satellite structures. A real-frequency, steady-state implementation with a non-crossing approximation (NCA) impurity solver avoids analytic continuation and resolves spectra on a $64\times64$ momentum grid.

What would settle it

Take the half-filled $U=4$, $t'=0$, $T=0.25$ case and replace the NCA double occupation in the TPSC sum rules with a value from a high-precision local or cluster solver; if $U_{\rm sp}$ and $U_{\rm ch}$ change appreciably and the antinodal spectral-weight suppression at $X$ disappears, the pseudogap is a solver artifact rather than nonlocal spin physics. Repeat the test at 10% doping by comparing the computed Fermi arc length with an independent cluster result.

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Extended reading notes

Core claim

The central claim is that the one-shot TPSC+DMFT construction is already sufficient for a meaningful description of both moderately and strongly correlated regimes of the square-lattice Hubbard model. In the half-filled $U=4$ system at $T=0.25$, adding the TPSC nonlocal self-energy to the DMFT local self-energy suppresses spectral weight at the antinodal $X$ point while leaving the nodal region metallic, producing a pseudogap that plain DMFT lacks and that FLEX is too smooth to form. For $U=6$ and $U=8$ at half-filling, the same construction keeps the Mott gap essentially intact, with nonlocal corrections only redistributing spectral weight. In the doped Mott regime ($U=8$, $t'=-0.3t$, at 5%, 10%, and 20% hole doping), the method yields a Fermi pocket at low doping, a clear Fermi arc at 10%, and an almost full Fermi surface at 20%, in line with cluster methods and photoemission studies. The paper also reports that self-consistent variants (TPSC+GG and TPSC+GG+DMFT) lose the pseudogap and collapse toward the DMFT solution, so the one-shot character is not a numerical shortcut but part of the mechanism.

Load-bearing premise

The construction stands or falls on the DMFT double occupation computed with the non-crossing approximation (NCA) solver being accurate enough to fix the effective vertices $U_{\rm sp}$ and $U_{\rm ch}$; the authors themselves note that NCA tends to underestimate this double occupation and they deliberately avoid an adjustable compensation, so a quantitatively wrong value could turn the pseudogap and Fermi arcs into solver artifacts.

Editorial extensions

If this is right

  • At half-filling with $U=4$, TPSC+DMFT predicts a pseudogap whose origin is nonlocal spin fluctuations at the antinode, while DMFT alone and FLEX do not produce this feature.
  • In the strongly correlated half-filled regime ($U=6$ and $U=8$), the nonlocal self-energy corrections do not destroy the Mott gap, indicating that the DMFT local physics dominates there.
  • In the doped Mott insulator, the method predicts a Fermi pocket at low doping that evolves into a Fermi arc and then a nearly full Fermi surface as doping increases, a sequence that can be compared directly with photoemission arc-length measurements.
  • Self-consistent feedback variants are contraindicated: iterating the interacting Green's function back into TPSC erases the nonlocal correlations and the pseudogap, leaving a solution close to plain DMFT.
  • The real-frequency implementation supplies momentum- and frequency-resolved spectra without analytic continuation, so the same setup can produce high-resolution benchmarks for other approaches.

Reading between the lines

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

  • A testable extension of the paper's own caveat: if the NCA double occupation is corrected by a more accurate impurity solver, the same one-shot construction could be pushed to lower temperatures or stronger couplings, where the pseudogap is sharper and the Fermi arcs shorter.
  • The steep rise of the charge vertex $U_{\rm ch}$ with underdoping (134.9 at 5% doping versus 32.5 at 20%) suggests that the upper-band distortions seen in the doped spectra are charge-channel artifacts; a natural test is to keep only the transverse spin channel below 10% doping and compare the resulting spectral functions.
  • The real-frequency Keldysh setup is ready in principle for nonequilibrium studies, and the missing ingredient the paper identifies is self-consistency; a partially self-consistent scheme that preserves the one-shot nonlocal vertex would open photo-doped Mott states to this method.
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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

4 major / 4 minor

Summary. The paper presents a real-frequency implementation of TPSC+DMFT for the square-lattice Hubbard model. In the proposed scheme, a DMFT calculation with an NCA impurity solver provides the local self-energy and the double occupation; the double occupation fixes the TPSC spin and charge vertices through the sum rules, and the TPSC nonlocal self-energy is added one-shot to the DMFT Green's function. Results are reported for half-filling at U=4, 6, 8 and for hole doping from 5% to 20% with t'=-0.3t and U=8. The authors observe pseudogap features at moderate coupling, a Mott gap at strong coupling, and a Fermi-pocket-to-arc evolution upon doping, and they conclude that one-shot TPSC+DMFT is a versatile and efficient bridge between weak- and strong-coupling descriptions of the Hubbard model.

Significance. If the claims are correct, the paper provides a computationally efficient real-frequency hybrid that avoids analytic continuation and offers direct access to spectral functions and Fermi surfaces. The strengths are the real-frequency implementation, the use of sum rules rather than fitted parameters, and the systematic comparison of TPSC, DMFT, FLEX, and TPSC+DMFT within a common setup. However, the central claim rests on the accuracy of the NCA double occupation and on qualitative comparisons with cluster DMFT and ARPES. No same-parameter quantitative benchmark is provided, and the acknowledged NCA bias could change the vertices and the nonlocal self-energy enough to alter the pseudogap and Fermi-arc features. The paper is therefore a useful methodological contribution whose main physical conclusions are not yet fully established.

major comments (4)
  1. [§III A, Table I, Eq. (12)] The sum rules in Eq. (12) are closed with the DMFT double occupation, and the resulting vertices Usp and Uch enter directly into the nonlocal self-energy in Eq. (17). In Section III A the authors state that the NCA solver 'tends to underestimate the double occupation' and overestimates correlation strength, yet they deliberately do not compensate by reducing U. For half-filling at U=8 the charge vertex is Uch=269.89 (Table I), and for 5% hole doping it is 134.88 (Table II). Because the charge vertex is so large, a moderate error in the double occupation can change the susceptibilities and the nonlocal self-energy substantially. The authors should quantify this sensitivity, for example by benchmarking the NCA double occupation against a higher-order steady-state impurity solver (refs. 40,41) or by repeating the TPSC+DMFT calculation with a DMFT U reduced within the stated NCA bias and showing that the pseudogap and Fermi-arc features are stable.
  2. [§III A–C, Figs. 3, 6, 9, 10] The paper's central claim that one-shot TPSC+DMFT gives a meaningful description is supported only by visual comparison with cluster DMFT and ARPES, not by a quantitative same-parameter benchmark. For example, the Fermi surface of the 5% doped system in Fig. 10(c) shows discontinuities, and the text acknowledges that TPSC+DMFT 'struggles in the underdoped regime'; the comparison with cluster DMFT results (Refs. 46,53) is qualitative. A direct benchmark at the same U, T, t', and doping—e.g., the antinodal spectral weight as a function of doping or the momentum dependence of the self-energy along the Fermi surface—would establish whether the pseudogap and Fermi-arc evolution are quantitatively reliable or mainly artifacts of the approach. Without such a benchmark, the main physical conclusions remain suggestive.
  3. [§III B, text before Fig. 7] For U=6 and U=8, the authors note that there may be two pairs of solutions for Usp and Uch and select the solution satisfying Usp<U and continuously connected to U=4. Since the selected vertices determine the nonlocal self-energy through Eq. (17), the results in Fig. 7 depend on this branch choice. The authors should justify why the discarded branch is unphysical and show that the main features (Mott gap, pseudogap, satellite structures) are robust with respect to the branch selection.
  4. [§II D, Appendix A] The one-shot character is a central methodological assumption: iterating the loop with NCA fails to converge, and the self-consistent variants in Appendix A lose the pseudogap and almost coincide with DMFT. This leaves open whether the pseudogap is a robust feature of the combined scheme or an artifact of stopping before feedback. A useful test would be to replace G(1) in Eq. (17) with the DMFT Green's function in the TPSC self-energy and check whether the qualitative conclusions survive; alternatively, the authors could provide a convergence diagnostic or a physical argument for why the one-shot stopping point is preferred.
minor comments (4)
  1. [§III B, text after Fig. 7] The text says 'TPSC+DMFT spectra, shown in Figs. 7(d), 7(f), and 7(i)', but the TPSC+DMFT panels in Fig. 7 are (c), (f), and (i), not (d), (f), and (i). Please correct the cross-reference.
  2. [§II F] The bath parameters Γ=0.05 and D=32 are introduced to stabilize the chemical potential, but their effect on the presented spectra is not discussed. A brief statement that the results are insensitive to these parameters, or a plot for a different Γ value, would clarify that the bath does not alter the physics.
  3. [Fig. 4 caption] The caption says TPSC+DMFT '(same as TPSC)', which is correct because the nonlocal self-energy is identical, but readers may be confused about whether the local self-energy is included. Please state explicitly in the text that Fig. 4 shows only the nonlocal part, with the local part subtracted.
  4. [General] Several equations contain apparent rendering artifacts in the arXiv text (e.g., Eq. (1) and Eq. (9)); please ensure the published version uses clean typeset notation.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: TPSC+DMFT's vertices are fixed by the DMFT double occupation, but the pseudogap and Fermi-arc outputs are not fed back into the fit, and validation rests on external cluster-DMFT and ARPES comparisons.

full rationale

The paper's derivation chain is self-contained and not circular in the sense prohibited by the review rules. The central construction is: (i) solve DMFT with the NCA impurity solver, (ii) take the DMFT double occupation and solve the TPSC sum rules, Eq. (12), for U_sp and U_ch, and (iii) use these vertices in the one-shot nonlocal self-energy, Eq. (17), to compute spectral functions and Fermi surfaces. The claimed outputs—pseudogap, Mott gap, Fermi pockets, and Fermi arcs—are not used as inputs anywhere in this chain. U_sp and U_ch are fixed by a local quantity (double occupation) plus the TPSC sum rules, not by the target momentum-resolved features. The pseudogap therefore emerges from the nonlocal spin-fluctuation self-energy rather than being imposed. The authors explicitly acknowledge the main accuracy limitation, stating that the NCA solver 'tends to underestimate the double occupation' and that reducing U in DMFT could compensate, but they choose not to introduce adjustable parameters. This is a correctness risk, not circularity. The Fermi-pocket-to-arc comparison is made against externally computed cluster DMFT results and ARPES data, not against the paper's own fitted parameters. Self-citations to Refs. [20] and [23] provide the TPSC+DMFT framework and prior applications, but the load-bearing claim that self-consistent feedback fails is also supported by the paper's own Appendix A, which shows that TPSC+GG and TPSC+GG+DMFT miss the pseudogap. No specific equation reduces to its own target, and no parameter is renamed as a prediction. The paper thus merits a low circularity score, with only minor, non-load-bearing self-citation present.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central results rest on the TPSC ladder approximation, the DMFT local self-energy from NCA, and the one-shot merging of nonlocal TPSC self-energy with DMFT. The bath parameters are numerical stabilizers. No new physical entities are introduced.

free parameters (2)
  • bath coupling Γ = 0.05
    Chosen to stabilize the chemical potential and accelerate convergence in the steady-state implementation; it introduces a finite broadening and may affect the spectral functions.
  • bath half-bandwidth D = 32
    Chosen as a wide, weakly-coupled bath to avoid artificial finite-size effects; a numerical stabilizer rather than a physical parameter.
assumptions (5)
  • domain assumption TPSC susceptibility ladder form with renormalized vertices U_sp and U_ch (Eq. 11)
    The nonlocal self-energy is computed from RPA-like susceptibilities, which is an uncontrolled approximation for strong interactions; central to the method.
  • domain assumption The TPSC sum rules (Eq. 12) closed with the DMFT double occupation instead of the original ansatz (Eq. 13)
    Assumes the DMFT double occupation is more accurate than the TPSC ansatz; this is the key input that links DMFT and TPSC.
  • ad hoc to paper One-shot combination: the TPSC susceptibilities are not updated with the final self-energy
    The authors note that self-consistent variants failed to converge with NCA; therefore the one-shot procedure is a pragmatic but unjustified approximation.
  • domain assumption NCA impurity solver provides a qualitatively correct local self-energy and double occupation
    NCA is known to overestimate correlations; the authors acknowledge this and rely on it for the DMFT component, affecting all final results.
  • domain assumption The single-band Hubbard model with t'=-0.3 is a valid minimal model for cuprate physics
    Used to motivate the doping study and compare with ARPES; the model choice is standard but not directly validated in this paper.

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Cite this review

Pith. "Pith review of Real-frequency TPSC+DMFT investigation of the square-lattice Hubbard model." pith.science (2026). https://pith.science/paper/OPZBTHUT

@misc{pith2026250105346,
  author       = {Pith},
  title        = {Pith review of: Real-frequency TPSC+DMFT investigation of the square-lattice Hubbard model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OPZBTHUT}},
  note         = {Machine review of arXiv:2501.05346}
}
read the original abstract

We investigate the two-dimensional Hubbard model using a real-frequency implementation of the TPSC+DMFT approach. This hybrid method combines the nonlocal correlations captured by the Two-Particle Self-Consistent (TPSC) approach with the local dynamical correlations of Dynamical Mean-Field Theory (DMFT). The results demonstrate that TPSC+DMFT effectively describes pseudogap physics and nonlocal fluctuations in the moderately correlated regime, while also reproducing the Mott insulating state at larger interaction strengths. For doped Mott insulators, we find that TPSC+DMFT captures the evolution of Fermi pockets into Fermi arcs, consistent with the results from cluster DMFT and photoemission studies. These findings highlight the capability of TPSC+DMFT to bridge the gap between weak and strong coupling physics in Hubbard models, providing insights into spin and charge fluctuations, as well as their role in the pseudogap formation.

Figures

Figures reproduced from arXiv: 2501.05346 by the authors.

Figure 1
Figure 1. FIG. 1. Feynman diagrams for the self-energy Σ [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Feynman diagrams for the self-energy Σ [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Spectral functions for the half-filled Hubbard model [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Imaginary part of the nonlocal self-energy as a funct [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Imaginary parts of the susceptibilities as a functio [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Fermi surface of the half-filled Hubbard model with [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Spectral functions for the half-filled Hubbard model [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Imaginary parts of the nonlocal self-energies for th [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Spectral functions for the hole-doped Hubbard model [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Fermi surface of the hole-doped Hubbard model with [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. FLEX spectral function (a) and Fermi surface (b) [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Spectral function for the half-filled Hubbard model [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [1]

    Conse- quently, finding accurate solutions to the Hubbard model has remained an important problem in theoretical con- densed matter physics. In one dimension, the exact so- lution for the ground state of the Hubbard model was provided by Lieb and Wu using the Bethe Ansatz 2, and extensions to few-leg ladders have been studied using the density matrix renor...

    work page

  2. [2]

    Self-energies and susceptibilities To explore how the nonlocal self-energy improves the results of DMFT, we plot the nonlocal self-energy ob- tained from TPSC and FLEX in Fig

    work page

  3. [3]

    However, for more realistic two- and three-dimensional systems, obtaining accurate solutions and clarifying the physics of the Hubbard model remains a formidable challenge 6

    In the limit of infinite dimensions, dynamical mean-field the- ory (DMFT) has been established as an exact numeri- cal framework 4,5. However, for more realistic two- and three-dimensional systems, obtaining accurate solutions and clarifying the physics of the Hubbard model remains a formidable challenge 6. Over the years, various numerical approaches have ...

    work page

  4. [4]

    We use the nearest- neighbor hopping t = 1 as the unit of energy throughout the paper and set the lattice spacing to unity

    is often expressed in momentum space as ∑k,σ ǫkc† k,σ ck,σ , where ǫk = −2t [cos(kx) + cos(ky)] − 4t′ cos(kx) cos(ky) (2) is the noninteracting dispersion. We use the nearest- neighbor hopping t = 1 as the unit of energy throughout the paper and set the lattice spacing to unity. In the two-dimensional Hubbard model, nonlocal cor- relations play a significa...

    work page

  5. [5]

    DMFT is exact in the limit of infinite dimensions 4 but provides only an approx- imate description of (interacting) two-dimensional sys- tems

    This enables DMFT to describe a Mott-insulating state. DMFT is exact in the limit of infinite dimensions 4 but provides only an approx- imate description of (interacting) two-dimensional sys- tems. By assuming a spatially local self-energy Σ (ω) 26, the lattice model is mapped onto a single-site Anderson impurity model, where the hoppings between the impu-...

    work page

  6. [6]

    (6) In this way, we avoid the explicit calculation of the self- energy, which may have significant numerical errors at high frequencies

    via the Dyson equation G−1 ato(ω) = G−1 imp(ω) + ∆ (ω), G−1(k, ω) = G−1 ato(ω) − ǫk. (6) In this way, we avoid the explicit calculation of the self- energy, which may have significant numerical errors at high frequencies

    work page

  7. [8]

    This trend explains the enhanced pseudogap observed for larger U

    The data re- veal that the intensity of the mirrored band of the self- energy near the Fermi surface increases as U increases. This trend explains the enhanced pseudogap observed for larger U . Additionally, the contribution from χch, which is responsible for the satellite structures in the spectral functions, shifts farther away from the contributions of...

    work page

  8. [9]

    The TPSC+DMFT results combine the shorter quasi-particle lifetimes of DMFT (width and intensity of the signal) with the momentum- dependent lifetimes of TPSC

    However, while the DMFT results show a uniform intensity along the Fermi surface, the TPSC results display arcs concentrated near the nodal point, due to the large imaginary self-energy compo- nents at the antinodal points. The TPSC+DMFT results combine the shorter quasi-particle lifetimes of DMFT (width and intensity of the signal) with the momentum- dep...

    work page

Show all 29 references
  1. [10]

    (14) To derive a more detailed expression, the reducible ver- tex must be expanded into irreducible vertices

    It can be divided into two parts: the Hartree term and the contribution from the fully reducible vertex Σ C (represented by the gray box in the figure): Σ (2) i,j (t, t′) = −iU G(1) i,i (t, t) + Σ C i,j (t, t′). (14) To derive a more detailed expression, the reducible ver- tex ...

  2. [11]

    These are generally referred to as the longitudinal channel and the transversal chan- nel, respectively

    into ladder diagrams. These are generally referred to as the longitudinal channel and the transversal chan- nel, respectively. Since the contribution from the reducible vertex can be expressed as a functional derivative with respect to an external field, we can couple either a ...

  3. [12]

    one-shot

    and ( 11), since the double occupation ⟨ˆni, ↑(t)ˆni, ↓(t)⟩ remains unknown. In practice, it is com- mon to adopt the following Ansatz, inspired by Ref. 32, U ⟨ˆni, ↑(t)ˆni, ↓(t)⟩ = Usp ⟨ˆni, ↑(t)⟩ ⟨ˆni, ↓(t)⟩ . (13) Without considering this specific Ansatz, as long as the doub...

  4. [13]

    Furthermore, using this Ansatz can cause the charge susceptibility to diverge in the strong- interaction regime, leading to unphysical results

    becomes unreliable deep inside the renormalized classical regime. Furthermore, using this Ansatz can cause the charge susceptibility to diverge in the strong- interaction regime, leading to unphysical results. As an alternative, we can use the double occupation calculated with...

  5. [14]

    We attempted to iterate the above procedure, as done in Ref

    to compute the final Green’s function G(2), without further self-consistent it- erations. We attempted to iterate the above procedure, as done in Ref. 23, but the process failed to converge when using the NCA solver adopted in this study. An alternative approach is to combine v...

  6. [16]

    Spectral functions The local and k-resolved spectral functions obtained with the various discussed methods are presented in Fig

  7. [17]

    All the TPSC results shown in this figure and the fig- ures below refer to TPSC combined with the double oc- cupation from DMFT, but without the local self-energy substitution

    The path for the k-resolved spectra connects the high-symmetry points in the first Brillouin zone: Γ = (0, 0) →X = (π, 0) →M = (π, π) →Γ = (0, 0). All the TPSC results shown in this figure and the fig- ures below refer to TPSC combined with the double oc- cupation from DMFT, but ...

  8. [18]

    Also, as mentioned before, DMFT is expected to provide a bet- ter estimate of the double occupation

    lacks solutions for Usp or Uch for most parameter sets considered in this paper. Also, as mentioned before, DMFT is expected to provide a bet- ter estimate of the double occupation. In the case shown in Fig. 3(b), the substitution of the double occupation only affects Uch, chan...

  9. [19]

    Another point worth noting is that for U = 6 and U = 8, there may exist two pairs of solutions for Usp and Uch

    To mitigate the errors associ- ated with the TPSC solution in this regime, we increase the temperature to T = 0.4 in the subsequent calcula- tions. Another point worth noting is that for U = 6 and U = 8, there may exist two pairs of solutions for Usp and Uch. We select the mor...

  10. [21]

    one- shot

    TPSC itself is however only ap- plicable to systems with weak or moderate correlation strength. To extend it deeper into the correlated regime, TPSC has recently been combined with DMFT 22–24, ei- ther by employing the DMFT double occupation in the TPSC sum rules, or by additi...

  11. [22]

    The nonlocal self-energy shown in Fig

    Note that TPSC and TPSC+DMFT share the same nonlocal self-energy, while the DMFT self-energy is momentum independent. The nonlocal self-energy shown in Fig. 4(a) corre- sponds to the average of the transversal and longitu- dinal channels, while Fig. 4(b) presents the transvers...

  12. [23]

    The k-resolved spectral functions averaged over ω ∈ [−0.2, 0.2] are presented in Fig

    Fermi surfaces Next, we focus on the electronic structure near the Fermi level. The k-resolved spectral functions averaged over ω ∈ [−0.2, 0.2] are presented in Fig. 6. In the non- interacting limit, the Fermi surface forms a square con- necting the four X points in the first B...

  13. [24]

    The FLEX results, shown in Fig

    The nodal points ( k = (± π 2 , ± π 2 )) are relatively unaffected by spin fluctuations. The FLEX results, shown in Fig. 6(b), ex- hibit a similar pattern, but the suppression of weight at the antinodal points is significantly weaker. In contrast, the DMFT result in Fig. 6(c) dis...

  14. [26]

    The spin and charge vertices calculated by TPSC decrease with increasing hole doping, as shown in Table II

    Spectral functions We consider different hole-doping levels of 5%, 10%, and 20%. The spin and charge vertices calculated by TPSC decrease with increasing hole doping, as shown in Table II. The spectral functions along the indicated path connecting high-symmetry points, calculat...

  15. [27]

    Fermi surfaces The Fermi surfaces of the doped Mott insulators serve as an excellent benchmark for validating our method, as they can be directly compared with ARPES exper- iments 56–58. Previous studies using DMFT extensions that incorporate nonlocal fluctuations, such as clus...

  16. [28]

    To avoid adjustable parame- ters, we however prefer to use the same U as in TPSC, keeping in mind the bias from the NCA

    This could be compensated by reducing U in the DMFT calculations. To avoid adjustable parame- ters, we however prefer to use the same U as in TPSC, keeping in mind the bias from the NCA. Recently devel- oped higher-order steady-state solvers 40,41 will eliminate this uncertainty

  17. [29]

    ( 4) and the hybridiza- tion function ∆ (ω) is updated using the local Green’s function ∆ (ω) = G−1 ato(ω) − G−1 loc(ω)

    Finally, the local Green’s function Gloc(ω) can be obtained from the momentum-resolved Green’s function G(k, ω) by Eq. ( 4) and the hybridiza- tion function ∆ (ω) is updated using the local Green’s function ∆ (ω) = G−1 ato(ω) − G−1 loc(ω). (7) This loop is iterated until the G...

  18. [30]

    Two-particle-self-consistent approa ch for the hubbard model,

    FLEX fails to converge at lower doping lev- els and produces a poor spectral function (without Hub- bard band features) at 20% doping. In Fig. 11(b), the Fermi surface exhibits a non-uniform structure, similar to TPSC and TPSC+DMFT. However, the spectral func- tion in Fig. 11(...

    1995 arXiv

  19. [31]

    In this manner, additional degrees of freedom are introduced to ensure that the Pauli exclu- sion principle is satisfied

    The TPSC approach addresses this issue by employing distinct vertex functions for the spin and charge channels, χsp = χ0 1 − 1 2 Uspχ0 , χch = χ0 1 + 1 2 Uchχ0 , (11) where Usp and Uch are in general different from the Hub- bard interaction U . In this manner, additional degree...

  20. [36]

    Moreover, it is incapable of describing the Mott insulat- ing state

    However, FLEX cannot accurately capture the physics of the pseudogap, unlike TPSC 19,37. Moreover, it is incapable of describing the Mott insulat- ing state. F. Steady-state implementation In contrast to previous studies, our simulations are implemented directly on the real fr...

  21. [43]

    Since the interac- tion U is only half the bandwidth, there is no Mott gap in this system. It is interesting to note that the DMFT spectral func- tion looks like the superposition of a weakly and strongly renormalized band, with the former featuring prominent weight near Γ and...

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Reviewed August 10, 2026 · model on record in the stance chip above.