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REVIEW 3 major objections 5 minor 84 references

Dynamical Charge Susceptibility in the Hubbard Model

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Pith's one-line read Charge fluctuations show no signature of the pseudogap in the eight-site Hubbard model, the authors argue, with the dynamical charge susceptibility reducing to a single featureless peak.

desk verdict First DCA calculation of the dynamical charge susceptibility in the Hubbard model's pseudogap regime; clean negative result for an eight-site cluster, but cluster-size convergence at low temperature is the key caveat. read the letter →

arxiv 1908.04776 v2 pith:E6LMJ63A submitted 2019-08-13 cond-mat.str-el

Xinyang Dong , Xi Chen , Emanuel Gull This is my paper · ORCID
classification cond-mat.str-el PACS 71.10.Fd71.27.+a74.72.-h
keywords HubbardmodeldynamicalchargesusceptibilitypseudogapclusterapproximationcupratesfluctuationsfluctuationdiagnosticsM-EELS
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 asks whether charge fluctuations can explain the pseudogap in the two-dimensional Hubbard model, the standard theoretical stand-in for the cuprate high-temperature superconductors. Using an eight-site dynamical cluster approximation, the authors compute the dynamical charge susceptibility across momenta, dopings, and temperatures, including the pseudogap regime. They find that the susceptibility is featureless: a single peak at a characteristic frequency with almost no momentum or temperature dependence and only a mild doping dependence. They conclude that charge fluctuations are not a good way to describe pseudogap physics in the parameter range studied, and that short-ranged antiferromagnetic spin fluctuations, not charge modes, carry the self-energy changes associated with the pseudogap.

What carries the argument

The central object is the dynamical charge susceptibility $\chi_{ch}(Q,\Omega)$, defined from the two-particle Green's function in the density channel and obtained from the eight-site dynamical cluster approximation (DCA) with a numerically exact continuous-time auxiliary-field impurity solver. The DCA coarse-grains momentum space into cluster momenta, giving four independent momentum transfers: $Q=(0,0)$, $(\pi/2,\pi/2)$, $(\pi,0)$, and $(\pi,\pi)$. The paper also uses fluctuation diagnostics, which expresses the single-particle self-energy in terms of two-particle quantities through the exact equation of motion, separating the self-energy into charge and magnetic fluctuation contributions; this is the machinery that lets the authors attribute the pseudogap to spin rather than charge fluctuations.

What would settle it

Compute the dynamical charge susceptibility $\chi_{ch}(Q,\Omega)$ on a 16-site or larger DCA cluster at the same $U/t=7$, $t'/t=-0.15$ parameters and at low temperatures ($\beta \gtrsim 10$) inside the pseudogap regime; if a significant momentum-dependent low-frequency peak or a $Q=(\pi/2,\pi/2)$ or commensurate charge response emerges that is absent on the eight-site cluster, the negative finding is a cluster-size artifact. A corresponding experiment would be to measure M-EELS on an underdoped cuprate near $\delta \approx 0.1$ and look for a sharp low-energy charge mode at a nonzero momentum that the single-peak prediction of this paper would miss.

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Extended reading notes

Core claim

The central claim is that in the eight-site dynamical cluster approximation of the two-dimensional Hubbard model at U/t = 7 and t'/t = -0.15, the dynamical charge susceptibility chi_ch(Q, $\Omega$) shows no clear signature of the pseudogap. As a function of frequency it is well represented by a single peak at a characteristic frequency; the peak position and weight show little momentum or temperature dependence, while doping shifts the peak to lower frequencies and sharpens it. The static uniform susceptibility rises with doping and is not suppressed upon entering the pseudogap, in contrast to the magnetic susceptibility. Vertex corrections are essential: without them the bare susceptibility has a dominant (pi,pi) contribution, whereas the full vertex-suppressed result shows comparable weight at all momenta. Fluctuation diagnostics decompose the self-energy into charge and spin channels; the pseudogap is well described by short-ranged Q = (pi,pi) magnetic fluctuations, while charge modes contribute from all momenta and many frequencies with comparable strength. The authors therefore state that charge fluctuations are not a good way to describe pseudogap physics in the entire parameter range studied here.

Load-bearing premise

The conclusion rests on the assumption that the eight-site DCA cluster gives enough momentum resolution to see the charge fluctuations that matter for the pseudogap, an assumption the authors themselves qualify by noting that the method is insensitive to stripes with periods larger than the cluster.

Editorial extensions

If this is right

  • If the central claim is correct, the pseudogap in the cuprate-relevant Hubbard model should not be interpreted as a charge-fluctuation phenomenon, and theories that build the pseudogap from charge order or charge modes would need to explain why the eight-site susceptibility shows no accompanying charge signature.
  • The computed $\chi_{ch}(Q,\Omega)$ provides a direct target for momentum-resolved electron energy-loss spectroscopy (M-EELS) and resonant inelastic X-ray scattering (RIXS) on cuprates, and a mismatch between those experiments and the single-peak prediction would indicate physics beyond the one-band Hubbard model at these parameters.
  • Because vertex corrections are essential in suppressing the $(\pi,\pi)$ charge response, low-order diagrammatic approximations that neglect vertex corrections will overestimate charge fluctuations and their role in the self-energy; conserving approximations are required for quantitative statements.
  • The fluctuation diagnostics result implies that any description of the pseudogap in terms of bosonic modes should be dominated by short-ranged antiferromagnetic fluctuations, and that charge-mode descriptions would require summing contributions from all momenta and a broad frequency range, making them impractical.
  • The near temperature independence of $\chi_{ch}$ across the pseudogap crossover means that charge susceptibility measurements alone are unlikely to serve as a pseudogap thermometer, unlike the magnetic susceptibility.

Reading between the lines

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

  • The paper's negative finding is a constraint on cluster size: a larger cluster that resolves stripe-like charge order with periods longer than eight sites could reveal charge fluctuations tied to the pseudogap that this calculation cannot see, so the claim is only as strong as the cluster momentum resolution.
  • A testable extension would be to compute the same quantities on larger clusters (e.g., 16- or 32-site DCA) at the low temperatures where the pseudogap opens, checking whether the single-peak structure and the absence of momentum dependence survive when longer-wavelength charge modulations are resolvable.
  • The similar magnitude of $\chi_{ch}$ at $(\pi,0)$ and $(\pi/2,\pi/2)$ suggests an approximate isotropy of charge fluctuations in momentum space that could be a generic feature of the doped Mott insulator, but confirming this would require systematic study at other $U$ and $t'$ values.
  • The authors' conclusion that the doping evolution of $\chi_{ch}$ is not caused by the pseudogap implies that the doping-dependent peak shift could instead track the single-particle bandwidth renormalization or the Mott scale, an attribution they do not make explicitly.
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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 / 5 minor

Summary. This paper presents dynamical cluster approximation (DCA) calculations of the charge susceptibility χch(Q,Ω) for the two-dimensional Hubbard model at U/t=7, t'/t=-0.15, using a numerically exact continuous-time auxiliary-field impurity solver on an eight-site cluster. The authors study the momentum, doping, and temperature dependence of the susceptibility and report that it can be described by a single peak at a characteristic frequency, with little momentum and temperature dependence, while doping shifts the peak to lower frequencies. They observe no clear signature of the pseudogap in the charge channel, in contrast to the magnetic channel, and use fluctuation diagnostics to argue that charge fluctuations are not a good way to describe pseudogap physics. The paper also compares with the bare (vertex-free) susceptibility, showing that vertex corrections suppress and flatten the momentum dependence, and suggests the results are relevant for momentum-resolved electron energy-loss spectroscopy.

Significance. If the central claims hold, this paper provides a useful reference calculation of the dynamical charge susceptibility in a regime relevant to cuprate high-temperature superconductors, filling a gap noted by the authors: previous theoretical results were available mostly at high temperature, while the pseudogap regime was unexplored in this observable. The use of an exact impurity solver and the explicit discussion of vertex corrections are strengths, as is the proposal of a direct comparison with M-EELS experiments. The fluctuation-diagnostics analysis also connects two-particle observables to the single-particle self-energy, which is a valuable methodological contribution. The main importance lies in the negative result: if charge fluctuations are not a good descriptor of the pseudogap at accessible momenta, this constrains theoretical scenarios linking charge order and the pseudogap.

major comments (3)
  1. [Section 2 (paragraph beginning 'Eight-site DCA yields...') and Conclusion] The central negative claim—that charge fluctuations show no pseudogap signature—is supported only at the four independent cluster momenta Q=(0,0), (π/2,π/2), (π,0), and (π,π). The authors themselves note in the Conclusion that DCA is insensitive to stripes with periods larger than the cluster size, and the 4-site and 16-site checks mentioned in Section 2 are described only at 'select points and high temperature', not at the low temperatures where the pseudogap develops. Since the pseudogap regime is the regime in which the negative claim is made (e.g., β=15 at δ=-0.05), the cluster-size dependence of the central result is not established. Please extend the 16-site (or at least a systematic 4- vs 8-site) comparison to β=15 for the underdoped and optimally doped cases, or explicitly restrict the main conclusion to the momentum resolution of the eight-site cluster throughout the abstract and conclusion.
  2. [Figs. 2–5 and related text] No statistical error bars are reported for χch(Q,iΩ_n) or for the analytically continued real-frequency spectra. The statements 'little temperature dependence' (Fig. 4) and 'no clear signature of the pseudogap' are based on differences of order 10–20% between β=5 and β=15 (e.g., the low-frequency Matsubara values in Fig. 4). Without error bars or tabulated values with uncertainties, the reader cannot assess whether the observed differences are statistically significant or whether the apparent absence of a pseudogap signature is an artifact of noise. Please provide error bars at least for the static zero-frequency susceptibilities and for the peak positions of the analytically continued spectra, and state the Monte Carlo statistics used for the two-particle Green's functions.
  3. [Fig. 6 and Eqs. (6)–(7)] The fluctuation diagnostics that motivate the conclusion 'charge fluctuations are not a good way to describe pseudogap physics' are presented only at β=10. According to the text around Fig. 4 and the phase diagram in Fig. 1, at δ=-0.05 the system 'gradually enter[s] the pseudogap regime' on cooling from β=5 to β=15, so β=10 may be on the boundary of, rather than fully inside, the pseudogap regime. To support the conclusion in the pseudogap region, please show the same diagnostics at β=15 for an underdoped doping (or explicitly state that β=10 is inside the pseudogap phase and justify the representativeness). In addition, the pie charts in Fig. 6 sum only the first 10 Matsubara frequencies; please justify that higher-frequency contributions do not change the qualitative conclusion.
minor comments (5)
  1. [Throughout] Typographical inconsistencies: 'Figure. 2', 'Figure. 4', and similar should be 'Fig. 2', 'Fig. 4', etc.; 'Pi chart' in the Fig. 6 caption should be 'Pie chart'.
  2. [Page 3, left column (Ward identity sentence)] The sentence 'A Ward identity requires the frequency dependence to be identically zero in systems that conserve total charge' is unclear. What vanishes for Ω≠0 at Q=(0,0) is χch(Q=0,iΩ_n), not its frequency dependence in general; please rephrase to state the Q=(0,0) result explicitly.
  3. [Abstract and closing paragraph] The authors state the results 'should be directly measurable' in M-EELS experiments, but M-EELS typically measures the loss function Im[-1/ε], which involves the dielectric function rather than the bare charge susceptibility. A brief statement relating χch to the loss function or to the dielectric function would make the proposed comparison more concrete.
  4. [Page 4, bottom (t' dependence)] The sentence 'approximately δ → −δ for t' → −t''' is not explained; consider adding a one-sentence clarification of the approximate particle-hole relation used here.
  5. [Fig. 6 caption and text] It is not clear whether the magnetic contributions shown in Fig. 6 (lower panels) are new results from this paper or taken from reference 29; please clarify the source of the spin-channel data.

Circularity Check

0 steps flagged · score 1.0 of 10

No material circularity: the central chi_ch(Q,Omega) result is a direct, parameter-free DCA/CT-AUX computation, with self-citations only providing method and phase-diagram context.

full rationale

The central claim — that chi_ch(Q,Omega) is a single-peak function with little momentum/temperature dependence and no clear pseudogap signature — is obtained by solving the DCA equations for Eq. (1) with a numerically exact CT-AUX impurity solver and then evaluating Eq. (5). No parameter appearing in chi_ch is fitted to chi_ch itself; inputs (U/t=7, t'/t=-0.15) are standard model parameters from the cuprate literature. The self-citations (Refs. 48, 53-58, 63-65, 74) supply the DCA formalism, the eight-site phase diagram, the CT-AUX solver, and the fluctuation-diagnostics decomposition; these are external method/reference results, not restatements of the paper's susceptibility output. The fluctuation-diagnostics argument uses the exact equation-of-motion identity (Eq. (6)-(7)) already published, and the conclusion that charge fluctuations are not convenient for describing the pseudogap follows from the computed broad, multi-momentum contributions rather than being assumed. The paper's own caveat — 'due to the limited momentum resolution, DCA is insensitive to stripes with periods larger than our cluster size' — is a limitation on the strength of a negative claim, not a circular derivation. Thus the central computation has independent content and no fitted input is renamed as a prediction.

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

No parameters were fitted to the charge susceptibility data; the model parameters and approximations are standard inputs. The central claim is a direct output of a controlled numerical simulation, so the ledger contains only the modeling and approximation assumptions that the result inherits.

assumptions (5)
  • domain assumption The one-band Hubbard model with U/t=7 and t'/t=-0.15 describes the relevant low-energy physics of cuprate superconductors.
    The paper relies on this to connect its results to experiments on cuprates; Refs 9, 10, 54, 55 support this modeling choice, but it remains a domain assumption.
  • domain assumption DCA with an eight-site cluster is a controlled approximation that becomes exact as cluster size Nc goes to infinity.
    Invoked throughout the paper; Refs 48 and 60-62 establish the method, but the finite cluster size is an approximation that limits momentum resolution.
  • domain assumption The paramagnetic solution is sufficient; magnetic order is not considered.
    The authors state 'Our solution is restricted to the paramagnetic phase,' which excludes competing ordered states that could affect the charge response.
  • domain assumption The maximum entropy analytical continuation produces reliable real-frequency spectra for the main peak of the charge susceptibility.
    The authors note that analytical continuation is uncontrolled, but they argue that different default models give the same main peak; this is an assumption about the reliability of the continuation for the central spectral feature.
  • standard math Charge conservation (Ward identity) requires the uniform (Q=0) dynamical charge susceptibility to vanish at non-zero frequency.
    The paper uses this identity as a consistency check on the DCA calculation, citing Ref 69; it is a standard theorem that does not require proof here.

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Cite this review

Pith. "Pith review of Dynamical Charge Susceptibility in the Hubbard Model." pith.science (2026). https://pith.science/paper/E6LMJ63A

@misc{pith2026190804776,
  author       = {Pith},
  title        = {Pith review of: Dynamical Charge Susceptibility in the Hubbard Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E6LMJ63A}},
  note         = {Machine review of arXiv:1908.04776}
}
read the original abstract

We compute the dynamical charge susceptibility in the two-dimensional Hubbard model within the dynamical cluster approximation. In order to understand the connection between charge susceptibility and pseudogap, we investigate the momentum, doping, and temperature dependence. We find that as a function of frequency, the dynamical charge susceptibility is well represented by a single peak at a characteristic frequency. It shows little momentum or temperature dependence, while the doping dependence is more evident, and no clear signature of the pseudogap is observed. Data for the doping evolution of the static susceptibility and for fluctuation diagnostics are presented. Our susceptibilities should be directly measurable in future Momentum-resolved electron energy-loss spectroscopy experiments.

Figures

Figures reproduced from arXiv: 1908.04776 by the authors.

Figure 1
Figure 1. FIG. 1. eight-site DCA phase diagram of the Hubbard model, [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Charge susceptibility at [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Charge susceptibility at [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Charge susceptibility at different temperatures with momentum transfer [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Static charge susceptibility and (b) static charge [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Fluctuation diagnostics [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

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Works this paper leans on

84 extracted references · 43 canonical work pages

  1. [1]

    Ishii , author T

    author author K. Ishii , author T. Tohyama , author S. Asano , author K. Sato , author M. Fujita , author S. Wakimoto , author K. Tustsui , author S. Sota , author J. Miyawaki , author H. Niwa , author Y. Harada , author J. Pelliciari , author Y. Huang , author T. Schmitt , author Y. Yamamoto , \ and\ author J. Mizuki ,\ 10.1103/PhysRevB.96.115148 journal...

    work page doi:10.1103/physrevb.96.115148

  2. [2]

    Suzuki , author M

    author author H. Suzuki , author M. Minola , author Y. Lu , author Y. Peng , author R. Fumagalli , author E. Lefran c ois , author T. Loew , author J. Porras , author K. Kummer , author D. Betto , author S. Ishida , author H. Eisaki , author C. Hu , author X. Zhou , author M. W. \ Haverkort , author N. B. \ Brookes , author L. Braicovich , author G. Ghiri...

    work page doi:10.1038/s41535-018-0139-7

  3. [3]

    author author E. H. \ da Silva Neto , author M. Minola , author B. Yu , author W. Tabis , author M. Bluschke , author D. Unruh , author H. Suzuki , author Y. Li , author G. Yu , author D. Betto , author K. Kummer , author F. Yakhou , author N. B. \ Brookes , author M. Le Tacon , author M. Greven , author B. Keimer , \ and\ author A. Damascelli ,\ 10.1103/...

    work page doi:10.1103/physrevb.98.161114

  4. [4]

    Hepting , author L

    author author M. Hepting , author L. Chaix , author E. W. \ Huang , author R. Fumagalli , author Y. Y. \ Peng , author B. Moritz , author K. Kummer , author N. B. \ Brookes , author W. C. \ Lee , author M. Hashimoto , author T. Sarkar , author J.-F. \ He , author C. R. \ Rotundu , author Y. S. \ Lee , author R. L. \ Greene , author L. Braicovich , author ...

    doi:10.1038/s41586-018-0648-3

  5. [5]

    Miao , author R

    author author H. Miao , author R. Fumagalli , author M. Rossi , author J. Lorenzana , author G. Seibold , author F. Yakhou-Harris , author K. Kummer , author N. B. \ Brookes , author G. D. \ Gu , author L. Braicovich , author G. Ghiringhelli , \ and\ author M. P. M. \ Dean ,\ 10.1103/PhysRevX.9.031042 journal journal Phys. Rev. X \ volume 9 ,\ pages 03104...

    work page doi:10.1103/physrevx.9.031042

  6. [6]

    Ishii , author M

    author author K. Ishii , author M. Kurooka , author Y. Shimizu , author M. Fujita , author K. Yamada , \ and\ author J. Mizuki ,\ 10.7566/JPSJ.88.075001 journal journal Journal of the Physical Society of Japan \ volume 88 ,\ pages 075001 ( year 2019 ) NoStop

    doi:10.7566/jpsj.88.075001

  7. [7]

    Vig , author A

    author author S. Vig , author A. Kogar , author M. Mitrano , author A. A. \ Husain , author V. Mishra , author M. S. \ Rak , author L. Venema , author P. D. \ Johnson , author G. D. \ Gu , author E. Fradkin , author M. R. \ Norman , \ and\ author P. Abbamonte ,\ 10.21468/SciPostPhys.3.4.026 journal journal SciPost Phys. \ volume 3 ,\ pages 026 ( year 2017...

    doi:10.21468/scipostphys.3.4.026

  8. [8]

    author author A. A. \ Husain , author M. Mitrano , author M. S. \ Rak , author S. I. \ Rubeck , author B. Uchoa , author J. Schneeloch , author R. Zhong , author G. D. \ Gu , \ and\ author P. Abbamonte ,\ @noop title Crossover of Charge Fluctuations across the Strange Metal Phase Diagram , \ ( year 2019 ),\ http://arxiv.org/abs/1903.04038 arXiv:1903.04038 NoStop

    arXiv 2019

Show all 84 references
  1. [9]

    author author P. W. \ Anderson ,\ 10.1126/science.235.4793.1196 journal journal Science \ volume 235 ,\ pages 1196 ( year 1987 ) NoStop

    1987

  2. [10]

    Scalapino ,\ in\ @noop booktitle Handbook of High-Temperature Superconductivity ,\ editor edited by\ editor J

    author author D. Scalapino ,\ in\ @noop booktitle Handbook of High-Temperature Superconductivity ,\ editor edited by\ editor J. Schrieffer \ and\ editor J. Brooks \ ( publisher Springer New York ,\ year 2007 )\ pp.\ pages 495--526 NoStop

    2007

  3. [11]

    Huscroft , author M

    author author C. Huscroft , author M. Jarrell , author T. Maier , author S. Moukouri , \ and\ author A. N. \ Tahvildarzadeh ,\ 10.1103/PhysRevLett.86.139 journal journal Phys. Rev. Lett. \ volume 86 ,\ pages 139 ( year 2001 ) NoStop

    2001 doi

  4. [12]

    Civelli , author M

    author author M. Civelli , author M. Capone , author S. S. \ Kancharla , author O. Parcollet , \ and\ author G. Kotliar ,\ 10.1103/PhysRevLett.95.106402 journal journal Phys. Rev. Lett. \ volume 95 ,\ eid 106402 ( year 2005 ) NoStop

    2005 doi

  5. [13]

    author author T. D. \ Stanescu \ and\ author G. Kotliar ,\ 10.1103/PhysRevB.74.125110 journal journal Phys. Rev. B \ volume 74 ,\ pages 125110 ( year 2006 ) NoStop

    2006 doi

  6. [14]

    Park , author K

    author author H. Park , author K. Haule , \ and\ author G. Kotliar ,\ 10.1103/PhysRevLett.101.186403 journal journal Phys. Rev. Lett. \ volume 101 ,\ pages 186403 ( year 2008 ) NoStop

    2008 doi

  7. [15]

    Liebsch \ and\ author N.-H

    author author A. Liebsch \ and\ author N.-H. \ Tong ,\ 10.1103/PhysRevB.80.165126 journal journal Phys. Rev. B \ volume 80 ,\ pages 165126 ( year 2009 ) NoStop

    2009 doi

  8. [16]

    Ferrero , author P

    author author M. Ferrero , author P. S. \ Cornaglia , author L. De Leo , author O. Parcollet , author G. Kotliar , \ and\ author A. Georges ,\ 10.1103/PhysRevB.80.064501 journal journal Phys. Rev. B \ volume 80 ,\ pages 064501 ( year 2009 a ) NoStop

    doi

  9. [17]

    Ferrero , author P

    author author M. Ferrero , author P. S. \ Cornaglia , author L. D. \ Leo , author O. Parcollet , author G. Kotliar , \ and\ author A. Georges ,\ 10.1209/0295-5075/85/57009 journal journal EPL (Europhysics Letters) \ volume 85 ,\ pages 57009 ( year 2009 b ) NoStop

    doi

  10. [18]

    Sakai , author Y

    author author S. Sakai , author Y. Motome , \ and\ author M. Imada ,\ 10.1103/PhysRevLett.102.056404 journal journal Phys. Rev. Lett. \ volume 102 ,\ pages 056404 ( year 2009 ) NoStop

    2009 doi

  11. [19]

    Sakai , author Y

    author author S. Sakai , author Y. Motome , \ and\ author M. Imada ,\ 10.1103/PhysRevB.82.134505 journal journal Phys. Rev. B \ volume 82 ,\ pages 134505 ( year 2010 ) NoStop

    2010 doi

  12. [20]

    Lin , author E

    author author N. Lin , author E. Gull , \ and\ author A. J. \ Millis ,\ 10.1103/PhysRevB.82.045104 journal journal Phys. Rev. B \ volume 82 ,\ pages 045104 ( year 2010 ) NoStop

    2010 doi

  13. [21]

    author author L. B. \ Ioffe \ and\ author A. J. \ Millis ,\ 10.1103/PhysRevB.61.9077 journal journal Phys. Rev. B \ volume 61 ,\ pages 9077 ( year 2000 ) NoStop

    doi

  14. [22]

    Toschi , author M

    author author A. Toschi , author M. Capone , \ and\ author C. Castellani ,\ 10.1103/PhysRevB.72.235118 journal journal Phys. Rev. B \ volume 72 ,\ pages 235118 ( year 2005 ) NoStop

    2005 doi

  15. [23]

    author author A. J. \ Millis , author A. Zimmers , author R. P. S. M. \ Lobo , author N. Bontemps , \ and\ author C. C. \ Homes ,\ 10.1103/PhysRevB.72.224517 journal journal Phys. Rev. B \ volume 72 ,\ pages 224517 ( year 2005 ) NoStop

    doi

  16. [24]

    Comanac , author L

    author author A. Comanac , author L. de' Medici , author M. Capone , \ and\ author A. J. \ Millis ,\ https://doi.org/10.1038/nphys883 journal journal Nature Physics \ volume 4 ,\ pages 287 EP ( year 2008 ) NoStop

    2008 doi

  17. [25]

    Ferrero , author O

    author author M. Ferrero , author O. Parcollet , author A. Georges , author G. Kotliar , \ and\ author D. N. \ Basov ,\ 10.1103/PhysRevB.82.054502 journal journal Phys. Rev. B \ volume 82 ,\ pages 054502 ( year 2010 ) NoStop

    2010 doi

  18. [26]

    Bergeron , author V

    author author D. Bergeron , author V. Hankevych , author B. Kyung , \ and\ author A.-M. S. \ Tremblay ,\ 10.1103/PhysRevB.84.085128 journal journal Phys. Rev. B \ volume 84 ,\ pages 085128 ( year 2011 ) NoStop

    2011 doi

  19. [27]

    Lin , author E

    author author N. Lin , author E. Gull , \ and\ author A. J. \ Millis ,\ 10.1103/PhysRevLett.109.106401 journal journal Phys. Rev. Lett. \ volume 109 ,\ pages 106401 ( year 2012 ) NoStop

    2012 doi

  20. [28]

    Gull \ and\ author A

    author author E. Gull \ and\ author A. J. \ Millis ,\ 10.1103/PhysRevB.88.075127 journal journal \ volume 88 ,\ eid 075127 ( year 2013 ) ,\ http://arxiv.org/abs/1304.6406 1304.6406 NoStop

    2013 arXiv

  21. [29]

    author author J. P. F. \ LeBlanc , author S. Li , author X. Chen , author R. Levy , author A. E. \ Antipov , author A. J. \ Millis , \ and\ author E. Gull ,\ 10.1103/PhysRevB.100.075123 journal journal Phys. Rev. B \ volume 100 ,\ pages 075123 ( year 2019 ) NoStop

    doi

  22. [30]

    Macridin , author M

    author author A. Macridin , author M. Jarrell , author T. Maier , author P. R. C. \ Kent , \ and\ author E. D'Azevedo ,\ 10.1103/PhysRevLett.97.036401 journal journal Phys. Rev. Lett. \ volume 97 ,\ pages 036401 ( year 2006 ) NoStop

    2006 doi

  23. [31]

    Chen , author J

    author author X. Chen , author J. P. F. \ LeBlanc , \ and\ author E. Gull ,\ https://doi.org/10.1038/ncomms14986 journal journal Nature Communications \ volume 8 ,\ pages 14986 EP ( year 2017 ) ,\ note article NoStop

    2017 doi

  24. [32]

    author author S. R. \ White \ and\ author D. J. \ Scalapino ,\ 10.1103/PhysRevLett.91.136403 journal journal Phys. Rev. Lett. \ volume 91 ,\ pages 136403 ( year 2003 ) NoStop

    2003 doi

  25. [33]

    Hager , author G

    author author G. Hager , author G. Wellein , author E. Jeckelmann , \ and\ author H. Fehske ,\ 10.1103/PhysRevB.71.075108 journal journal Phys. Rev. B \ volume 71 ,\ pages 075108 ( year 2005 ) NoStop

    2005 doi

  26. [34]

    \ Chang \ and\ author S

    author author C.-C. \ Chang \ and\ author S. Zhang ,\ 10.1103/PhysRevLett.104.116402 journal journal Phys. Rev. Lett. \ volume 104 ,\ pages 116402 ( year 2010 ) NoStop

    2010 doi

  27. [35]

    author author D. J. \ Scalapino ,\ 10.1103/RevModPhys.84.1383 journal journal Rev. Mod. Phys. \ volume 84 ,\ pages 1383 ( year 2012 ) NoStop

    2012 doi

  28. [36]

    \ Zheng \ and\ author G

    author author B.-X. \ Zheng \ and\ author G. K.-L. \ Chan ,\ 10.1103/PhysRevB.93.035126 journal journal Phys. Rev. B \ volume 93 ,\ pages 035126 ( year 2016 ) NoStop

    2016 doi

  29. [37]

    author author E. W. \ Huang , author C. B. \ Mendl , author S. Liu , author S. Johnston , author H.-C. \ Jiang , author B. Moritz , \ and\ author T. P. \ Devereaux ,\ 10.1126/science.aak9546 journal journal Science \ volume 358 ,\ pages 1161 ( year 2017 ) NoStop

    doi

  30. [38]

    \ Zheng , author C.-M

    author author B.-X. \ Zheng , author C.-M. \ Chung , author P. Corboz , author G. Ehlers , author M.-P. \ Qin , author R. M. \ Noack , author H. Shi , author S. R. \ White , author S. Zhang , \ and\ author G. K.-L. \ Chan ,\ 10.1126/science.aam7127 journal journal Science \ vo...

    doi

  31. [39]

    author author E. W. \ Huang , author C. B. \ Mendl , author H.-C. \ Jiang , author B. Moritz , \ and\ author T. P. \ Devereaux ,\ 10.1038/s41535-018-0097-0 journal journal npj Quantum Materials \ volume 3 ,\ eid 22 ( year 2018 ) ,\ http://arxiv.org/abs/1709.02398 1709.02398 NoStop

    arXiv

  32. [40]

    author author T. I. \ Vanhala \ and\ author P. T\"orm\"a ,\ 10.1103/PhysRevB.97.075112 journal journal Phys. Rev. B \ volume 97 ,\ pages 075112 ( year 2018 ) NoStop

    2018 doi

  33. [41]

    author author A. S. \ Darmawan , author Y. Nomura , author Y. Yamaji , \ and\ author M. Imada ,\ 10.1103/PhysRevB.98.205132 journal journal Phys. Rev. B \ volume 98 ,\ pages 205132 ( year 2018 ) NoStop

    2018 doi

  34. [42]

    author author L. F. \ Tocchio , author A. Montorsi , \ and\ author F. Becca ,\ 10.21468/SciPostPhys.7.2.021 journal journal SciPost Phys. \ volume 7 ,\ pages 21 ( year 2019 ) NoStop

    2019 doi

  35. [43]

    Zimmermann , author R

    author author W. Zimmermann , author R. Fr\'esard , \ and\ author P. W\"olfle ,\ 10.1103/PhysRevB.56.10097 journal journal Phys. Rev. B \ volume 56 ,\ pages 10097 ( year 1997 ) NoStop

    1997 doi

  36. [44]

    Sherman \ and\ author M

    author author A. Sherman \ and\ author M. Schreiber ,\ 10.1103/PhysRevB.77.155117 journal journal Phys. Rev. B \ volume 77 ,\ pages 155117 ( year 2008 ) NoStop

    2008 doi

  37. [45]

    Hochkeppel , author F

    author author S. Hochkeppel , author F. F. \ Assaad , \ and\ author W. Hanke ,\ 10.1103/PhysRevB.77.205103 journal journal Phys. Rev. B \ volume 77 ,\ pages 205103 ( year 2008 ) NoStop

    2008 doi

  38. [46]

    author author Y. F. \ Kung , author E. A. \ Nowadnick , author C. J. \ Jia , author S. Johnston , author B. Moritz , author R. T. \ Scalettar , \ and\ author T. P. \ Devereaux ,\ 10.1103/PhysRevB.92.195108 journal journal Phys. Rev. B \ volume 92 ,\ pages 195108 ( year 2015 ) NoStop

    doi

  39. [47]

    author author J. Q. \ Lin , author J. Yuan , author K. Jin , author Z. P. \ Yin , author G. Li , author K.-J. \ Zhou , author X. Lu , author M. Dantz , author T. Schmitt , author H. Ding , author H. Guo , author M. P. M. \ Dean , \ and\ author X. Liu ,\ @noop title Doping evol...

    2019 arXiv

  40. [48]

    author author T. A. \ Maier , author M. Jarrell , author T. Pruschke , \ and\ author M. Hettler ,\ 10.1103/RevModPhys.77.1027 journal journal Rev. Mod. Phys. \ volume 77 ,\ eid 1027 ( year 2005 a ) NoStop

    2005 doi

  41. [49]

    author author O. K. \ Andersen , author O. Jepsen , author A. I. \ Liechtenstein , \ and\ author I. I. \ Mazin ,\ 10.1103/PhysRevB.49.4145 journal journal Phys. Rev. B \ volume 49 ,\ pages 4145 ( year 1994 ) NoStop

    1994 doi

  42. [50]

    Andersen , author A

    author author O. Andersen , author A. Liechtenstein , author O. Jepsen , \ and\ author F. Paulsen ,\ https://doi.org/10.1016/0022-3697(95)00269-3 journal journal Journal of Physics and Chemistry of Solids \ volume 56 ,\ pages 1573 ( year 1995 ) ,\ note proceedings of the Confe...

    doi

  43. [51]

    Kim , author P

    author author C. Kim , author P. J. \ White , author Z.-X. \ Shen , author T. Tohyama , author Y. Shibata , author S. Maekawa , author B. O. \ Wells , author Y. J. \ Kim , author R. J. \ Birgeneau , \ and\ author M. A. \ Kastner ,\ 10.1103/PhysRevLett.80.4245 journal journal P...

    doi

  44. [52]

    author author M. Z. \ Hasan , author E. D. \ Isaacs , author Z.-X. \ Shen , author L. L. \ Miller , author K. Tsutsui , author T. Tohyama , \ and\ author S. Maekawa ,\ 10.1126/science.288.5472.1811 journal journal Science \ volume 288 ,\ pages 1811 ( year 2000 ) NoStop

  45. [53]

    Gull , author O

    author author E. Gull , author O. Parcollet , author P. Werner , \ and\ author A. J. \ Millis ,\ 10.1103/PhysRevB.80.245102 journal journal Phys. Rev. B \ volume 80 ,\ pages 245102 ( year 2009 ) NoStop

    2009 doi

  46. [54]

    Gull \ and\ author A

    author author E. Gull \ and\ author A. J. \ Millis ,\ 10.1103/PhysRevB.86.241106 journal journal Phys. Rev. B \ volume 86 ,\ pages 241106(R) ( year 2012 ) NoStop

    2012 doi

  47. [55]

    Gull , author O

    author author E. Gull , author O. Parcollet , \ and\ author A. J. \ Millis ,\ 10.1103/PhysRevLett.110.216405 journal journal Phys. Rev. Lett. \ volume 110 ,\ pages 216405 ( year 2013 ) NoStop

    2013 doi

  48. [56]

    Gull \ and\ author A

    author author E. Gull \ and\ author A. J. \ Millis ,\ 10.1103/PhysRevB.90.041110 journal journal Phys. Rev. B \ volume 90 ,\ pages 041110(R) ( year 2014 ) NoStop

    2014 doi

  49. [57]

    Gull \ and\ author A

    author author E. Gull \ and\ author A. J. \ Millis ,\ 10.1103/PhysRevB.91.085116 journal journal Phys. Rev. B \ volume 91 ,\ pages 085116 ( year 2015 ) NoStop

    2015 doi

  50. [58]

    Gull , author M

    author author E. Gull , author M. Ferrero , author O. Parcollet , author A. Georges , \ and\ author A. J. \ Millis ,\ 10.1103/PhysRevB.82.155101 journal journal Phys. Rev. B \ volume 82 ,\ pages 155101 ( year 2010 ) NoStop

    2010 doi

  51. [59]

    Fuhrmann , author S

    author author A. Fuhrmann , author S. Okamoto , author H. Monien , \ and\ author A. J. \ Millis ,\ 10.1103/PhysRevB.75.205118 journal journal Phys. Rev. B \ volume 75 ,\ pages 205118 ( year 2007 ) NoStop

    2007 doi

  52. [60]

    author author T. A. \ Maier , author M. Jarrell , author T. C. \ Schulthess , author P. R. C. \ Kent , \ and\ author J. B. \ White ,\ 10.1103/PhysRevLett.95.237001 journal journal Phys. Rev. Lett. \ volume 95 ,\ pages 237001 ( year 2005 b ) NoStop

    doi

  53. [61]

    Fuchs , author E

    author author S. Fuchs , author E. Gull , author L. Pollet , author E. Burovski , author E. Kozik , author T. Pruschke , \ and\ author M. Troyer ,\ @noop journal journal Phys. Rev. Lett. \ volume 106 ,\ pages 030401 ( year 2011 ) NoStop

    2011

  54. [62]

    author author J. P. F. \ LeBlanc , author A. E. \ Antipov , author F. Becca , author I. W. \ Bulik , author G. K.-L. \ Chan , author C.-M. \ Chung , author Y. Deng , author M. Ferrero , author T. M. \ Henderson , author C. A. \ Jim\'enez-Hoyos , author E. Kozik , author X.-W. ...

  55. [63]

    Gull , author P

    author author E. Gull , author P. Werner , author O. Parcollet , \ and\ author M. Troyer ,\ 10.1209/0295-5075/82/57003 journal journal Europhys. Lett. \ volume 82 ,\ pages 57003 ( year 2008 ) NoStop

    2008 doi

  56. [64]

    Gull , author A

    author author E. Gull , author A. J. \ Millis , author A. I. \ Lichtenstein , author A. N. \ Rubtsov , author M. Troyer , \ and\ author P. Werner ,\ 10.1103/RevModPhys.83.349 journal journal Rev. Mod. Phys. \ volume 83 ,\ pages 349 ( year 2011 a ) NoStop

    doi

  57. [65]

    Gull , author P

    author author E. Gull , author P. Staar , author S. Fuchs , author P. Nukala , author M. S. \ Summers , author T. Pruschke , author T. C. \ Schulthess , \ and\ author T. Maier ,\ 10.1103/PhysRevB.83.075122 journal journal Phys. Rev. B \ volume 83 ,\ pages 075122 ( year 2011 b ) NoStop

    doi

  58. [66]

    Rohringer , author A

    author author G. Rohringer , author A. Valli , \ and\ author A. Toschi ,\ 10.1103/PhysRevB.86.125114 journal journal Phys. Rev. B \ volume 86 ,\ pages 125114 ( year 2012 ) NoStop

    2012 doi

  59. [67]

    Jarrell \ and\ author J

    author author M. Jarrell \ and\ author J. E. \ Gubernatis ,\ http://www.sciencedirect.com/science/article/B6TVP-3VTNGGS-3/2/5f7d5c3a6ee0870df8970b51e47592d2 journal journal Physics Reports \ volume 269 ,\ pages 133 ( year 1996 ) NoStop

    1996

  60. [68]

    Levy , author J

    author author R. Levy , author J. P. F. \ LeBlanc , \ and\ author E. Gull ,\ 10.1016/j.cpc.2017.01.018 journal journal Computer Physics Communications \ volume 215 ,\ pages 149 ( year 2017 ) ,\ http://arxiv.org/abs/1606.00368 1606.00368 NoStop

    2017 arXiv

  61. [69]

    Hafermann , author E

    author author H. Hafermann , author E. G. C. P. \ van Loon , author M. I. \ Katsnelson , author A. I. \ Lichtenstein , \ and\ author O. Parcollet ,\ 10.1103/PhysRevB.90.235105 journal journal Phys. Rev. B \ volume 90 ,\ pages 235105 ( year 2014 ) NoStop

    doi

  62. [70]

    Bohm \ and\ author D

    author author D. Bohm \ and\ author D. Pines ,\ 10.1103/PhysRev.82.625 journal journal Phys. Rev. \ volume 82 ,\ pages 625 ( year 1951 ) NoStop

    1951 doi

  63. [71]

    Hedin ,\ 10.1103/PhysRev.139.A796 journal journal Phys

    author author L. Hedin ,\ 10.1103/PhysRev.139.A796 journal journal Phys. Rev. \ volume 139 ,\ pages A796 ( year 1965 ) NoStop

    1965 doi

  64. [72]

    author author N. E. \ Bickers ,\ title Self-consistent many-body theory for condensed matter systems , \ in\ 10.1007/0-387-21717-7_6 booktitle Theoretical Methods for Strongly Correlated Electrons \ ( publisher Springer New York ,\ address New York, NY ,\ year 2004 )\ pp.\ pag...

    doi

  65. [73]

    Brener , author H

    author author S. Brener , author H. Hafermann , author A. N. \ Rubtsov , author M. I. \ Katsnelson , \ and\ author A. I. \ Lichtenstein ,\ 10.1103/PhysRevB.77.195105 journal journal Phys. Rev. B \ volume 77 ,\ pages 195105 ( year 2008 ) NoStop

    doi

  66. [74]

    Gunnarsson , author T

    author author O. Gunnarsson , author T. Sch\"afer , author J. P. F. \ LeBlanc , author E. Gull , author J. Merino , author G. Sangiovanni , author G. Rohringer , \ and\ author A. Toschi ,\ 10.1103/PhysRevLett.114.236402 journal journal Phys. Rev. Lett. \ volume 114 ,\ pages 23...

    doi

  67. [75]

    Wu , author M

    author author W. Wu , author M. Ferrero , author A. Georges , \ and\ author E. Kozik ,\ 10.1103/PhysRevB.96.041105 journal journal Phys. Rev. B \ volume 96 ,\ pages 041105(R) ( year 2017 ) NoStop

    2017 doi

  68. [76]

    Arzhang , author A

    author author B. Arzhang , author A. E. \ Antipov , \ and\ author J. P. F. \ LeBlanc ,\ @noop title Fluctuation diagnostics of the finite temperature quasi-antiferromagnetic regime of the 2D Hubbard model , \ ( year 2019 ),\ http://arxiv.org/abs/1905.07462 arXiv:1905.07462 NoStop

    2019 arXiv

  69. [77]

    Gunnarsson , author J

    author author O. Gunnarsson , author J. Merino , author T. Sch\"afer , author G. Sangiovanni , author G. Rohringer , \ and\ author A. Toschi ,\ 10.1103/PhysRevB.97.125134 journal journal Phys. Rev. B \ volume 97 ,\ pages 125134 ( year 2018 ) NoStop

    doi

  70. [78]

    Abbamonte , author E

    author author P. Abbamonte , author E. Demler , author J. S. \ Davis , \ and\ author J.-C. \ Campuzano ,\ https://doi.org/10.1016/j.physc.2012.04.006 journal journal Physica C: Superconductivity \ volume 481 ,\ pages 15 ( year 2012 ) ,\ note stripes and Electronic Liquid Cryst...

    2012 doi

  71. [79]

    Kohsaka , author C

    author author Y. Kohsaka , author C. Taylor , author K. Fujita , author A. Schmidt , author C. Lupien , author T. Hanaguri , author M. Azuma , author M. Takano , author H. Eisaki , author H. Takagi , author S. Uchida , \ and\ author J. C. \ Davis ,\ 10.1126/science.1138584 jou...

    doi

  72. [80]

    Fujita , author H

    author author M. Fujita , author H. Goka , author K. Yamada , author J. M. \ Tranquada , \ and\ author L. P. \ Regnault ,\ 10.1103/PhysRevB.70.104517 journal journal Phys. Rev. B \ volume 70 ,\ pages 104517 ( year 2004 ) NoStop

    2004 doi

  73. [81]

    author author C. V. \ Parker , author P. Aynajian , author E. H. \ da Silva Neto , author A. Pushp , author S. Ono , author J. Wen , author Z. Xu , author G. Gu , \ and\ author A. Yazdani ,\ 10.1038/nature09597 journal journal Nature \ volume 468 ,\ pages 677 ( year 2010 ) NoStop

    doi

  74. [82]

    author author M. J. \ Lawler , author K. Fujita , author J. Lee , author A. R. \ Schmidt , author Y. Kohsaka , author C. K. \ Kim , author H. Eisaki , author S. Uchida , author J. C. \ Davis , author J. P. \ Sethna , \ and\ author E.-A. \ Kim ,\ https://doi.org/10.1038/nature0...

    doi

  75. [83]

    Hashimoto , author R.-H

    author author M. Hashimoto , author R.-H. \ He , author K. Tanaka , author J.-P. \ Testaud , author W. Meevasana , author R. G. \ Moore , author D. Lu , author H. Yao , author Y. Yoshida , author H. Eisaki , author T. P. \ Devereaux , author Z. Hussain , \ and\ author Z.-X. \ ...

    doi

  76. [84]

    Daou , author J

    author author R. Daou , author J. Chang , author D. Leboeuf , author O. Cyr-Choini \`e re , author F. Lalibert \'e , author N. Doiron-Leyraud , author B. J. \ Ramshaw , author R. Liang , author D. A. \ Bonn , \ and\ author W. N. \ Hardy ,\ 10.1038/nature08716 journal journal \...

    arXiv

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