Pith. sign in

REVIEW 3 major objections 6 minor 1 cited by

The pseudogap in high-$T_c$ superconductors from SU(2) gauge symmetry and dynamic correlation effects

T0 review · 3 major / 6 minor · reviewed 2026-07-12 · grok-4.5

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read DMFT-ordered chargons plus spinon fluctuations turn hole pockets into Fermi arcs at low doping in the Hubbard model.

desk verdict Solid incremental spectral result: at low doping, spinon dressing turns nearly symmetric DMFT chargon pockets into arcs while leaving the pockets as QO candidates. read the letter →

arxiv 2606.02838 v2 pith:TS6O45IT submitted 2026-06-01 cond-mat.str-el cond-mat.supr-con

I. A. Goremykin , A. A. Katanin This is my paper · ORCID
classification cond-mat.str-elcond-mat.supr-con
keywords pseudogapFermiarcsHubbardmodelSU(2)gaugetheorychargonsspinonsDMFTholepockets
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

The paper sets out to show how the pseudogap of underdoped cuprates can arise from the two-dimensional Hubbard model once electrons are fractionalized into chargons and spinons under an SU(2) gauge theory. Dynamical mean-field theory is used to place the chargon sector in a magnetically ordered state that already reconstructs the spectrum into hole pockets; long-wavelength spinon fluctuations are then convolved on top of those pockets. The central demonstration is that this combination is essential for the strong asymmetry in damping between the inner and outer sides of the pockets, which converts them into Fermi arcs at low hole doping (around 4 percent). Pure DMFT chargon spectra alone do not produce that asymmetry at this doping. A sympathetic reader cares because the same framework simultaneously accounts for the arcs seen in photoemission and the closed pockets inferred from quantum oscillations, without invoking static density-wave order of the physical electrons.

What carries the argument

The fractionalization map c = R ψ together with the convolution of the DMFT chargon spectral function with the gapped spinon propagator (Eq. 38). The map supplies magnetically ordered chargon pockets and a Luttinger surface; the convolution transfers spectral weight asymmetrically, converting pockets into arcs.

What would settle it

At x approximately 4 percent and temperatures around 0.1 t, compute or measure whether the outer side of the nodal pocket is more strongly damped than the inner side once long-wavelength magnetic fluctuations are included; if the damping remains symmetric, the claimed arc-formation mechanism fails.

Watch

Extended reading notes

Core claim

DMFT treatment of long-range magnetic order in the chargon sector, when supplemented by long-wavelength spinon fluctuations, is essential for the asymmetry in damping between the inner and outer regions of the hole pockets and the resulting formation of Fermi arcs in the underdoped regime, especially at low hole doping. The underlying chargon pockets remain the objects that would be seen in quantum-oscillation measurements.

Load-bearing premise

The physical electron can be faithfully rewritten as a chargon that lives in a long-range magnetically ordered state plus a weakly fluctuating SU(2) rotation whose dynamics are captured by a local, static spin-stiffness model.

Editorial extensions

If this is right

  • Photoemission should see Fermi arcs while quantum oscillations continue to report the area of the underlying chargon hole pockets.
  • A uniform magnetic field couples mainly to the chargon sector, so oscillation frequencies track chargon-pocket areas even when spinons are present.
  • Spinons reduce oscillation amplitude by spectral smearing but do not shift the frequencies themselves.
  • At higher doping the same asymmetry can already appear inside pure DMFT; the spinon correction is therefore most decisive at the lowest dopings.

Reading between the lines

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

  • If the ordered-chargon-plus-spinon decomposition is correct, temperature-dependent ARPES should show the outer-arc weight recovering faster than the inner-arc weight as spinon gap softens.
  • The same convolution that destroys the outer pocket should leave a residual spectral continuum whose high-energy tails could be checked against existing ARPES intensity maps near the M point.
  • A controlled comparison of local versus nonlocal spin stiffness at fixed DMFT chargon solution would quantify how much of the arc asymmetry is locked to the static-stiffness approximation.
Share X Bluesky LinkedIn Reddit HN

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. The paper studies the spectral properties of the two-dimensional Hubbard model in the underdoped regime within an SU(2) gauge theory that fractionalizes electrons into chargons and spinons. Long-range antiferromagnetic order in the chargon sector is treated by DMFT in a local spin frame (Eqs. 5–10), producing a singular self-energy contribution and a Luttinger surface (Eq. 21). Long-wavelength spinon fluctuations are described by a CP1/NLSM mean-field theory with spin stiffnesses extracted from the chargon sector (Eqs. 30–36, Appendix A). Physical electron spectra are obtained by convolving chargon spectral functions with the gapped spinon propagator (Eq. 38). At hole doping x≈4% the authors find that pure DMFT chargon spectra form nearly symmetric hole pockets, while spinon dressing generates an inner/outer damping asymmetry and Fermi-arc-like features; the chargon pockets are proposed as the origin of quantum-oscillation signals.

Significance. The work addresses a central experimental tension in underdoped cuprates—ARPES Fermi arcs versus small closed pockets in quantum oscillations—within a single fractionalization framework. A concrete, falsifiable mechanism is proposed: nearly symmetric chargon hole pockets (visible to quantum oscillations) are converted into arcs by anisotropic effective mass and asymmetric spectral-weight transfer under gapped spinon dressing. Spin stiffnesses ρ and χ are computed from the chargon sector rather than fitted to produce arcs, and Appendix C supplies a semi-analytical anisotropic-mass model that clarifies the directional dependence. If the ordered-chargon plus weakly fluctuating spinon decomposition is faithful, the paper provides a useful microscopic route to unify the two probes and clarifies why pure DMFT at low doping is insufficient for arc asymmetry.

major comments (3)
  1. The ultraviolet cutoff Λ=0.5 enters the spinon sum rule (Eq. 36) and thereby fixes the magnetic gap Δ (and ξ) that controls whether the hole pocket is only partially gapped into an arc or fully suppressed. No sensitivity of the arc morphology (Figs. 3c,d and 4c,d) to Λ is reported. Because the central claim is that spinon dressing produces arcs rather than a full gap, a brief scan over a plausible range of Λ (or an explicit statement of how Δ and the arc endpoints change) is needed to show the result is not an artifact of this single choice.
  2. The abstract and Sec. I assert that “DMFT supplemented by long-wavelength magnetic fluctuations is essential,” contrasting with prior static mean-field treatments of the chargon sector. The paper demonstrates that pure DMFT chargon pockets at x=4% are nearly symmetric (Figs. 3a,b and 4a,b) and that spinon convolution produces arcs, but it does not show the corresponding static-mean-field chargon spectrum dressed by the same spinons. Without that comparison (or a softened claim limited to what is actually computed), the load-bearing assertion that dynamical DMFT correlations—not merely ordered chargons plus spinons—are required for the inner/outer asymmetry remains incompletely supported.
  3. Appendix B shows that the temporal stiffness χ_ωn retains visible frequency dependence even though it is weaker than in static mean-field theory; the main text nevertheless uses a fully static, local J (Eqs. 34–35). Because the convolution (Eq. 38) and the gap scale ω0=√(Δ/χ) depend on this approximation, the manuscript should either quantify how a frequency-dependent χ would broaden or shift the arc features, or state more carefully that the static-χ NLSM is an uncontrolled but improved approximation relative to prior work.
minor comments (6)
  1. Fig. 2 caption: “chansons” should be “chargons”.
  2. Sec. III, paragraph introducing electronic spectra: “functrions” → “functions”.
  3. Introduction: “ord-density-wave order” appears to be a typo for “charge-density-wave order”.
  4. The blue dotted line of Eq. (21) is central to the non-Fermi-liquid diagnostics (Figs. 2 and 5) but is only briefly defined; a short reminder in the figure captions would help readers.
  5. Sec. IV’s quantum-oscillation argument (uniform B acts mainly on chargons) is plausible but purely qualitative; a sentence clarifying that no Landau-level calculation is performed would avoid over-reading.
  6. Notation for the physical Green’s function switches between G^g and G; a single consistent symbol would improve readability.

Circularity Check

1 steps flagged · score 2.0 of 10

Mild self-citation of the authors' prior DMFT+SU(2) pipeline; the reported inner/outer arc asymmetry is a computed output of the spinon convolution, not forced by construction or by a fitted parameter.

  1. self citation load bearing [Introduction, paragraph on DMFT treatment of chargons; also Sec. II.A]
    "To account for dynamical correlation effects associated with the formation of well-defined local magnetic moments, the DMFT treatment of spin symmetry breaking in the chargon subsystem65,67,68 seems more preferable. Compared to the static mean-field description, this approach improves the applicability of the non-linear sigma model (NLSM) for describing long-wavelength fluctuations of the magnetic moments and enables one to capture effects related to quasiparticle damping."

    The claim that DMFT (rather than static mean-field) is essential for the asymmetry rests on the authors' own earlier works that introduced and validated the local-frame DMFT formalism for ordered chargons. The present calculation inherits that pipeline wholesale; without those self-citations the methodological premise is not independently re-derived here. The spectral output itself, however, remains a fresh computation and is not forced by the citations alone.

full rationale

The derivation chain is: (i) fractionalization ansatz c=Rψ (Eq. 2) with long-range order assumed only in the chargon sector; (ii) DMFT self-energies for ordered chargons in the local frame, transformed to the global frame yielding the singular contribution (Eq. 10) and Luttinger surface (Eq. 21); (iii) spin stiffnesses ρ,χ extracted from the same chargon DMFT (spin-current correlators and dynamical susceptibility); (iv) mean-field spinon propagator Dq (Eqs. 30–35) with gap fixed by the sum rule; (v) physical spectral function obtained by the convolution (Eq. 38). Steps (ii)–(v) are explicit numerical calculations whose output (asymmetric damping of the outer side of the hole pocket, Figs. 3–4) is not algebraically identical to any input. Spin stiffnesses are computed, not fitted to the arcs; the ultraviolet cutoff Λ=0.5 and static-χ approximation are stated choices whose effect is checked in Appendix B, not tuned to force the result. The only mild circularity is that the preference for DMFT-ordered chargons over static mean-field, and the concrete implementation of the local-frame formalism, rest on the authors' own prior papers (Refs. 65,67,68). That dependence is load-bearing for the method but does not make the spectral asymmetry a tautology. No self-definitional loop, no fitted-input-called-prediction, and no uniqueness theorem imported from the same authors appear. Score 2 reflects ordinary methodological self-citation without reduction of the central claim.

Assumptions & free parameters 5 free parameters · 6 assumptions · 2 invented entities

The central claim rests on the SU(2) fractionalization ansatz, DMFT for ordered chargons, a mean-field/local NLSM for spinons, and a handful of model parameters (U, t′, cutoff, doping, temperatures). Chargons and spinons are inherited from the existing gauge-theory literature rather than invented here; free parameters are mostly standard Hubbard-model choices plus the UV cutoff and static-stiffness approximation. No collider-style independent handle is offered for the fractionalization map itself.

free parameters (5)
  • Hubbard U = 5.6t
    Set to U=5.6t without a scan; controls moment formation and the strength of the singular self-energy that creates the Luttinger surface.
  • next-nearest hopping t' = 0.3t
    Fixed at t'=0.3t; shapes the bare Fermi surface and hole-pocket location.
  • NLSM ultraviolet cutoff Λ = 0.5
    Set to Λ=0.5 following prior work; enters the spinon sum rule that fixes the magnetic gap Δ.
  • hole doping x (Wu point) = 0.04
    Focus doping x=0.04 chosen as the ‘Wu point’; results for arc formation are reported primarily there.
  • temperatures T = 0.1t, 0.2t
    Main spectra at T=0.1t and 0.2t; spin stiffnesses and gaps are temperature-dependent outputs but the temperatures themselves are chosen.
assumptions (6)
  • domain assumption Physical electrons fractionalize as c_x = R_x ψ_x with chargons ψ carrying long-range magnetic order and R_x fluctuations restoring SU(2).
    Stated in Sec. II as the basis of the approach; without it the chargon/spinon split and Eq. (38) do not describe physical electrons.
  • domain assumption Antiferromagnetic order with Q=(π,π) and only transverse spin stiffnesses nonzero (Eq. 24).
    Restricts the gauge-field content and the CP1 spinon parametrization used throughout.
  • ad hoc to paper Spin stiffnesses may be taken fully local in space and time, J_μ,x;ν,x' ≈ J_μν δ_{x,x'}, with static temporal stiffness χ.
    Sec. II.B and Appendix B; justified as better than prior mean-field but still an approximation that closes the spinon theory.
  • domain assumption Physical electron Green function is the momentum convolution of chargon G with spinon propagator D (Eq. 37–38).
    Direct consequence of the fractionalization map under the mean-field spinon treatment; this is how arcs are obtained from pockets.
  • domain assumption DMFT self-consistency in the local spin frame correctly captures the ordered chargon self-energy including the singular piece that defines the Luttinger surface (Eqs. 5–21).
    Standard DMFT assumption for magnetically ordered states; nonlocal vertex corrections beyond static Q are deferred to spinons.
  • standard math Standard Matsubara DMFT / Anderson-impurity and lattice Green-function identities on the square lattice.
    Used throughout Sec. II.A without modification.
invented entities (2)
  • Chargons (ψ) as ordered fermionic charge carriers in the SU(2) decomposition
    purpose: Carry the reconstructed hole pockets and the singular self-energy / Luttinger surface under DMFT magnetic order.
    Not invented in this paper; inherited from the SU(2) gauge theory of the Hubbard model. No new independent experimental handle is added here beyond associating pockets with quantum oscillations.
  • Spinons (z / R field) as gapped bosonic SU(2) rotations
    purpose: Restore physical spin symmetry and, via convolution, asymmetrically damp outer pocket sides into Fermi arcs.
    Again standard in the cited gauge-theory literature; the paper’s novelty is their dynamical effect on spectra, not the entity itself.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The pseudogap in high-$T_c$ superconductors from SU(2) gauge symmetry and dynamic correlation effects." pith.science (2026). https://pith.science/paper/TS6O45IT

@misc{pith2026260602838,
  author       = {Pith},
  title        = {Pith review of: The pseudogap in high-$T_c$ superconductors from SU(2) gauge symmetry and dynamic correlation effects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TS6O45IT}},
  note         = {Machine review of arXiv:2606.02838}
}
abstract

We consider the spectral properties of the two-dimensional Hubbard model, describing the electronic properties of high-$T_c$ compounds, within the SU(2) gauge theory, which assumes the separation of electronic degrees of freedom into those of spinon and chargon subsystems. We use the dynamic mean-field theory (DMFT) approach to describe magnetic long-range order in the chargon subsystem while also treating spinon fluctuations on top of this state. We show that DMFT supplemented by long-wavelength magnetic fluctuations is essential for describing the asymmetry in the damping between the inner and outer regions of the hole pockets and the resulting formation of Fermi arcs in the underdoped regime, especially at low hole doping. The underlying hole pockets in the chargon subsystem can be associated with those observed in quantum oscillation measurements.

Figures

Figures reproduced from arXiv: 2606.02838 by the authors.

Figure 1
Figure 1. FIG. 1. Spectral functions of chargons with pseudogap for [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Imaginary part of the difference of self-energies of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Spectral density of chargons (a,b) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The same as Fig [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Imaginary part of self-energies differences of electrons [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Evidence of the Cooper-Pair Field with Gaussian Memory Kernel in Unconventional Superconductors

    cond-mat.supr-con 2026-07 conditional novelty 5.0 of 10

    Cuprate superconductivity reorganizes a Gaussian-memory Cooper-pair continuum between incoherent pseudogap and coherent Bogoliubov channels, with Raman/ARPES/tunneling as complementary PCF projections.

Reference graph

Works this paper leans on

129 extracted references · 1 canonical work pages · cited by 1 Pith paper

  1. [1]

    Berthier, M

    C. Berthier, M. H. Julien, M. Horvatić, Y. Berthier, NMR studies of the normal state of high temperature superconductors, J. Phys. I France 6, 2205 (1996) https://dx.doi.org/10.1051/jp1:1996209

    doi:10.1051/jp1:1996209 1996

  2. [2]

    Asayama, Y

    K. Asayama, Y. Kitaoka, G. Q. Zheng, K. Ishida, NMR studies of high T_c superconductors, Prog. Nucl. Magn. Reson. Spectrosc. 28, 221 (1996) https://dx.doi.org/10.1016/0079-6565(95)01025-4

    doi:10.1016/0079-6565(95)01025-4 1996

  3. [3]

    Plakida, High-Temperature Cuprate Superconductors (Springer, Heidelberg, 2010)

    N. Plakida, High-Temperature Cuprate Superconductors (Springer, Heidelberg, 2010)

    2010

  4. [4]

    Y. Ando, Y. Kurita, S. Komiya, S. Ono, K. Segawa, Evolution of the Hall coefficient and the peculiar electronic structure of the cuprate superconductors, Phys. Rev. Lett. 92, 197001 (2004) https://dx.doi.org/10.1103/PhysRevLett.92.197001

    doi:10.1103/physrevlett.92.197001 2004

  5. [5]

    Y. Ando, S. Komiya, K. Segawa, S. Ono, Y. Kurita, Electronic phase diagram of high- T_c cuprate superconductors from a mapping of the in-plane resistivity curvature, Phys. Rev. Lett. 93, 267001 (2004) https://dx.doi.org/10.1103/PhysRevLett.93.267001

    doi:10.1103/physrevlett.93.267001 2004

  6. [6]

    S. Ono, S. Komiya, Y. Ando, Strong charge fluctuations manifested in the high-temperature Hall coefficient of high- T_c cuprates, Phys. Rev. B 75, 024515 (2007) https://dx.doi.org/10.1103/PhysRevB.75.024515

    doi:10.1103/physrevb.75.024515 2007

  7. [7]

    Damascelli, Z.-X

    A. Damascelli, Z.-X. Shen, and Z. Hussain, Angle-resolved photoemission studies of the cuprate superconductors, Rev. Mod. Phys. 75, 473 (2003) https://doi.org/10.1103/RevModPhys.75.473

    doi:10.1103/revmodphys.75.473 2003

  8. [8]

    Hashimoto, T

    M. Hashimoto, T. Yoshida, H. Yagi, M. Takizawa, A. Fujimori, M. Kubota, K. Ono, K. Tanaka, D. H. Lu, Z.-X. Shen, S. Ono, and Y. Ando, Doping evolution of the electronic structure in the single-layer cuprates Bi _2 Sr _ 2-x La _x CuO _ 6+ : Comparison with other single-layer cuprates, Phys. Rev. B 77, 094516 (2008) https://doi.org/10.1103/PhysRevB.77.094516

    doi:10.1103/physrevb.77.094516 2008

Show all 129 references
  1. [9]

    Yoshida, X

    T. Yoshida, X. J. Zhou, K. Tanaka, W. L. Yang, Z. Hussain, Z.-X. Shen, A. Fujimori, S. Komiya, Y. Ando, H. Eisaki, T. Kakeshita, and S. Uchida, Systematic doping evolution of the underlying Fermi surface of La _ 2-x Sr _x CuO _4 , Phys. Rev. B 74, 224510 (2006) https://doi.org...

    2006 doi

  2. [10]

    K. M. Shen, F. Ronning, D. H. Lu, F. Baumberger, N. J. C. Ingle, W. S. Lee, W. Meevasana, Y. Kohsaka, M. Azuma, M. Takano, H. Takagi, and Z.-X. Shen, Nodal quasiparticles and antinodal charge ordering in Ca _ 2-x Na _x CuO _2 Cl _2 , Science 307, 901 (2005) https://doi.org/10....

    2005 doi

  3. [11]

    M. R. Norman, H. Ding, M. Randeria, J. C. Campuzano, T. Yokoya, T. Takeuchi, T. Takahashi, T. Mochiku, K. Kadowaki, P. Guptasarma, D. G. Hinks, Destruction of the Fermi surface in underdoped high- T_c superconductors, Nature 392, 157 (1998) https://dx.doi.org/10.1038/32366

    1998 doi

  4. [12]

    Kanigel, M

    A. Kanigel, M. R. Norman, M. Randeria, U. Chatterjee, S. Souma, A. Kaminski, H. M. Fretwell, S. Rosenkranz, M. Shi, T. Sato, et. al., Evolution of the pseudogap from Fermi arcs to the nodal liquid, Nature Physics 2, 447 (2006) https://dx.doi.org/10.1038/nphys334

    2006 doi

  5. [13]

    Doiron-Leyraud, C

    N. Doiron-Leyraud, C. Proust, D. LeBoeuf, J. Levallois, J.-B. Bonnemaison, R. Liang, D. A. Bonn, W. N. Hardy, L. Taillefer, Quantum oscillations and the Fermi surface in an underdoped high- T_c superconductor, Nature 447, 565 (2007) https://dx.doi.org/10.1038/nature05872

    2007 doi

  6. [14]

    LeBoeuf, N

    D. LeBoeuf, N. Doiron-Leyraud, J. Levallois, R. Daou, J.-B. Bonnemaison, N. E. Hussey, L. Balicas, B. J. Ramshaw, R. Liang, D. A. Bonn, W. N. Hardy, S. Adachi, C. Proust, L. Taillefer, Electron pockets in the Fermi surface of hole-doped high- T_c superconductors, Nature 450, 5...

    2007 doi

  7. [15]

    Doiron-Leyraud, S

    N. Doiron-Leyraud, S. Badoux, S. René de Cotret, D. LeBoeuf, N. E. Hussey, H. Chang, B. J. Ramshaw, R. Liang, D. A. Bonn, W. N. Hardy, L. Taillefer, Evidence for a small hole pocket in the Fermi surface of underdoped YBa _2 Cu _3 O _y , Nature Communications 6, 6034 (2015) htt...

    2015 doi

  8. [16]

    E. A. Yelland, J. Singleton, C. H. Mielke, N. Harrison, F. F. Balakirev, B. Dabrowski, J. R. Cooper, Quantum oscillations in the underdoped cuprate YBa _2 Cu _4 O _8 , Phys. Rev. Lett. 100, 047003 (2008) https://dx.doi.org/10.1103/PhysRevLett.100.047003

    2008 doi

  9. [17]

    S. E. Sebastian, N. Harrison, M. M. Altarawneh, R. Liang, D. A. Bonn, W. N. Hardy, G. G. Lonzarich, Chemical potential oscillations from nodal Fermi surface pocket in the underdoped high-temperature superconductor YBa _2 Cu _3 O _ 6+x , Nature Communications 2, 471 (2011) http...

    2011 doi

  10. [18]

    Barišić, S

    N. Barišić, S. Badoux, M. K. Chan, C. Dorow, W. Tabis, B. Vignolle, G. Yu, J. Béard, X. Zhao, C. Proust, and M. Greven, Universal quantum oscillations in the underdoped cuprate superconductors, Nature Physics 9, 761 (2013) https://dx.doi.org/10.1038/nphys2792

    2013 doi

  11. [19]

    M. K. Chan, N. Harrison, R. D. McDonald, B. J. Ramshaw, K. A. Modic, N. Barišić, M. Greven, Single reconstructed Fermi surface pocket in an underdoped single-layer cuprate superconductor, Nature Communications 7, 12244 (2016) https://dx.doi.org/10.1038/ncomms12244

    2016 doi

  12. [20]

    S. E. Sebastian and C. Proust, Quantum oscillations in hole-doped cuprates, Annual Review of Condensed Matter Physics 6, 411 (2015) https://dx.doi.org/10.1146/annurev-conmatphys-030212-184305

    2015 doi

  13. [21]

    H.-B. Yang, J. D. Rameau, Z.-H. Pan, G. D. Gu, P. D. Johnson, H. Claus, D. G. Hinks, and T. E. Kidd, Reconstructed Fermi Surface of Underdoped Bi _2 Sr _2 CaCu _2 O _ 8+ Cuprate Superconductors, http://dx.doi.org/10.1103/PhysRevLett.107.047003 Phys. Rev. Lett. 107 , 047003 (2011)

    2011 doi

  14. [22]

    J. Meng, G. Liu, W. Zhang, L. Zhao, H. Liu, X. Jia, D. Mu, S. Liu, X. Dong, W. Lu, G. Wang, Y. Zhou, Y. Zhu, X. Wang, Z. Xu, C. Chen, and X. J. Zhou, Coexistence of Fermi arcs and Fermi pockets in a high- T_c copper oxide superconductor, Nature 462, 335 (2009) https://doi.org/...

    2009 doi

  15. [23]

    S. E. Sebastian, N. Harrison, E. Palm, T. P. Murphy, C. H. Mielke, R. Liang, D. A. Bonn, W. N. Hardy, G. G. Lonzarich, A multi-component Fermi surface in the vortex state of an underdoped high- T_c superconductor, Nature 454, 200 (2008) https://doi.org/10.1038/nature07095

    2008 doi

  16. [24]

    A. J. Millis, M. R. Norman, Antiphase stripe order as the origin of electron pockets observed in 1/8-hole-doped cuprates, Phys. Rev. B 76, 220503(R) (2007) https://doi.org/10.1103/PhysRevB.76.220503

    2007 doi

  17. [25]

    Chakravarty, H.-Y

    S. Chakravarty, H.-Y. Kee, Fermi pockets and quantum oscillations of the Hall coefficient in high-temperature superconductors, PNAS 105, 8835 (2008) https://doi.org/10.1073/pnas.0804002105

    2008 doi

  18. [26]

    A. J. Millis, H. Monien, and D. Pines, Phenomenological model of nuclear relaxation in the normal state of YBa _2 Cu _3 O _7 , https://doi.org/10.1103/PhysRevB.42.167 Phys.\ Rev.\ B 42, 167 (1990)

    1990 doi

  19. [27]

    A. J. Millis, Spin fluctuations in high-temperature superconductors, https://doi.org/10.1103/PhysRevB.50.16052 Phys.\ Rev.\ B 50, 16052--16055 (1994)

    1994 doi

  20. [28]

    Y. Zha, V. Barzykin, and D. Pines, NMR and neutron-scattering experiments on the cuprate superconductors: A critical reexamination, https://doi.org/10.1103/PhysRevB.54.7561 Phys.\ Rev.\ B 54, 7561--7574 (1996)

    1996 doi

  21. [29]

    A. V. Chubukov, D. Pines, and B. P. Stojkovi\'c, Temperature crossovers in cuprates, https://doi.org/10.1088/0953-8984/8/48/021 J.\ Phys.: Condens.\ Matter 8, 10017--10036 (1996)

    1996 doi

  22. [30]

    Schmalian, D

    J. Schmalian, D. Pines, and B. Stojkovi\'c, Microscopic theory of weak pseudogap behavior in the underdoped cuprate superconductors: General theory and quasiparticle properties, https://doi.org/10.1103/PhysRevB.60.667 Phys.\ Rev.\ B 60, 667 (1999)

    1999 doi

  23. [31]

    E. Z. Kuchinskii and M. V. Sadovskii, Models of the pseudogap state of two-dimensional systems, https://doi.org/10.1134/1.558879 JETP 88, 968 (1999)

    1999 doi

  24. [33]

    Sokol and D

    A. Sokol and D. Pines, Toward a unified magnetic phase diagram of the cuprate superconductors, https://doi.org/10.1103/PhysRevLett.71.2813 Phys.\ Rev.\ Lett. 71, 2813 (1993)

    1993 doi

  25. [34]

    A. V. Chubukov, D. Pines, and B. P. Stojkovi\'c, Crossover and scaling in a nearly antiferromagnetic Fermi liquid in two dimensions, https://doi.org/10.1103/PhysRevB.51.14874 Phys.\ Rev.\ B 51, 14874 (1995)

    1995 doi

  26. [35]

    M. S. Scheurer, S. Chatterjee, W. Wu, M. Ferrero, A. Georges, and S. Sachdev, Topological order in the pseudogap metal, PNAS 115, E3665 (2018) https://doi.org/10.1073/pnas.1720580115

    2018 doi

  27. [37]

    Iskakov, M

    S. Iskakov, M. I. Katsnelson, A. I. Lichtenstein, Perturbative solution of fermionic sign problem in quantum Monte Carlo computations, https://doi.org/10.1038/s41524-024-01221-w npj Computational Materials 10 , 36 (2024)

    2024 doi

  28. [38]

    Stepanov, S

    E.A. Stepanov, S. Iskakov, M.I. Katsnelson, A.I. Lichtenstein, Superconductivity of Bad Fermions: Origin of Two Gaps in HTSC Cuprates, https://doi.org/10.1038/s42005-026-02532-8 Comm. Physics 9, 91 (2026)

    2026 doi

  29. [39]

    A.-M. S. Tremblay, B. Kyung, and D. S\'en\'echal, Pseudogap and high-temperature superconductivity from weak to strong coupling. Towards a quantitative theory, https://doi.org/10.1063/1.2199446 Low Temp. Phys. 32 , 424 (2006)

    2006 doi

  30. [40]

    E. Gull, O. Parcollet, and A. J. Millis, Superconductivity and the Pseudogap in the two-dimensional Hubbard model, https://doi.org/10.1103/PhysRevLett.110.216405 Phys. Rev. Lett. 110 , 216405 (2013)

    2013 doi

  31. [41]

    Gunnarsson, T

    O. Gunnarsson, T. Schäfer, J. P. F. LeBlanc, E. Gull, J. Merino, G. Sangiovanni, G. Rohringer, A. Toschi, Fluctuation diagnostics of the electron self-energy: Origin of the pseudogap physics, https://doi.org/10.1103/PhysRevLett.114.236402 Phys. Rev. Lett. 114 , 236402 (2015)

    2015 doi

  32. [42]

    Y. Yu, S. Iskakov, E. Gull, K. Held, and F. Krien, Unambiguous Fluctuation Decomposition of the Self-Energy: Pseudogap Physics beyond Spin Fluctuations, Phys. Rev. Lett. 132, 216501 (2024) https://doi.org/10.1103/PhysRevLett.132.216501; Pairing boost from enhanced spin-fermion...

    2024 doi

  33. [43]

    W. Wu, M. S. Scheurer, S. Chatterjee, S. Sachdev, A. Georges, and M. Ferrero, Pseudogap and Fermi-Surface Topology in the Two-Dimensional Hubbard Model, Phys. Rev. X 8, 021048 (2018) https://doi.org/10.1103/PhysRevX.8.021048

    2018 doi

  34. [44]

    Krien, P

    F. Krien, P. Worm, P. Chalupa, A. Toschi, and K. Held, Explaining the pseudogap through damping and antidamping on the Fermi surface by imaginary spin scattering, 10.1038/s42005-022-01117-5 Commun. Phys. 5 , 336 (2022)

    2022 doi

  35. [45]

    J.-M. Lihm, D. Kiese, S.-S. B. Lee, F. B. Kugler, The finite-difference parquet method: Enhanced electron-paramagnon scattering opens a pseudogap, https://doi.org/10.1073/pnas.2525308123 PNAS 123 , e2525308123 (2026)

    2026 doi

  36. [46]

    Dagotto, Correlated electrons in high-temperature superconductors, https://doi.org/10.1103/RevModPhys.66.763 Rev

    E. Dagotto, Correlated electrons in high-temperature superconductors, https://doi.org/10.1103/RevModPhys.66.763 Rev. Mod. Phys. 66 , 763 (1994)

    1994 doi

  37. [47]

    Imada, A

    M. Imada, A. Fujimori, and Y. Tokura, Metal-insulator transitions, https://doi.org/10.1103/RevModPhys.70.1039 Rev. Mod. Phys. 70 , 1039 (1998)

    1998 doi

  38. [48]

    P. A. Lee, N. Nagaosa, and X.-G. Wen, Doping a Mott insulator: Physics of high-temperature superconductivity, https://doi.org/10.1103/RevModPhys.78.17 Rev. Mod. Phys. 78 , 17 (2006)

    2006 doi

  39. [49]

    A. I. Milstein and O. P. Sushkov, Effective action, magnetic excitations, and quantum fluctuations in lightly doped single-layer cuprates, Phys. Rev. B 78, 014501 https://doi.org/10.1103/PhysRevB.78.014501 (2008)

    2008 doi

  40. [50]

    Nikolaenko, J

    A. Nikolaenko, J. von Milczewski, D. G. Joshi, and S. Sachdev, Spin density wave, Fermi liquid, and fractionalized phases in a theory of antiferromagnetic metals using paramagnons and bosonic spinons, https://doi.org/10.1103/PhysRevB.108.045123 Phys. Rev. B 108 , 045123 (2023)

    2023 doi

  41. [51]

    Zhang and S

    Y.-H. Zhang and S. Sachdev, Deconfined criticality and ghost Fermi surfaces at the onset of antiferromagnetism in a metal, https://doi.org/10.1103/PhysRevB.102.155124 Phys. Rev. B 102 , 155124 (2020)

    2020 doi

  42. [52]

    Sachdev, H

    S. Sachdev, H. D. Scammell, M. S. Scheurer, and G. Tarnopolsky, Gauge theory for the cuprates near optimal doping, https://doi.org/10.1103/PhysRevB.99.054516 Phys. Rev. B 99 , 054516 (2019)

    2019 doi

  43. [53]

    Sachdev, E

    S. Sachdev, E. Berg, S. Chatterjee, and Y. Schattner, Spin density wave order, topological order, and Fermi surface reconstruction, https://doi.org/10.1103/PhysRevB.94.115147 Phys. Rev. B 94 , 115147 (2016)

    2016 doi

  44. [54]

    Sachdev, M

    S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, Fluctuating spin density waves in metals, Phys. Rev. B 80, 155129 (2009) https://doi.org/10.1103/PhysRevB.80.155129

    2009 doi

  45. [55]

    Qi and S

    Y. Qi and S. Sachdev, Effective theory of Fermi pockets in fluctuating antiferromagnets, https://doi.org/10.1103/PhysRevB.81.115129 Phys. Rev. B 81 , 115129 (2010)

    2010 doi

  46. [56]

    Chowdhury and S

    D. Chowdhury and S. Sachdev, Higgs criticality in a two-dimensional metal, https://doi.org/10.1103/PhysRevB.91.115123 Phys. Rev. B 91 , 115123 (2015)

    2015 doi

  47. [57]

    Chatterjee, S

    S. Chatterjee, S. Sachdev, and A. Eberlein, Thermal and electrical transport in metals and superconductors across antiferromagnetic and topological quantum transitions, https://doi.org/10.1103/PhysRevB.96.075103 Phys. Rev. B 96 , 075103 (2017)

    2017 doi

  48. [58]

    Chatterjee, S

    S. Chatterjee, S. Sachdev, and M. Scheurer, Intertwining topological order and broken symmetry in a theory of fluctuating spin density waves, https://doi.org/10.1103/PhysRevLett.119.227002 Phys. Rev. Lett. 119 , 227002 (2017)

    2017 doi

  49. [59]

    E. A. Stepanov, S. Brener, V. Harkov, M. I. Katsnelson, and A. I. Lichtenstein, Spin dynamics of itinerant electrons: local magnetic moment formation and Berry phase, https://doi.org/10.1103/PhysRevB.105.155151 Phys. Rev. B 105 , 155151 (2022)

    2022 doi

  50. [60]

    Vilardi and P

    D. Vilardi and P. M. Bonetti, SC ^* superconductivity and spin stiffnesses in the SU(2) gauge theory of the two-dimensional Hubbard model, arXiv:2511.03436 [cond-mat.str-el] (2025) https://doi.org/10.48550/arXiv.2511.03436

    2025 doi

  51. [61]

    M\"uller-Groeling, P

    H. M\"uller-Groeling, P. M. Bonetti, P. Forni, and W. Metzner, SU(2) gauge theory of fluctuating stripe order in the two-dimensional Hubbard model, arXiv:2603.13071 [cond-mat.str-el] (2026) https://doi.org/10.48550/arXiv.2603.13071

    2026 doi arXiv

  52. [62]

    Vilardi, P

    D. Vilardi, P. M. Bonetti, and W. Metzner, Spin stiffnesses and stability of magnetic order in the lightly doped two-dimensional Hubbard model, https://doi.org/10.1103/x7qr-f6lm Phys. Rev. B 112 , 245149 (2025)

    2025 doi

  53. [63]

    Forni, P

    P. Forni, P. M. Bonetti, H. M\"uller-Groeling, D. Vilardi, and W. Metzner, Spin susceptibility in a pseudogap state with fluctuating spiral magnetic order, https://doi.org/10.1103/zm7b-jdzf Phys. Rev. B 113 , 045144 (2026)

    2026 doi

  54. [64]

    P. M. Bonetti and W. Metzner, SU(2) gauge theory of the pseudogap phase in the two-dimensional Hubbard model, Phys. Rev. B 106, 205152 (2022) https://doi.org/10.1103/PhysRevB.106.205152

    2022 doi

  55. [65]

    I. A. Goremykin and A. A. Katanin, Antiferromagnetic and spin spiral correlations in the doped two-dimensional Hubbard model: gauge symmetry, Ward identities, and dynamical mean-field theory analysis, https://dx.doi.org/10.1103/PhysRevB.110.085153 Phys. Rev. B 110 , 085153 (2024)

    2024 doi

  56. [66]

    I. A. Goremykin and A. A. Katanin, Frequency dependence of temporal spin stiffness and short-range magnetic order in the doped two-dimensional Hubbard model, https://dx.doi.org/10.1103/w4vc-n5l6 Phys. Rev. B 112 , L060405 (2025)

    2025 doi

  57. [67]

    I. A. Goremykin and A. A. Katanin, Commensurate and spiral magnetic order in the doped two-dimensional Hubbard model: Dynamical mean-field theory analysis, Phys. Rev. B 107, 245104 (2023) https://doi.org/10.1103/PhysRevB.107.245104

    2023 doi

  58. [68]

    P. M. Bonetti, J. Mitscherling, D. Vilardi, and W. Metzner, Charge carrier drop at the onset of pseudogap behavior in the two-dimensional Hubbard model, https://doi.org/10.1103/PhysRevB.101.165142 Phys. Rev. B 101 , 165142 (2020)

    2020 doi

  59. [69]

    H. J. Schulz, Effective action for strongly correlated fermions from functional integrals, Phys. Rev. Lett. 65, 2462 (1990) https://doi.org/10.1103/PhysRevLett.65.2462; H. J. Schulz, Functional Integrals for Correlated Electrons, Proceedings of NATO Advanced Research Workshop ...

    1990 doi

  60. [70]

    Z. Y. Weng, C. S. Ting, and T. K. Lee, Path-integral approach to the Hubbard model, Phys. Rev. B 43, 3790 (1991) https://doi.org/10.1103/PhysRevB.43.3790

    1991 doi

  61. [71]

    Sengupta and N

    K. Sengupta and N. Dupuis, Effective action and collective modes in quasi-one-dimensional spin-density-wave systems, Phys. Rev. B 61, 13493 (2000) https://doi.org/10.1103/PhysRevB.61.13493; Y. Tomio, N. Dupuis, and Y. Suzumura, Effect of nearest- and next-nearest neighbor inte...

    2000 doi

  62. [72]

    Dupuis, Spin fluctuations and pseudogap in the two-dimensional half-filled Hubbard model at weak coupling, Phys

    N. Dupuis, Spin fluctuations and pseudogap in the two-dimensional half-filled Hubbard model at weak coupling, Phys. Rev. B 65, 245118 (2002) https://doi.org/10.1103/PhysRevB.65.245118; K. Borejsza and N. Dupuis, Antiferromagnetism and single-particle properties in the two-dime...

    2002 doi

  63. [73]

    Fleck, A

    M. Fleck, A. I. Liechtenstein, A. M. Ole\'s, L. Hedin, and V. I. Anisimov, Dynamical Mean-Field Theory for Doped Antiferromagnets, https://dx.doi.org/10.1103/PhysRevLett.80.2393 Phys. Rev. Lett. 80 , 2393 (1998)

    1998 doi

  64. [74]

    S. Goto, S. Kurihara, and D. Yamamoto, Incommensurate spiral magnetic order on anisotropic triangular lattice: Dynamical mean-field study in a spin-rotating frame, https://dx.doi.org/10.1103/PhysRevB.94.245145 Phys. Rev. B 94 , 245145 (2016)

    2016 doi

  65. [75]

    Luttinger, Fermi Surface and Some Simple Equilibrium Properties of a System of Interacting Fermions, Phys

    J.M. Luttinger, Fermi Surface and Some Simple Equilibrium Properties of a System of Interacting Fermions, Phys. Rev. 119, 1153 (1960) https://doi.org/10.1103/PhysRev.119.1153

    1960 doi

  66. [76]

    Dzyaloshinskii, Extended Van-Hove Singularity and Related Non-Fermi Liquids, https://dx.doi.org/10.1051/jp1:1996127 J

    I. Dzyaloshinskii, Extended Van-Hove Singularity and Related Non-Fermi Liquids, https://dx.doi.org/10.1051/jp1:1996127 J. Phys. I France 6 119 (1996) ; Some consequences of the Luttinger theorem: The Luttinger surfaces in non-Fermi liquids and Mott insulators, https://doi.org/...

    1996 doi

  67. [77]

    Kitatani, Y

    M. Kitatani, Y. Nomura, S. Sakai, and R. Arita, Luttinger surface and exchange splitting induced by ferromagnetic fluctuations, arXiv:2509.21034 (2025) https://doi.org/10.48550/arXiv.2509.21034

    2025 doi

  68. [78]

    P. Worm, M. Reitner, K. Held, and A. Toschi, Fermi and Luttinger Arcs: Two Concepts, Realized on One Surface, https://doi.org/10.1103/PhysRevLett.133.166501 Phys. Rev. Lett. 133 , 166501 (2024)

    2024 doi

  69. [79]

    Watzenb\"ock, M

    C. Watzenb\"ock, M. Fellinger, K. Held, and A. Toschi, Long-term memory magnetic correlations in the Hubbard model: A dynamical mean-field theory analysis, https://dx.doi.org/10.21468/SciPostPhys.12.6.184 SciPost Phys. 12 , 184 (2022)

    2022 doi

  70. [80]

    J. J. Wagman, G. Van Gastel, K. A. Ross, Z. Yamani, Y

  71. [81]

    Cheong, G

    S-W. Cheong, G. Aeppli, T. E. Mason, H. Mook, S. M. Hayden,

  72. [82]

    T. E. Mason, G. Aeppli, S. M. Hayden, A. P. Ramirez, and H

  73. [83]

    Matsuda, K

    M. Matsuda, K. Yamada, Y. Endoh, T. R. Thurston, G. Shirane,

  74. [84]

    Yamada, C

    K. Yamada, C. H. Lee, K. Kurahashi, J. Wada, S. Wakimoto, S. Ueki, H. Kimura, Y. Endoh, S. Hosoya, G. Shirane, R. J. Birgeneau, M. Greven, M. A. Kastner, and Y. J. Kim, Doping dependence of the spatially modulated dynamical spin correlations and the superconducting-transition ...

    1998 doi

  75. [85]

    Wakimoto, G

    S. Wakimoto, G. Shirane,

  76. [86]

    Katano, M

    S. Katano, M. Sato, K. Yamada, T. Suzuki, and T. Fukase, Enhancement of static antiferromagnetic correlations by magnetic field in a superconductor La_ 2-x Sr_x Cu O_4 with x 0.12 , https://link.aps.org/doi/10.1103/PhysRevB.62.R14677 Phys. Rev. B 62 , R14677 (2000)

    2000 doi

  77. [87]

    Khaykovich, Y

    B. Khaykovich, Y. S. Lee, R. W. Erwin, S.-H. Lee, S. Wakimoto, K. J. Thomas, M. A. Kastner, and R. J. Birgeneau, Enhancement of long-range magnetic order by magnetic field in superconducting La_2 Cu O_ 4+y , https://link.aps.org/doi/10.1103/PhysRevB.66.014528 Phys. Rev. B 66 ,...

    2002 doi

  78. [88]

    B. Lake, H. M. R nnow, N. B. Christensen, G. Aeppli, K. Lefmann, D. F. McMorrow, P. Vorderwisch, P. Smeibidl, N. Mangkorntong, T. Sasagawa, et. al., Antiferromagnetic order induced by an applied magnetic field in a high-temperature superconductor, https://www.nature.com/articl...

    2002

  79. [89]

    Khaykovich, R

    B. Khaykovich, R. J. Birgeneau, F. C. Chou, R. W. Erwin, M. A. Kastner, S.-H. Lee, Y. S. Lee, P. Smeibidl, P. Vorderwisch, and S. Wakimoto, Effect of a magnetic field on long-range magnetic order in stage-4 and stage-6 superconducting La_2 Cu O_ 4+y , https://link.aps.org/doi/...

    2003 doi

  80. [90]

    Khaykovich, S

    B. Khaykovich, S. Wakimoto, R. J. Birgeneau, M. A. Kastner, Y. S. Lee, P. Smeibidl, P. Vorderwisch, and K. Yamada, Field-induced transition between magnetically disordered and ordered phases in underdoped La_ 2-x Sr_x Cu O_4 , https://link.aps.org/doi/10.1103/PhysRevB.71.22050...

    2005 doi

  81. [91]

    Chang, Ch

    J. Chang, Ch. Niedermayer, R. Gilardi, N. B. Christensen, H. M. R nnow, D. F. McMorrow, M. Ay, J. Stahn, O. Sobolev, A. Heiss, et al., Tuning competing orders in La_ 2-x Sr_x Cu O_4 cuprate superconductors by the application of an external magnetic field, https://link.aps.org/...

    2008 doi

  82. [92]

    Frachet, I

    M. Frachet, I. Vinograd, R. Zhou, S. Benhabib, S. Wu, H. Mayaffre, S. Kr\"amer, S. K. Ramakrishna, A. P. Reyes, J. Debray, et. al., Hidden magnetism at the pseudogap critical point of a cuprate superconductor, https://doi.org/10.1038/s41567-020-0950-5 Nature Physics 16 , 1064 (2020)

    2020 doi

  83. [93]

    Vinograd, R

    I. Vinograd, R. Zhou, H. Mayaffre, S. Kr\"amer, S. K. Ramakrishna, A. P. Reyes, T. Kurosawa, N. Momono, M. Oda, S. Komiya, et. al., Competition between spin ordering and superconductivity near the pseudogap boundary in La _ 2-x Sr _x CuO _4 : Insights from NMR, https://doi.org...

    2022 doi

  84. [94]

    D. J. Campbell, M. Frachet, V. Oliviero, T. Kurosawa, N. Momono, M. Oda, J. Chang, D. Vignolles, C. Proust, and D. LeBoeuf, Strange metal from spin fluctuations in a cuprate superconductor, ArXiv: 2412.03720 https://arxiv.org/abs/2412.03720

  85. [95]

    Gunnarsson, T

    O. Gunnarsson, T. Sch\"afer, J. P. F. LeBlanc, E

  86. [96]

    E. A. Stepanov, L. Peters, I. S. Krivenko, A. I

  87. [97]

    Krien, P

    F. Krien, P. Worm, P. Chalupa-Gantner, A. Toschi,

  88. [98]

    Vilardi, P

    D. Vilardi, P. M. Bonetti, and W. Metzner, Dynamical functional renormalization group computation of order parameters and critical temperatures in the two-dimensional Hubbard model, https://doi.org/10.1103/PhysRevB.102.245128 Phys

    doi

  89. [99]

    A. M. Polyakov, Phys. Lett. B 59, 79 (1975)

    1975

  90. [100]

    Auerbach, Interacting electrons and quantum magnetism

    A. Auerbach, Interacting electrons and quantum magnetism

  91. [101]

    D. R. Nelson and R. A. Pelcovits, Momentum-shell recursion relations, anisotropic spins, and liquid crystals in 2+ dimensions, Phys. Rev. B 16, 2191 (1977) https://doi.org/10.1103/PhysRevB.16.2191

    1977 doi

  92. [102]

    F. D. M. Haldane, Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets: Semiclassically Quantized Solitons of the One-Dimensional Easy-Axis Néel State, https://doi.org/10.1103/PhysRevLett.50.1153 Phys. Rev. Lett. 50 , 1153 (1983)

    1983 doi

  93. [103]

    Chakravarty, B

    S. Chakravarty, B. I. Halperin, and D. R. Nelson, Low-temperature behavior of two-dimensional quantum antiferromagnets, https://doi.org/10.1103/PhysRevLett.60.1057 Phys. Rev. Lett. 60 , 1057 (1988) ; Two-dimensional quantum Heisenberg antiferromagnet at low temperatures https:...

    1988 doi

  94. [104]

    Chubukov, S

    A. Chubukov, S. Sachdev, and J. Ye, Theory of Two-Dimensional Quantum Heisenberg Antiferromagnets with a Nearly Critical Ground State, https://doi.org/10.1103/PhysRevB.49.11919 Phys. Rev. B 49 , 11919 (1994)

    1994 doi

  95. [105]

    Dombre and N

    T. Dombre and N. Read, Nonlinear models for triangular quantum antiferromagnets, Phys. Rev. B 39, 6797 (1989) https://doi.org/10.1103/PhysRevB.39.6797

    1989 doi

  96. [106]

    Azaria, B

    P. Azaria, B. Delamotte, and T. Jolicoeur, Nonuniversality in helical and canted-spin systems, Phys. Rev. Lett. 64, 3175 (1990) https://doi.org/10.1103/PhysRevLett.64.3175; P. Azaria, B. Delamotte, and D. Mouhanna, Low-temperature properties of two-dimensional frustrated quant...

    1990 doi

  97. [107]

    Sachdev and N

    S. Sachdev and N. Read, Large N expansion for frustrated and doped quantum antiferromagnets, https://doi.org/10.1142/S0217979291000158 Int. J. Mod. Phys. B 5 , 219 (1991)

    1991 doi

  98. [108]

    A. V. Chubukov, T. Senthil, and S. Sachdev, Universal Magnetic Properties of Frustrated Quantum Antiferromagnets in Two Dimensions, https://doi.org/10.1103/PhysRevLett.72.2089 Phys. Rev. Lett. 72 , 2089 (1994) ; A. V. Chubukov, S. Sachdev, and T. Senthil, Quantum Phase Transit...

    doi

  99. [109]

    Azaria, P

    P. Azaria, P. Lecheminant, and D. Mouhanna, The massive CP^ N-1 model for frustrated spin systems, Nucl. Phys. B 455, 648 (1995) https://doi.org/10.1016/0550-3213(95)00514-S

    1995 doi

  100. [110]

    P. M. Bonetti, Local Ward identities for collective excitations in fermionic systems with spontaneously broken symmetries, Phys. Rev. B 106, 155105 (2022) https://doi.org/10.1103/PhysRevB.106.155105

    2022 doi

  101. [111]

    Bonetti, Erratum: Local Ward identities for collective excitations in fermionic systems with spontaneously broken

    P. Bonetti, Erratum: Local Ward identities for collective excitations in fermionic systems with spontaneously broken

  102. [112]

    P. M. Bonetti, Long-range order, bosonic fluctuations, and pseudogap in strongly correlated electron systems, PhD thesis https://doi.org/10.18419/opus-12480, Universit\"at Stuttgart, 2022; ArXiv2210.08889 https://doi.org/10.48550/arXiv.2210.08889

    2022 doi arXiv

  103. [113]

    P. M. Bonetti and W. Metzner, Spin stiffness, spectral weight, and Landau damping of magnons in metallic spiral magnets, https://dx.doi.org/10.1103/PhysRevB.105.134426 Phys. Rev. B 105 , 134426 (2022)

    2022 doi

  104. [114]

    A. V. Syromyatnikov, Collective excitations in spin-1/2 magnets through bond-operator formalism designed both for paramagnetic and ordered phases, https://doi.org/10.1103/PhysRevB.98.184421 Phys. Rev. B 98 , 184421 (2018) ; A. V. Syromyatnikov and A. Yu. Aktersky, Elementary e...

    2018 doi

  105. [115]

    SSDW1,SSDW2,SSDWIc1,SSDWIc2,SSDWIc3,SSDW3,SSDW4,SSDWOur,OurFirst,FrequencyAndJumpIssues1

    See Supplemental Material for the derivation of the nonlinear sigma model, explicit form of the momentum cutoff, mean field equations, details of calculation of spin susceptibility and current correlation functions, as well as additional results for spatial, temporal spin stif...

  106. [116]

    A. A. Katanin, H. Yamase, and V. Yu. Irkhin, Ferromagnetic instability and finite-temperature properties of two-dimensional electron systems with van Hove singularities, https://doi.org/10.1143/JPSJ.80.063702 J. Phys. Soc. Jpn. 80 , 063702 (2011)

    2011 doi

  107. [117]

    P. A. Igoshev, M. A. Timirgazin, V. F. Gilmutdinov, A. K. Arzhnikov, and V. Yu. Irkhin, Spiral magnetism in the single-band Hubbard model: the Hartree-Fock and slave-boson approaches, https://doi.org/10.1088/0953-8984/27/44/446002 J. Phys.: Cond. Matt. 27 , 446002 (2015) ; V. ...

    2015 doi

  108. [118]

    D. K. Singh, A. Go, H.-Y. Choi, and Y. Bang, The stability of hole-doped antiferromagnetic state in a two-orbital model, New J. Phys. 22, 063048 (2020) https://iopscience.iop.org/article/10.1088/1367-2630/ab84b7

    2020 doi

  109. [119]

    Scholle, P

    R. Scholle, P. M. Bonetti, D. Vilardi, and W. Metzner, Comprehensive mean-field analysis of magnetic and charge orders in the two-dimensional Hubbard model, https://doi.org/10.1103/PhysRevB.108.035139 Phys. Rev. B 108 , 035139 (2023)

    2023 doi

  110. [120]

    Radaelli, O

    J. Radaelli, O. J. Lipscombe, M. Zhu, J. R. Stewart, A. A. Patel, S. Sachdev, and S. M. Hayden, Critical spin fluctuations across the superconducting dome in La _ 2-x Sr _x CuO _4 , ArXiv: 2503.13600 https://arxiv.org/abs/2503.13600

    arXiv

  111. [121]

    Schmalian, D

    J. Schmalian, D. Pines, and B. Stojkovic, Weak Pseudogap Behavior in the Underdoped Cuprate Superconductors, Phys. Rev. Lett. 80, 3839 (1998) https://doi.org/10.1103/PhysRevLett.80.3839; Microscopic theory of weak pseudogap behavior in the underdoped cuprate superconductors: G...

    1998 doi

  112. [122]

    Onufrieva, P

    F. Onufrieva, P. Pfeuty, and M. Kiselev, New Scenario for High- T_c Cuprates: Electronic Topological Transition as a Motor for Anomalies in the Underdoped Regime, Phys. Rev. Lett. 82, 2370 (1999) https://doi.org/10.1103/PhysRevLett.82.2370; F. Onufrieva and P. Pfeuty, Normal S...

    1999 doi

  113. [123]

    Brezin and J

    E. Brezin and J. Zinn-Justin, Renormalization of the Nonlinear Model in 2+ Dimensions -- Application to the Heisenberg Ferromagnets, Phys. Rev. Lett. 13, 691 (1976) https://doi.org/10.1103/PhysRevLett.36.691; Spontaneous breakdown of continuous symmetries near two dimensions, ...

    1976 doi

  114. [124]

    J. R. Schrieffer, X. G. Wen, and S. C. Zhang, Dynamic spin fluctuations and the bag mechanism of high- T_c superconductivity, https://dx.doi.org/10.1103/PhysRevB.39.11663 Phys. Rev. B 39 , 11663 (1989)

    1989 doi

  115. [125]

    A. V. Chubukov and D. M. Frenkel, Renormalized perturbation theory of magnetic instabilities in the two-dimensional Hubbard model at small doping, https://dx.doi.org/10.1103/PhysRevB.46.11884 Phys. Rev. B 46 11884 (1992)

    1992 doi

  116. [126]

    Dzierzawa, Hartree-Fock theory of spiral magnetic order in the 2-d Hubbard model, https://dx.doi.org/10.1007/BF01323546 Z

    M. Dzierzawa, Hartree-Fock theory of spiral magnetic order in the 2-d Hubbard model, https://dx.doi.org/10.1007/BF01323546 Z. Phys. B 86 , 49 (1992)

    1992 doi

  117. [127]

    A. P. Kampf and W. Brenig, Charge dynamics and spin order in doped Hubbard models, https://dx.doi.org/10.1007/BF00754949 J. Low Temp. Phys. 95 , 335 (1994) ; W. Brenig, Spiral magnetism and collective excitations in doped Hubbard models, https://dx.doi.org/10.1007/BF00752301 i...

    1994 doi

  118. [128]

    C\^ot\'e and A

    R. C\^ot\'e and A. M. S. Tremblay, Spiral Magnets as Gapless Mott Insulators, https://dx.doi.org/10.1209/0295-5075/29/1/007 Europhys. Lett. 29 , 37 (1995)

    1995 doi

  119. [129]

    A. V. Chubukov and K. A. Musaelian, Magnetic phases of the two-dimensional Hubbard model at low doping, 10.1103/PhysRevB.51.12605 Phys. Rev. B 51 , 12605 (1995)

    1995 doi

  120. [130]

    P. A. Igoshev, M. A. Timirgazin, A. A. Katanin, A. K. Arzhnikov, and V. Yu. Irkhin, Incommensurate magnetic order and phase separation in the two-dimensional Hubbard model with nearest- and next-nearest-neighbor hopping, 10.1103/PhysRevB.81.094407 Phys. Rev. B 81 , 094407 (2010)

    2010 doi

  121. [131]

    R. K. Kaul, Y. B. Kim, S. Sachdev, and T. Senthil, Algebraic charge liquids, https://doi.org/10.1038/nphys790 Nature Physics 4 , 28 (2008)

    2008 doi

Pith tools

Reviewed July 12, 2026 · model on record in the stance chip above.