pith. sign in

arxiv: 2606.02838 · v1 · pith:TS6O45ITnew · submitted 2026-06-01 · ❄️ cond-mat.str-el · cond-mat.supr-con

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

I. A. Goremykin , A. A. Katanin This is my paper

Pith reviewed 2026-06-28 12:21 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.supr-con
keywords pseudogapFermi arcsHubbard modelhigh-Tc superconductorsSU(2) gauge theorydynamic mean-field theorymagnetic fluctuations
0
0 comments X

The pith

SU(2) gauge theory with DMFT and magnetic fluctuations accounts for Fermi arcs in underdoped high-Tc superconductors through asymmetric damping.

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

The paper examines the spectral properties of the two-dimensional Hubbard model for high-Tc compounds using an SU(2) gauge theory that separates electronic degrees of freedom into spinon and chargon subsystems. It applies dynamic mean-field theory to capture magnetic long-range order in the chargon subsystem and adds spinon fluctuations. This combination is shown to be necessary for reproducing the damping asymmetry between inner and outer parts of hole pockets, which leads to Fermi arc formation in the underdoped regime at low hole doping. The approach links the chargon hole pockets to features seen in quantum oscillation experiments. A sympathetic reader would care because it provides a mechanism for the pseudogap phenomenon observed in cuprate superconductors.

Core claim

Within the SU(2) gauge theory, dynamic mean-field theory 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 correspond to those observed in quantum oscillation measurements.

What carries the argument

The SU(2) gauge symmetry allowing separation of electronic degrees of freedom into spinon and chargon subsystems, with DMFT handling chargon magnetic order and additional treatment of spinon fluctuations.

If this is right

  • The model produces Fermi arcs specifically at low hole doping due to damping differences.
  • Hole pockets from the chargon subsystem match quantum oscillation data.
  • This framework describes the pseudogap in high-Tc compounds.
  • Asymmetry in damping is key to arc formation rather than other effects.

Where Pith is reading between the lines

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

  • If the approach is correct, calculations omitting long-wavelength fluctuations should fail to produce the observed damping asymmetry.
  • Similar gauge theory methods might apply to describing other aspects of the pseudogap phase.
  • The link to quantum oscillations suggests a way to connect ARPES and quantum oscillation data in underdoped cuprates.

Load-bearing premise

The electronic degrees of freedom can be separated into spinon and chargon subsystems via SU(2) gauge symmetry, and DMFT accurately captures the magnetic long-range order in the chargon subsystem on top of which spinon fluctuations are treated.

What would settle it

A spectral function calculation without long-wavelength magnetic fluctuations that nonetheless exhibits the inner-outer damping asymmetry and Fermi arcs would show they are not essential.

Figures

Figures reproduced from arXiv: 2606.02838 by A. A. Katanin, I. A. Goremykin.

Figure 1
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. 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. Figure 3: FIG. 3. Spectral density of chargons (a,b) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The same as Fig [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
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] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 1 minor

Summary. The paper claims that the spectral properties of the 2D Hubbard model for high-Tc cuprates can be described via an SU(2) gauge theory that separates electronic degrees of freedom into spinon and chargon subsystems; DMFT is used to capture magnetic long-range order in the chargon sector, with spinon fluctuations added on top, and this combination is essential to produce the inner/outer damping asymmetry of hole pockets that yields Fermi arcs in the underdoped regime (especially at low doping). The underlying chargon hole pockets are identified with those seen in quantum-oscillation experiments.

Significance. If the central approximation holds, the work would supply a concrete mechanism linking gauge-theory separation, DMFT magnetic order, and long-wavelength fluctuations to the pseudogap and Fermi-arc phenomenology, while also connecting to quantum-oscillation data. No machine-checked proofs, reproducible code, or parameter-free derivations are reported.

major comments (2)
  1. [Abstract / central claim] The central claim that DMFT plus spinon fluctuations is 'essential' for the damping asymmetry and Fermi arcs rests on the SU(2) gauge decomposition into decoupled spinon and chargon sectors. No derivation or error estimate is supplied showing that gauge-field fluctuations and the local constraint remain negligible at low doping; this is load-bearing for the reported spectral functions.
  2. [Abstract / method description] The manuscript asserts that DMFT accurately captures chargon magnetic long-range order in two dimensions, yet provides no comparison to methods that respect the Mermin-Wagner theorem or to exact diagonalization / quantum Monte Carlo benchmarks that would test whether the reported inner/outer damping asymmetry survives when the gauge ansatz is relaxed.
minor comments (1)
  1. Notation for the spinon and chargon Green's functions is introduced without an explicit definition of the gauge-fixing procedure or the form of the constraint term.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful review and constructive feedback on our manuscript. We address each major comment below, providing clarifications on the framework and approximations used while noting where revisions can strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract / central claim] The central claim that DMFT plus spinon fluctuations is 'essential' for the damping asymmetry and Fermi arcs rests on the SU(2) gauge decomposition into decoupled spinon and chargon sectors. No derivation or error estimate is supplied showing that gauge-field fluctuations and the local constraint remain negligible at low doping; this is load-bearing for the reported spectral functions.

    Authors: The SU(2) gauge theory provides an established framework for separating spin and charge degrees of freedom in the Hubbard model, with the local constraint enforced on average via the gauge structure (as in prior literature on this approach). Within this ansatz, gauge-field fluctuations are treated at the saddle-point level, and our calculations focus on the resulting chargon spectral functions supplemented by spinon fluctuations. We agree that an explicit discussion of the validity regime and references to error estimates from related gauge-theory studies would improve clarity. We will revise the manuscript to include such a discussion in the methods and conclusions sections. revision: partial

  2. Referee: [Abstract / method description] The manuscript asserts that DMFT accurately captures chargon magnetic long-range order in two dimensions, yet provides no comparison to methods that respect the Mermin-Wagner theorem or to exact diagonalization / quantum Monte Carlo benchmarks that would test whether the reported inner/outer damping asymmetry survives when the gauge ansatz is relaxed.

    Authors: DMFT is applied to the chargon sector to capture local correlations and the tendency toward magnetic order, consistent with its standard use as an approximation in Hubbard-model studies; the added long-wavelength spinon fluctuations are intended to incorporate effects beyond strict mean-field. We acknowledge that DMFT does not enforce the Mermin-Wagner theorem in 2D and will add a clarifying note on this point with references to related discussions in the literature. Comprehensive benchmarks relaxing the full gauge ansatz against ED or QMC are not feasible within the current scope due to computational demands but represent a valuable direction for future work. revision: partial

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper applies an SU(2) gauge decomposition to separate electronic degrees of freedom into spinon and chargon subsystems, then employs DMFT to treat magnetic long-range order in the chargon sector while adding spinon fluctuations. The abstract and description present this as a calculational framework whose output (damping asymmetry and Fermi arcs) follows from the method rather than reducing to a fitted parameter or self-citation by construction. No load-bearing self-citations, self-definitional steps, or renamed known results are identifiable from the provided text; the central claim retains independent content from the chosen approximations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the SU(2) gauge symmetry separation and the validity of DMFT for the chargon magnetic state; both are domain assumptions imported from earlier literature rather than derived in the paper.

axioms (2)
  • domain assumption SU(2) gauge symmetry permits clean separation of electronic degrees of freedom into spinon and chargon subsystems
    Invoked at the outset of the abstract as the starting point for the spectral calculation.
  • domain assumption DMFT provides an adequate description of magnetic long-range order in the chargon subsystem
    Stated as the method used to treat the chargon sector before adding spinon fluctuations.

pith-pipeline@v0.9.1-grok · 5678 in / 1380 out tokens · 32559 ms · 2026-06-28T12:21:17.420078+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

129 extracted references · 113 canonical work pages · 1 internal anchor

  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

    work page 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

    work page 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

    work page 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

    work page 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

    work page 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

    work page 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

    work page doi:10.1103/physrevb.77.094516 2008

  9. [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/10.1103/PhysRevB.74.224510

    work page doi:10.1103/physrevb.74.224510 2006

  10. [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/10.1126/science.1103627

    work page doi:10.1126/science.1103627 2005

  11. [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

    work page doi:10.1038/32366 1998

  12. [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

    work page doi:10.1038/nphys334 2006

  13. [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

    work page doi:10.1038/nature05872 2007

  14. [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, 533 (2007) https://dx.doi.org/10.1038/nature06332

    work page doi:10.1038/nature06332 2007

  15. [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) https://dx.doi.org/10.1038/ncomms7034

    work page doi:10.1038/ncomms7034 2015

  16. [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

    work page doi:10.1103/physrevlett.100.047003 2008

  17. [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) https://dx.doi.org/10.1038/ncomms1468

    work page doi:10.1038/ncomms1468 2011

  18. [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

    work page doi:10.1038/nphys2792 2013

  19. [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

    work page doi:10.1038/ncomms12244 2016

  20. [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

    work page doi:10.1146/annurev-conmatphys-030212-184305 2015

  21. [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)

    work page doi:10.1103/physrevlett.107.047003 2011

  22. [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/10.1038/nature08521

    work page doi:10.1038/nature08521 2009

  23. [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

    work page doi:10.1038/nature07095 2008

  24. [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

    work page doi:10.1103/physrevb.76.220503 2007

  25. [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

    work page doi:10.1073/pnas.0804002105 2008

  26. [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)

    work page doi:10.1103/physrevb.42.167 1990

  27. [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)

    work page doi:10.1103/physrevb.50.16052 1994

  28. [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)

    work page doi:10.1103/physrevb.54.7561 1996

  29. [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)

    work page doi:10.1088/0953-8984/8/48/021 1996

  30. [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)

    work page doi:10.1103/physrevb.60.667 1999

  31. [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)

    work page doi:10.1134/1.558879 1999

  32. [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)

    work page doi:10.1103/physrevlett.71.2813 1993

  33. [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)

    work page doi:10.1103/physrevb.51.14874 1995

  34. [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

    work page doi:10.1073/pnas.1720580115 2018

  35. [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)

    work page doi:10.1038/s41524-024-01221-w 2024

  36. [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)

    work page doi:10.1038/s42005-026-02532-8 2026

  37. [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)

    work page doi:10.1063/1.2199446 2006

  38. [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)

    work page doi:10.1103/physrevlett.110.216405 2013

  39. [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)

    work page doi:10.1103/physrevlett.114.236402 2015

  40. [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 coupling in the pseudogap regime, Phys. Rev. B 112 , L041105 (2025)

    work page doi:10.1103/physrevlett.132.216501 2024

  41. [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

    work page doi:10.1103/physrevx.8.021048 2018

  42. [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)

    work page doi:10.1038/s42005-022-01117-5 2022

  43. [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)

    work page doi:10.1073/pnas.2525308123 2026

  44. [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)

    work page doi:10.1103/revmodphys.66.763 1994

  45. [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)

    work page doi:10.1103/revmodphys.70.1039 1998

  46. [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)

    work page doi:10.1103/revmodphys.78.17 2006

  47. [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)

    work page doi:10.1103/physrevb.78.014501 2008

  48. [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)

    work page doi:10.1103/physrevb.108.045123 2023

  49. [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)

    work page doi:10.1103/physrevb.102.155124 2020

  50. [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)

    work page doi:10.1103/physrevb.99.054516 2019

  51. [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)

    work page doi:10.1103/physrevb.94.115147 2016

  52. [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

    work page doi:10.1103/physrevb.80.155129 2009

  53. [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)

    work page doi:10.1103/physrevb.81.115129 2010

  54. [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)

    work page doi:10.1103/physrevb.91.115123 2015

  55. [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)

    work page doi:10.1103/physrevb.96.075103 2017

  56. [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)

    work page doi:10.1103/physrevlett.119.227002 2017

  57. [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)

    work page doi:10.1103/physrevb.105.155151 2022

  58. [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

    work page doi:10.48550/arxiv.2511.03436 2025

  59. [61]

    SU(2) gauge theory of fluctuating stripe order in the two-dimensional Hubbard model

    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

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2603.13071 2026

  60. [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)

    work page doi:10.1103/x7qr-f6lm 2025

  61. [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)

    work page doi:10.1103/zm7b-jdzf 2026

  62. [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

    work page doi:10.1103/physrevb.106.205152 2022

  63. [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)

    work page doi:10.1103/physrevb.110.085153 2024

  64. [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)

    work page doi:10.1103/w4vc-n5l6 2025

  65. [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

    work page doi:10.1103/physrevb.107.245104 2023

  66. [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)

    work page doi:10.1103/physrevb.101.165142 2020

  67. [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 on the Physics and Mathematical Physics of the Hubbard Model (Springer, New York...

    work page doi:10.1103/physrevlett.65.2462 1990

  68. [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

    work page doi:10.1103/physrevb.43.3790 1991

  69. [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 interactions on the spin-wave velocity of one-dimensional quarter-filled spin-densit...

    work page doi:10.1103/physrevb.61.13493 2000

  70. [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-dimensional half-filled Hubbard model: A nonlinear sigma model approach, Phys. Rev. ...

    work page doi:10.1103/physrevb.65.245118 2002

  71. [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)

    work page doi:10.1103/physrevlett.80.2393 1998

  72. [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)

    work page doi:10.1103/physrevb.94.245145 2016

  73. [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

    work page doi:10.1103/physrev.119.1153 1960

  74. [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/10.1103/PhysRevB.68.085113 Phys. Rev. B 68 , 085113 (2003)

    work page doi:10.1051/jp1:1996127 1996

  75. [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

    work page doi:10.48550/arxiv.2509.21034 2025

  76. [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)

    work page doi:10.1103/physrevlett.133.166501 2024

  77. [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)

    work page doi:10.21468/scipostphys.12.6.184 2022

  78. [80]

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

  79. [81]

    Cheong, G

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

  80. [82]

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

Showing first 80 references.