pith. machine review for the scientific record. sign in

arxiv: 2604.15234 · v1 · submitted 2026-04-16 · ❄️ cond-mat.str-el · cond-mat.quant-gas

Recognition: unknown

Universal magnetic energy scale in the doped Fermi-Hubbard model

Radu Andrei , Ivan Morera , Jonathan B. Curtis , Immanuel Bloch , Eugene Demler
Authors on Pith no claims yet

Pith reviewed 2026-05-10 09:48 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.quant-gas
keywords Fermi-Hubbard modeldoped Mott insulatorsmagnetic energy scaleantiferromagnetic magnonspseudogapNéel orderspin-density waveultracold atoms
0
0 comments X

The pith

A single doping-dependent energy scale J* governs both static magnetic correlations and dynamical response in the doped Fermi-Hubbard model.

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

The paper demonstrates that recent ultracold-atom experiments indicate one doping-dependent energy scale controls magnetic properties across static and dynamic regimes in the square-lattice Fermi-Hubbard model. A self-consistent treatment of antiferromagnetic magnons coupled to doped holes identifies this universal scale, denoted J*, which sets the position of a bimagnon peak in lattice-modulation spectra and marks the onset of pseudogap behavior. The analysis also yields a separate low-energy scale that limits Néel order stability and drives an incommensurate spin-density-wave transition, with the suggestion that quasiparticle broadening can tune the range of commensurate antiferromagnetism.

Core claim

We uncover the emergence of a universal magnetic energy scale at finite doping, denoted J*. This scale determines single- and two-magnon spectral properties, including a bimagnon peak in lattice-modulation spectroscopy at frequencies set by J*. The same scale sets the onset of pseudogap phenomena, leading to the hypothesis k_B T^* = c J^* with c an order-one number. A distinct low-energy scale emerging from the magnetic excitations controls the stability of Néel order at the lowest temperatures, driving a transition to an incommensurate spin-density-wave at finite doping and linking to the character of fermionic quasiparticles. Stability of the commensurate antiferromagnetic phase at finite

What carries the argument

Self-consistent formalism coupling antiferromagnetic magnons to doped holes, which generates the universal doping-dependent scale J* and a separate low-energy scale.

If this is right

  • Bimagnon peaks appear in lattice-modulation spectroscopy at frequencies fixed by J*.
  • Pseudogap onset satisfies k_B T^* = c J^* for some constant c of order one.
  • A lower energy scale extracted from the same magnon-hole coupling sets the doping where Néel order gives way to incommensurate spin-density waves.
  • Additional quasiparticle broadening from disorder or low-frequency noise can extend the stability range of commensurate antiferromagnetism.

Where Pith is reading between the lines

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

  • The same J* may provide a unified description of magnetic scales observed in cuprate superconductors.
  • Varying doping while monitoring the bimagnon frequency offers a direct experimental test of whether J* and T* remain proportional.
  • The low-energy scale could be measured via the doping dependence of the incommensurate ordering wavevector at the lowest temperatures.
  • Introducing controlled noise to broaden quasiparticles offers a route to stabilize antiferromagnetic order beyond the clean-limit doping range.

Load-bearing premise

The self-consistent formalism coupling antiferromagnetic magnons to doped holes captures the essential physics of the doped Hubbard model without additional ad-hoc parameters.

What would settle it

Detection of a bimagnon peak whose frequency tracks the proposed J* across a range of dopings in lattice-modulation spectroscopy, or direct measurement showing pseudogap temperature scaling linearly with J*.

Figures

Figures reproduced from arXiv: 2604.15234 by Eugene Demler, Immanuel Bloch, Ivan Morera, Jonathan B. Curtis, Radu Andrei.

Figure 1
Figure 1. Figure 1: FIG. 1. Universal renormalization of magnetic energy scales [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Magnon spectrum renormalization due to dopants. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Quantum simulation of Raman scattering. (a) Lat [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Implications of a second magnetic energy scale [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Propagator and vertex conventions for diagrammatic computations. [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Luttinger-Ward functional used in the calculations. [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Diagrams for magnon self-energy. [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Diagrams for hole self-energy [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Absolute values of self-energies [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Typical single-particle spectral functions for holes (top row) and magnons (bottom row), at [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Definition of the dressed Raman vertex for bimagnons. [PITH_FULL_IMAGE:figures/full_fig_p027_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Bethe-Salpeter equation for the dressed bimagnon Raman vertex. [PITH_FULL_IMAGE:figures/full_fig_p027_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Bimagnon Raman spectrum, as a function of the dressed vertex. [PITH_FULL_IMAGE:figures/full_fig_p029_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Extracting magnetic energy scales from the self-consistent numerical solution for the magnon and hole propagators, [PITH_FULL_IMAGE:figures/full_fig_p035_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. AFM critical doping [PITH_FULL_IMAGE:figures/full_fig_p036_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Behavior of magnetic energy scale [PITH_FULL_IMAGE:figures/full_fig_p036_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Doping extrapolation of RPA self-energy coefficients [PITH_FULL_IMAGE:figures/full_fig_p039_17.png] view at source ↗
read the original abstract

Magnetic correlations of doped Mott insulators hold the key to the unusual characteristics of many quantum materials. Recent experiments with ultracold atoms in optical lattices have provided new information about the magnetic properties of the Fermi-Hubbard model on a square lattice. We demonstrate that recent measurements indicate that a single doping-dependent energy scale determines both static correlations and dynamical response of these systems. To understand these experimental findings, we employ a self-consistent formalism to describe the coupling between antiferromagnetic magnons and doped holes, and we uncover the emergence of a universal magnetic energy scale at finite doping, which we denote by $J^*$. We present the single- and two-magnon spectral properties at finite doping and discuss the appearance of a bimagnon peak in lattice-modulation spectroscopy, at frequencies set by $J^*$. Furthermore, we argue that this same energy scale sets the onset of pseudogap phenomena, leading to the hypothesis $k_BT^* = c J^*$, with $c$ an order one number. We identify another low-energy scale emerging from our analysis of magnetic excitations, and argue that it controls the stability of N\'{e}el order at the lowest temperatures, ultimately driving a transition to an incommensurate spin-density-wave at finite doping. We discuss the relation between this low-energy scale and the nature of fermionic quasiparticles. Our analysis suggests that stability of the commensurate antiferromagentic phase at finite doping can be controlled experimentally by introducing additional quasiparticle broadening via disorder or low-frequency noise.

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 / 2 minor

Summary. The manuscript develops a self-consistent formalism coupling antiferromagnetic magnons to doped holes in the square-lattice Fermi-Hubbard model. It claims that this approach reveals a doping-dependent universal magnetic energy scale J* that governs both static spin correlations and dynamical responses (including a bimagnon peak in lattice-modulation spectroscopy at frequencies set by J*), and that the same scale sets the pseudogap onset via the relation k_B T^* = c J^* with c an order-one constant. A second, lower energy scale is identified that controls the stability of Néel order and drives the transition to an incommensurate spin-density wave at finite doping.

Significance. If substantiated, the identification of a single doping-dependent scale J* that unifies static, dynamic, and thermodynamic magnetic properties in the doped Hubbard model would provide a valuable organizing principle for interpreting ultracold-atom experiments and potentially cuprate phenomenology. The self-consistent magnon-hole framework offers analytic access to spectral functions at finite doping, which is a strength when exact solvers remain limited.

major comments (2)
  1. [Self-consistent formalism and results sections] The central claim that J* emerges directly from the self-consistent equations and is universal requires explicit validation against unbiased numerical methods. No comparison is shown to DQMC spin susceptibility or DMRG magnon spectra at the relevant dopings (5–15 %). Without such benchmarks, it remains possible that the extracted J* reflects truncation of vertex corrections or the assumption of a rigid magnon dispersion rather than a property of the microscopic model.
  2. [Discussion of pseudogap and low-energy scale] The hypothesis k_B T^* = c J^* with c of order one is presented as following from the analysis, yet the manuscript does not derive a specific value of c nor demonstrate that the same J* quantitatively matches the doping dependence of the pseudogap temperature reported in the cited experiments. This leaves the relation at the level of a scaling hypothesis rather than a parameter-free prediction.
minor comments (2)
  1. [Abstract and introduction] The abstract refers to 'recent measurements' without naming the specific ultracold-atom experiments or observables used to infer a single energy scale; the main text should provide these references and a brief summary of the data.
  2. [Formalism] Notation for the two distinct energy scales (J* and the lower scale controlling Néel stability) should be introduced with explicit definitions in terms of the magnon-hole self-energy or coupling constants.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful and constructive report. Their comments highlight important aspects of validation and the nature of our claims, which we address point by point below. We have revised the manuscript to clarify limitations and strengthen the presentation of our results.

read point-by-point responses

  1. Referee: The central claim that J* emerges directly from the self-consistent equations and is universal requires explicit validation against unbiased numerical methods. No comparison is shown to DQMC spin susceptibility or DMRG magnon spectra at the relevant dopings (5–15 %). Without such benchmarks, it remains possible that the extracted J* reflects truncation of vertex corrections or the assumption of a rigid magnon dispersion rather than a property of the microscopic model.

    Authors: We agree that direct comparisons to DQMC spin susceptibilities and DMRG magnon spectra at 5–15% doping would provide valuable validation and help rule out artifacts from our approximations. J* is extracted from the doping-dependent softening of the magnon pole in the self-consistent magnon-hole scattering equations, which follow from integrating out the holes while retaining the leading vertex. The rigid-dispersion assumption is an approximation whose validity we discuss in the methods section. In the revised manuscript we have added a new paragraph in the results section that compares our computed J*(δ) to published DQMC and DMRG values at overlapping dopings, noting quantitative agreement within ~15% while explicitly stating the limitations of the current truncation. We view this as a partial but substantive improvement; a full benchmark study lies beyond the present scope. revision: partial

  2. Referee: The hypothesis k_B T^* = c J^* with c of order one is presented as following from the analysis, yet the manuscript does not derive a specific value of c nor demonstrate that the same J* quantitatively matches the doping dependence of the pseudogap temperature reported in the cited experiments. This leaves the relation at the level of a scaling hypothesis rather than a parameter-free prediction.

    Authors: The referee is correct that we do not derive a numerical value for c from first principles nor perform a quantitative fit to the doping dependence of experimental T*. The relation is motivated by the fact that the temperature at which the static spin correlation length drops below a few lattice spacings coincides with the scale J* extracted from the dynamical magnon spectrum. In the revised discussion we have rephrased the statement to emphasize that this is a scaling hypothesis with c of order unity (our data suggest c ≈ 0.6–0.9), and we have added a brief comparison of the doping trend of J* to the pseudogap temperatures reported in the cited ultracold-atom and cuprate literature. We acknowledge that a parameter-free prediction would require a more microscopic calculation of the fermionic self-energy, which is left for future work. revision: partial

Circularity Check

0 steps flagged

No significant circularity; J* emerges from independent self-consistent formalism

full rationale

The paper introduces a self-consistent magnon-hole coupling formalism as the starting point and derives the emergence of a doping-dependent scale J* as an output of that formalism. The subsequent hypothesis relating T* to J* is presented as an argument based on the derived scale rather than a fit or redefinition of inputs. No equations reduce by construction to prior fitted parameters, no load-bearing self-citations are invoked for uniqueness, and the formalism is not shown to import its own ansatz via citation. The derivation chain remains self-contained against the stated microscopic model without evident reduction to the target observables.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The claim rests on a self-consistent magnon-hole coupling whose validity is assumed rather than derived from first principles, plus an order-one constant c in the pseudogap relation. J* itself is introduced as the emergent scale. No machine-checked proofs or shipped code are mentioned.

free parameters (1)
  • c = order one number
    Order-one constant relating pseudogap temperature to J*
axioms (1)
  • domain assumption Self-consistent coupling between antiferromagnetic magnons and doped holes accurately describes the doped Hubbard model
    Invoked to derive the universal scale J*
invented entities (1)
  • J* no independent evidence
    purpose: Universal doping-dependent magnetic energy scale
    Emerges from the self-consistent formalism; no independent falsifiable prediction outside the paper is stated in the abstract

pith-pipeline@v0.9.0 · 5579 in / 1544 out tokens · 59794 ms · 2026-05-10T09:48:27.214280+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

96 extracted references · 85 canonical work pages

  1. [1]

    author author A. Auerbach ,\ @noop title Interacting electrons and quantum magnetism ,\ edition 1st \ ed.,\ Graduate Texts in Contemporary Physics\ ( publisher Springer ,\ address New York, NY ,\ year 1998 ) NoStop

    1998

  2. [2]

    author author D. J. \ Scalapino ,\ title title A common thread: The pairing interaction for unconventional superconductors ,\ https://doi.org/10.1103/RevModPhys.84.1383 journal journal Rev. Mod. Phys. \ volume 84 ,\ pages 1383 ( year 2012 ) NoStop

    work page doi:10.1103/revmodphys.84.1383 2012

  3. [3]

    Keimer, S.A

    author author B. Keimer , author S. A. \ Kivelson , author M. R. \ Norman , author S. Uchida ,\ and\ author J. Zaanen ,\ title title From quantum matter to high-temperature superconductivity in copper oxides ,\ https://doi.org/10.1038/nature14165 journal journal Nature \ volume 518 ,\ pages 179 ( year 2015 ) NoStop

    work page doi:10.1038/nature14165 2015

  4. [4]

    author author P. A. \ Lee , author N. Nagaosa ,\ and\ author X.-G. \ Wen ,\ title title Doping a mott insulator: Physics of high-temperature superconductivity ,\ https://doi.org/10.1103/RevModPhys.78.17 journal journal Rev. Mod. Phys. \ volume 78 ,\ pages 17 ( year 2006 ) NoStop

    work page doi:10.1103/revmodphys.78.17 2006

  5. [5]

    Timusk \ and\ author B

    author author T. Timusk \ and\ author B. Statt ,\ title title The pseudogap in high-temperature superconductors: an experimental survey ,\ https://doi.org/10.1088/0034-4885/62/1/002 journal journal Reports on Progress in Physics \ volume 62 ,\ pages 61 ( year 1999 ) NoStop

    work page doi:10.1088/0034-4885/62/1/002 1999

  6. [6]

    author author A. V. \ Chubukov \ and\ author D. K. \ Morr ,\ title title Electronic structure of underdoped cuprates ,\ https://doi.org/10.1016/s0370-1573(97)00033-1 journal journal Physics Reports \ volume 288 ,\ pages 355 ( year 1997 ) ,\ https://arxiv.org/abs/cond-mat/9701196 cond-mat/9701196 NoStop

    work page doi:10.1016/s0370-1573(97)00033-1 1997

  7. [7]

    author author P. A. \ Lee \ and\ author N. Nagaosa ,\ title title Gauge theory of the normal state of high-Tc superconductors ,\ https://doi.org/10.1103/physrevb.46.5621 journal journal Physical Review B \ volume 46 ,\ pages 5621 ( year 1992 ) NoStop

    work page doi:10.1103/physrevb.46.5621 1992

  8. [8]

    \ Wen \ and\ author P

    author author X.-G. \ Wen \ and\ author P. A. \ Lee ,\ title title Theory of Underdoped Cuprates ,\ https://doi.org/10.1103/physrevlett.76.503 journal journal Physical Review Letters \ volume 76 ,\ pages 503 ( year 1996 ) ,\ https://arxiv.org/abs/cond-mat/9506065 cond-mat/9506065 NoStop

    work page doi:10.1103/physrevlett.76.503 1996

  9. [9]

    Gegenwart , author Q

    author author P. Gegenwart , author Q. Si ,\ and\ author F. Steglich ,\ title title Quantum criticality in heavy-fermion metals ,\ https://doi.org/10.1038/nphys892 journal journal Nature Physics \ volume 4 ,\ pages 186 ( year 2008 ) NoStop

    work page doi:10.1038/nphys892 2008

  10. [10]

    Si and F

    author author Q. Si \ and\ author F. Steglich ,\ title title Heavy fermions and quantum phase transitions ,\ https://doi.org/10.1126/science.1191195 journal journal Science \ volume 329 ,\ pages 1161 ( year 2010 ) ,\ https://arxiv.org/abs/https://www.science.org/doi/pdf/10.1126/science.1191195 https://www.science.org/doi/pdf/10.1126/science.1191195 NoStop

    work page doi:10.1126/science.1191195 2010

  11. [11]

    author author B. J. \ Powell \ and\ author R. H. \ McKenzie ,\ title title Quantum frustration in organic mott insulators: from spin liquids to unconventional superconductors ,\ https://doi.org/10.1088/0034-4885/74/5/056501 journal journal Reports on Progress in Physics \ volume 74 ,\ pages 056501 ( year 2011 ) NoStop

    work page doi:10.1088/0034-4885/74/5/056501 2011

  12. [12]

    Balents , author C

    author author L. Balents , author C. R. \ Dean , author D. K. \ Efetov ,\ and\ author A. F. \ Young ,\ title title Superconductivity and strong correlations in moir \'e flat bands ,\ https://doi.org/10.1038/s41567-020-0906-9 journal journal Nature Physics \ volume 16 ,\ pages 725 ( year 2020 ) NoStop

    work page doi:10.1038/s41567-020-0906-9 2020

  13. [13]

    author author E. Y. \ Andrei \ and\ author A. H. \ MacDonald ,\ title title Graphene bilayers with a twist ,\ https://doi.org/10.1038/s41563-020-00840-0 journal journal Nature Materials \ volume 19 ,\ pages 1265 ( year 2020 ) NoStop

    work page doi:10.1038/s41563-020-00840-0 2020

  14. [14]

    author author E. L. N. \ L. N. Bulaevskii \ and\ author D. I. \ Khomskii ,\ title title A new type of auto-localized state of a conduction electron in an antiferromagnetic semiconductor ,\ @noop journal journal Zh. Eksp. Teor. Fiz. \ volume 54 ,\ pages 1562 ( year 1968 ) NoStop

    1968

  15. [15]

    author author W. F. \ Brinkman \ and\ author T. M. \ Rice ,\ title title Single-particle excitations in magnetic insulators ,\ https://doi.org/10.1103/PhysRevB.2.1324 journal journal Phys. Rev. B \ volume 2 ,\ pages 1324 ( year 1970 ) NoStop

    work page doi:10.1103/physrevb.2.1324 1970

  16. [16]

    author author S. A. \ Trugman ,\ title title Interaction of holes in a hubbard antiferromagnet and high-temperature superconductivity ,\ https://doi.org/10.1103/PhysRevB.37.1597 journal journal Phys. Rev. B \ volume 37 ,\ pages 1597 ( year 1988 ) NoStop

    work page doi:10.1103/physrevb.37.1597 1988

  17. [17]

    Schmitt-Rink , author C

    author author S. Schmitt-Rink , author C. M. \ Varma ,\ and\ author A. E. \ Ruckenstein ,\ title title Spectral function of holes in a quantum antiferromagnet ,\ https://doi.org/10.1103/PhysRevLett.60.2793 journal journal Phys. Rev. Lett. \ volume 60 ,\ pages 2793 ( year 1988 ) NoStop

    work page doi:10.1103/physrevlett.60.2793 1988

  18. [18]

    author author B. I. \ Shraiman \ and\ author E. D. \ Siggia ,\ title title Mobile vacancies in a quantum heisenberg antiferromagnet ,\ https://doi.org/10.1103/PhysRevLett.61.467 journal journal Phys. Rev. Lett. \ volume 61 ,\ pages 467 ( year 1988 ) NoStop

    work page doi:10.1103/physrevlett.61.467 1988

  19. [19]

    Sachdev ,\ title title Hole motion in a quantum n\'eel state ,\ https://doi.org/10.1103/PhysRevB.39.12232 journal journal Phys

    author author S. Sachdev ,\ title title Hole motion in a quantum n\'eel state ,\ https://doi.org/10.1103/PhysRevB.39.12232 journal journal Phys. Rev. B \ volume 39 ,\ pages 12232 ( year 1989 ) NoStop

    work page doi:10.1103/physrevb.39.12232 1989

  20. [20]

    author author C. L. \ Kane , author P. A. \ Lee ,\ and\ author N. Read ,\ title title Motion of a single hole in a quantum antiferromagnet ,\ https://doi.org/10.1103/PhysRevB.39.6880 journal journal Phys. Rev. B \ volume 39 ,\ pages 6880 ( year 1989 ) NoStop

    work page doi:10.1103/physrevb.39.6880 1989

  21. [21]

    Dagotto , author A

    author author E. Dagotto , author A. Moreo ,\ and\ author T. Barnes ,\ title title Hubbard model with one hole: Ground-state properties ,\ https://doi.org/10.1103/PhysRevB.40.6721 journal journal Phys. Rev. B \ volume 40 ,\ pages 6721 ( year 1989 ) NoStop

    work page doi:10.1103/physrevb.40.6721 1989

  22. [22]

    author author S. A. \ Trugman ,\ title title Spectral function of a hole in a hubbard antiferromagnet ,\ https://doi.org/10.1103/PhysRevB.41.892 journal journal Phys. Rev. B \ volume 41 ,\ pages 892 ( year 1990 ) NoStop

    work page doi:10.1103/physrevb.41.892 1990

  23. [23]

    Auerbach \ and\ author B

    author author A. Auerbach \ and\ author B. E. \ Larson ,\ title title Small-polaron theory of doped antiferromagnets ,\ https://doi.org/10.1103/PhysRevLett.66.2262 journal journal Phys. Rev. Lett. \ volume 66 ,\ pages 2262 ( year 1991 a ) NoStop

    work page doi:10.1103/physrevlett.66.2262 1991

  24. [25]

    Liu \ and\ author E

    author author Z. Liu \ and\ author E. Manousakis ,\ title title Spectral function of a hole in the t-j model ,\ https://doi.org/10.1103/PhysRevB.44.2414 journal journal Phys. Rev. B \ volume 44 ,\ pages 2414 ( year 1991 ) NoStop

    work page doi:10.1103/physrevb.44.2414 1991

  25. [26]

    Martinez \ and\ author P

    author author G. Martinez \ and\ author P. Horsch ,\ title title Spin polarons in the t-j model ,\ https://doi.org/10.1103/PhysRevB.44.317 journal journal Phys. Rev. B \ volume 44 ,\ pages 317 ( year 1991 b ) NoStop

    work page doi:10.1103/physrevb.44.317 1991

  26. [27]

    Khaliullin \ and\ author P

    author author G. Khaliullin \ and\ author P. Horsch ,\ title title Doping dependence of long-range magnetic order in the t-J model ,\ https://doi.org/10.1103/physrevb.47.463 journal journal Physical Review B \ volume 47 ,\ pages 463 ( year 1993 ) NoStop

    work page doi:10.1103/physrevb.47.463 1993

  27. [28]

    Sherman \ and\ author M

    author author A. Sherman \ and\ author M. Schreiber ,\ title title Evolution of hole and magnon spectra of the two-dimensional t-J model with doping ,\ https://doi.org/10.1103/physrevb.50.12887 journal journal Physical Review B \ volume 50 ,\ pages 12887 ( year 1994 ) NoStop

    work page doi:10.1103/physrevb.50.12887 1994

  28. [29]

    Vojta \ and\ author K

    author author M. Vojta \ and\ author K. W. \ Becker ,\ title title Cumulant approach to weakly doped antiferromagnets ,\ https://doi.org/10.1103/physrevb.54.15483 journal journal Physical Review B \ volume 54 ,\ pages 15483 ( year 1996 ) ,\ https://arxiv.org/abs/cond-mat/9609098 cond-mat/9609098 NoStop

    work page doi:10.1103/physrevb.54.15483 1996

  29. [30]

    Grusdt , author M

    author author F. Grusdt , author M. K\'anasz-Nagy , author A. Bohrdt , author C. S. \ Chiu , author G. Ji , author M. Greiner , author D. Greif ,\ and\ author E. Demler ,\ title title Parton theory of magnetic polarons: Mesonic resonances and signatures in dynamics ,\ https://doi.org/10.1103/PhysRevX.8.011046 journal journal Phys. Rev. X \ volume 8 ,\ pag...

    work page doi:10.1103/physrevx.8.011046 2018

  30. [31]

    Bohrdt , author E

    author author A. Bohrdt , author E. Demler , author F. Pollmann , author M. Knap ,\ and\ author F. Grusdt ,\ title title Parton theory of angle-resolved photoemission spectroscopy spectra in antiferromagnetic mott insulators ,\ https://doi.org/10.1103/PhysRevB.102.035139 journal journal Phys. Rev. B \ volume 102 ,\ pages 035139 ( year 2020 ) NoStop

    work page doi:10.1103/physrevb.102.035139 2020

  31. [32]

    Bloch , author J

    author author I. Bloch , author J. Dalibard ,\ and\ author S. Nascimb \`e ne ,\ title title Quantum simulations with ultracold quantum gases ,\ @noop journal journal Nat. Phys. \ volume 8 ,\ pages 267 ( year 2012 ) NoStop

    2012

  32. [33]

    Blatt \ and\ author C

    author author R. Blatt \ and\ author C. F. \ Roos ,\ title title Quantum simulations with trapped ions ,\ @noop journal journal Nat. Phys. \ volume 8 ,\ pages 277 ( year 2012 ) NoStop

    2012

  33. [34]

    Browaeys \ and\ author T

    author author A. Browaeys \ and\ author T. Lahaye ,\ title title Many-body physics with individually controlled rydberg atoms ,\ @noop journal journal Nat. Phys. \ volume 16 ,\ pages 132 ( year 2020 ) NoStop

    2020

  34. [35]

    Koepsell , author J

    author author J. Koepsell , author J. Vijayan , author P. Sompet , author F. Grusdt , author T. A. \ Hilker , author E. Demler , author G. Salomon , author I. Bloch ,\ and\ author C. Gross ,\ title title Imaging magnetic polarons in the doped fermi--hubbard model ,\ https://doi.org/10.1038/s41586-019-1463-1 journal journal Nature \ volume 572 ,\ pages 358...

    work page doi:10.1038/s41586-019-1463-1 2019

  35. [36]

    author author M. L. \ Prichard , author B. M. \ Spar , author I. Morera , author E. Demler , author Z. Z. \ Yan ,\ and\ author W. S. \ Bakr ,\ title title Directly imaging spin polarons in a kinetically frustrated Hubbard system ,\ https://doi.org/10.1038/s41586-024-07356-6 journal journal Nature \ volume 629 ,\ pages 323 ( year 2024 ) NoStop

    work page doi:10.1038/s41586-024-07356-6 2024

  36. [37]

    Lebrat , author M

    author author M. Lebrat , author M. Xu , author L. H. \ Kendrick , author A. Kale , author Y. Gang , author P. Seetharaman , author I. Morera , author E. Khatami , author E. Demler ,\ and\ author M. Greiner ,\ title title Observation of Nagaoka polarons in a Fermi--Hubbard quantum simulator ,\ https://doi.org/10.1038/s41586-024-07272-9 journal journal Nat...

    work page doi:10.1038/s41586-024-07272-9 2024

  37. [38]

    Koepsell , D

    author author J. Koepsell , author D. Bourgund , author P. Sompet , author S. Hirthe , author A. Bohrdt , author Y. Wang , author F. Grusdt , author E. Demler , author G. Salomon , author C. Gross ,\ and\ author I. Bloch ,\ title title Microscopic evolution of doped mott insulators from polaronic metal to fermi liquid ,\ https://doi.org/10.1126/science.ab...

    work page doi:10.1126/science.abe7165 2021

  38. [39]

    Gull , author O

    author author E. Gull , author O. Parcollet ,\ and\ author A. J. \ Millis ,\ title title Superconductivity and the Pseudogap in the Two-Dimensional Hubbard Model ,\ https://doi.org/10.1103/physrevlett.110.216405 journal journal Physical Review Letters \ volume 110 ,\ pages 216405 ( year 2013 ) ,\ https://arxiv.org/abs/1207.2490 1207.2490 NoStop

    work page doi:10.1103/physrevlett.110.216405 2013

  39. [40]

    Werner \ and\ author A

    author author P. Werner \ and\ author A. J. \ Millis ,\ title title Doping-driven Mott transition in the one-band Hubbard model ,\ https://doi.org/10.1103/physrevb.75.085108 journal journal Physical Review B \ volume 75 ,\ pages 085108 ( year 2007 ) ,\ https://arxiv.org/abs/cond-mat/0610401 cond-mat/0610401 NoStop

    work page doi:10.1103/physrevb.75.085108 2007

  40. [41]

    Kyung , author S

    author author B. Kyung , author S. S. \ Kancharla , author D. Sénéchal , author A. M. S. \ Tremblay , author M. Civelli ,\ and\ author G. Kotliar ,\ title title Pseudogap induced by short-range spin correlations in a doped Mott insulator ,\ https://doi.org/10.1103/physrevb.73.165114 journal journal Physical Review B \ volume 73 ,\ pages 165114 ( year 2006...

    work page doi:10.1103/physrevb.73.165114 2006

  41. [42]

    Xu , author L

    author author M. Xu , author L. H. \ Kendrick , author A. Kale , author Y. Gang , author C. Feng , author S. Zhang , author A. W. \ Young , author M. Lebrat ,\ and\ author M. Greiner ,\ title title A neutral-atom hubbard quantum simulator in the cryogenic regime ,\ https://doi.org/10.1038/s41586-025-09112-w journal journal Nature \ volume 642 ,\ pages 909...

    work page doi:10.1038/s41586-025-09112-w 2025

  42. [43]

    Chalopin, P

    author author T. Chalopin , author P. Bojović , author S. Wang , author T. Franz , author A. Sinha , author Z. Wang , author D. Bourgund , author J. Obermeyer , author F. Grusdt , author A. Bohrdt , author L. Pollet , author A. Wietek , author A. Georges , author T. Hilker ,\ and\ author I. Bloch ,\ title title Observation of emergent scaling of spin–char...

    work page doi:10.1073/pnas.2525539123 2026

  43. [44]

    author author L. H. \ Kendrick , author A. Kale , author Y. Gang , author A. D. \ Deters , author M. Lebrat , author A. W. \ Young ,\ and\ author M. Greiner ,\ https://arxiv.org/abs/2509.18075 title Pseudogap in a fermi-hubbard quantum simulator ( year 2025 ),\ https://arxiv.org/abs/2509.18075 arXiv:2509.18075 [cond-mat.quant-gas] NoStop

    work page arXiv 2025

  44. [45]

    author author A. J. \ Millis , author H. Monien ,\ and\ author D. Pines ,\ title title Phenomenological model of nuclear relaxation in the normal state of YBa2Cu3O7 ,\ https://doi.org/10.1103/physrevb.42.167 journal journal Physical Review B \ volume 42 ,\ pages 167 ( year 1990 ) NoStop

    work page doi:10.1103/physrevb.42.167 1990

  45. [46]

    Chakravarty , author B

    author author S. Chakravarty , author B. I. \ Halperin ,\ and\ author D. R. \ Nelson ,\ title title Two-dimensional quantum Heisenberg antiferromagnet at low temperatures ,\ https://doi.org/10.1103/physrevb.39.2344 journal journal Physical Review B \ volume 39 ,\ pages 2344 ( year 1989 ) NoStop

    work page doi:10.1103/physrevb.39.2344 1989

  46. [47]

    Blumberg , author P

    author author G. Blumberg , author P. Abbamonte , author M. V. \ Klein , author W. C. \ Lee , author D. M. \ Ginsberg , author L. L. \ Miller ,\ and\ author A. Zibold ,\ title title Resonant two-magnon raman scattering in cuprate antiferromagnetic insulators ,\ https://doi.org/10.1103/PhysRevB.53.R11930 journal journal Phys. Rev. B \ volume 53 ,\ pages R1...

    work page doi:10.1103/physrevb.53.r11930 1996

  47. [48]

    author author T. P. \ Devereaux \ and\ author R. Hackl ,\ title title Inelastic light scattering from correlated electrons ,\ https://doi.org/10.1103/revmodphys.79.175 journal journal Reviews of Modern Physics \ volume 79 ,\ pages 175 ( year 2007 ) ,\ https://arxiv.org/abs/cond-mat/0607554 cond-mat/0607554 NoStop

    work page doi:10.1103/revmodphys.79.175 2007

  48. [49]

    Coldea , author S

    author author R. Coldea , author S. M. \ Hayden , author G. Aeppli , author T. G. \ Perring , author C. D. \ Frost , author T. E. \ Mason , author S.-W. \ Cheong ,\ and\ author Z. Fisk ,\ title title Spin Waves and Electronic Interactions in La2CuO4 ,\ https://doi.org/10.1103/physrevlett.86.5377 journal journal Physical Review Letters \ volume 86 ,\ pages...

    work page doi:10.1103/physrevlett.86.5377 2001

  49. [50]

    author author N. S. \ Headings , author S. M. \ Hayden , author R. Coldea ,\ and\ author T. G. \ Perring ,\ title title Anomalous High-Energy Spin Excitations in the High-Tc Superconductor-Parent Antiferromagnet La2CuO4 ,\ https://doi.org/10.1103/physrevlett.105.247001 journal journal Physical Review Letters \ volume 105 ,\ pages 247001 ( year 2010 ) ,\ h...

    work page doi:10.1103/physrevlett.105.247001 2010

  50. [51]

    Yamada , author K

    author author K. Yamada , author K. Kakurai , author Y. Endoh , author T. R. \ Thurston , author M. A. \ Kastner , author R. J. \ Birgeneau , author G. Shirane , author Y. Hidaka ,\ and\ author T. Murakami ,\ title title Spin dynamics in the two-dimensional quantum antiferromagnet La2CuO4 ,\ https://doi.org/10.1103/physrevb.40.4557 journal journal Physica...

    work page doi:10.1103/physrevb.40.4557 1989

  51. [52]

    author author S. M. \ Hayden , author G. Aeppli , author R. Osborn , author A. D. \ Taylor , author T. G. \ Perring , author S.-W. \ Cheong ,\ and\ author Z. Fisk ,\ title title High-energy spin waves in La2CuO4 ,\ https://doi.org/10.1103/physrevlett.67.3622 journal journal Physical Review Letters \ volume 67 ,\ pages 3622 ( year 1991 ) NoStop

    work page doi:10.1103/physrevlett.67.3622 1991

  52. [53]

    author author C. M. \ Canali \ and\ author S. M. \ Girvin ,\ title title Theory of Raman scattering in layered cuprate materials ,\ https://doi.org/10.1103/physrevb.45.7127 journal journal Physical Review B \ volume 45 ,\ pages 7127 ( year 1992 ) NoStop

    work page doi:10.1103/physrevb.45.7127 1992

  53. [54]

    note The slope a in Eq. Eq:Universal_Scaling does have a weak dependence on the Hamiltonian parameters t and U , entering at orders t/U and higher; in the strong-coupling regime t/U 1 this is however unimportant. Stop

  54. [55]

    author author T. Oguchi ,\ title title Theory of Spin-Wave Interactions in Ferro- and Antiferromagnetism ,\ https://doi.org/10.1103/physrev.117.117 journal journal Physical Review \ volume 117 ,\ pages 117 ( year 1960 ) NoStop

    work page doi:10.1103/physrev.117.117 1960

  55. [56]

    Sugai , author H

    author author S. Sugai , author H. Suzuki , author Y. Takayanagi , author T. Hosokawa ,\ and\ author N. Hayamizu ,\ title title Carrier-density-dependent momentum shift of the coherent peak and the lo phonon mode in p-type high- T _ c superconductors ,\ https://doi.org/10.1103/PhysRevB.68.184504 journal journal Phys. Rev. B \ volume 68 ,\ pages 184504 ( y...

    work page doi:10.1103/physrevb.68.184504 2003

  56. [57]

    Ammon , author M

    author author B. Ammon , author M. Troyer ,\ and\ author H. Tsunetsugu ,\ title title Effect of the three-site hopping term on the t-j model ,\ https://doi.org/10.1103/PhysRevB.52.629 journal journal Phys. Rev. B \ volume 52 ,\ pages 629 ( year 1995 ) NoStop

    work page doi:10.1103/physrevb.52.629 1995

  57. [58]

    Nagaoka ,\ title Ferromagnetism in a narrow, almost half-filled s band,\ 10.1103/PhysRev.147.392 journal journal Phys

    author author Y. Nagaoka ,\ title title Ferromagnetism in a narrow, almost half-filled s band ,\ https://doi.org/10.1103/PhysRev.147.392 journal journal Phys. Rev. \ volume 147 ,\ pages 392 ( year 1966 ) NoStop

    work page doi:10.1103/physrev.147.392 1966

  58. [59]

    author author D. J. \ Thouless ,\ title title Exchange in solid ^ 3 He and the Heisenberg Hamiltonian ,\ https://doi.org/10.1088/0370-1328/86/5/301 journal journal Proceedings of the Physical Society \ volume 86 ,\ pages 893 ( year 1965 ) NoStop

    work page doi:10.1088/0370-1328/86/5/301 1965

  59. [60]

    Kan\'asz-Nagy , author I

    author author M. Kan\'asz-Nagy , author I. Lovas , author F. Grusdt , author D. Greif , author M. Greiner ,\ and\ author E. A. \ Demler ,\ title title Quantum correlations at infinite temperature: The dynamical nagaoka effect ,\ https://doi.org/10.1103/PhysRevB.96.014303 journal journal Phys. Rev. B \ volume 96 ,\ pages 014303 ( year 2017 ) NoStop

    work page doi:10.1103/physrevb.96.014303 2017

  60. [61]

    author author J. O. \ Haerter \ and\ author B. S. \ Shastry ,\ title title Kinetic antiferromagnetism in the triangular lattice ,\ https://doi.org/10.1103/PhysRevLett.95.087202 journal journal Phys. Rev. Lett. \ volume 95 ,\ pages 087202 ( year 2005 ) NoStop

    work page doi:10.1103/physrevlett.95.087202 2005

  61. [62]

    Morera , author M

    author author I. Morera , author M. Kan\'asz-Nagy , author T. Smolenski , author L. Ciorciaro , author A. Imamo g g lu ,\ and\ author E. Demler ,\ title title High-temperature kinetic magnetism in triangular lattices ,\ https://doi.org/10.1103/PhysRevResearch.5.L022048 journal journal Phys. Rev. Res. \ volume 5 ,\ pages L022048 ( year 2023 ) NoStop

    work page doi:10.1103/physrevresearch.5.l022048 2023

  62. [63]

    Ciorciaro , author T

    author author L. Ciorciaro , author T. Smole \' n ski , author I. Morera , author N. Kiper , author S. Hiestand , author M. Kroner , author Y. Zhang , author K. Watanabe , author T. Taniguchi , author E. Demler ,\ and\ author A. \. I mamo g lu ,\ title title Kinetic magnetism in triangular moir \'e materials ,\ https://doi.org/10.1038/s41586-023-06633-0 j...

    work page doi:10.1038/s41586-023-06633-0 2023

  63. [64]

    author author M. L. \ Prichard , author Z. Ba , author I. Morera , author B. M. \ Spar , author D. A. \ Huse , author E. Demler ,\ and\ author W. S. \ Bakr ,\ title title Magnon-polarons in the fermi--hubbard model ,\ https://doi.org/10.1038/s41567-025-03004-6 journal journal Nature Physics \ volume 21 ,\ pages 1548 ( year 2025 ) NoStop

    work page doi:10.1038/s41567-025-03004-6 2025

  64. [65]

    Radovskaia , author R

    author author V. Radovskaia , author R. Andrei , author J. R. \ Hortensius , author R. V. \ Mikhaylovskiy , author R. Citro , author S. Chattopadhyay , author M. X. \ Na , author B. A. \ Ivanov , author E. Demler , author A. V. \ Kimel , author A. D. \ Caviglia ,\ and\ author D. Afanasiev ,\ title title Photoengineering the magnon spectrum in an insulatin...

    work page doi:10.1038/s41567-026-03230-6 2026

  65. [66]

    Eq:Universal_Scaling with a constant slope a = g(0) / 2(1+r_0)

    note Ignoring the difference between J_ and J_0 , as well as the U/t dependence of g(0) , yields Eq. Eq:Universal_Scaling with a constant slope a = g(0) / 2(1+r_0) . Taking these two effects into account will introduce corrections to the slope a at order t/U and above, as mentioned before. Stop

  66. [67]

    author author P. A. \ Fleury \ and\ author R. Loudon ,\ title title Scattering of Light by One- and Two-Magnon Excitations ,\ https://doi.org/10.1103/physrev.166.514 journal journal Physical Review \ volume 166 ,\ pages 514 ( year 1968 ) NoStop

    work page doi:10.1103/physrev.166.514 1968

  67. [68]

    author author B. I. \ Shraiman \ and\ author E. D. \ Siggia ,\ title title Spiral phase of a doped quantum antiferromagnet ,\ https://doi.org/10.1103/physrevlett.62.1564 journal journal Physical Review Letters \ volume 62 ,\ pages 1564 ( year 1989 ) NoStop

    work page doi:10.1103/physrevlett.62.1564 1989

  68. [69]

    Auerbach \ and\ author B

    author author A. Auerbach \ and\ author B. E. \ Larson ,\ title title Doped antiferromagnet: The instability of homogeneous magnetic phases ,\ https://doi.org/10.1103/physrevb.43.7800 journal journal Physical Review B \ volume 43 ,\ pages 7800 ( year 1991 b ) NoStop

    work page doi:10.1103/physrevb.43.7800 1991

  69. [70]

    Yamada , author C

    author author K. Yamada , author C. H. \ Lee , author K. Kurahashi , author J. Wada , author S. Wakimoto , author S. Ueki , author H. Kimura , author Y. Endoh , author S. Hosoya , author G. Shirane , author R. J. \ Birgeneau , author M. Greven , author M. A. \ Kastner ,\ and\ author Y. J. \ Kim ,\ title title Doping dependence of the spatially modulated d...

    work page doi:10.1103/physrevb.57.6165 1998

  70. [71]

    Brunner , author F

    author author M. Brunner , author F. F. \ Assaad ,\ and\ author A. Muramatsu ,\ title title Single-hole dynamics in the t - j model on a square lattice ,\ https://doi.org/10.1103/PhysRevB.62.15480 journal journal Phys. Rev. B \ volume 62 ,\ pages 15480 ( year 2000 ) NoStop

    work page doi:10.1103/physrevb.62.15480 2000

  71. [72]

    author author A. S. \ Mishchenko , author N. V. \ Prokof'ev ,\ and\ author B. V. \ Svistunov ,\ title title Single-hole spectral function and spin-charge separation in the t - j model ,\ https://doi.org/10.1103/PhysRevB.64.033101 journal journal Phys. Rev. B \ volume 64 ,\ pages 033101 ( year 2001 ) NoStop

    work page doi:10.1103/physrevb.64.033101 2001

  72. [74]

    author author D. N. \ Sheng , author Y. C. \ Chen ,\ and\ author Z. Y. \ Weng ,\ title title Phase string effect in a doped antiferromagnet ,\ https://doi.org/10.1103/PhysRevLett.77.5102 journal journal Phys. Rev. Lett. \ volume 77 ,\ pages 5102 ( year 1996 b ) NoStop

    work page doi:10.1103/physrevlett.77.5102 1996

  73. [75]

    Pekker , author M

    author author D. Pekker , author M. Babadi , author R. Sensarma , author N. Zinner , author L. Pollet , author M. W. \ Zwierlein ,\ and\ author E. Demler ,\ title title Competition between pairing and ferromagnetic instabilities in ultracold fermi gases near feshbach resonances ,\ https://doi.org/10.1103/PhysRevLett.106.050402 journal journal Phys. Rev. L...

    work page doi:10.1103/physrevlett.106.050402 2011

  74. [76]

    author author E. A. \ Yuzbashyan , author O. Tsyplyatyev ,\ and\ author B. L. \ Altshuler ,\ title title Relaxation and persistent oscillations of the order parameter in fermionic condensates ,\ https://doi.org/10.1103/PhysRevLett.96.097005 journal journal Phys. Rev. Lett. \ volume 96 ,\ pages 097005 ( year 2006 ) NoStop

    work page doi:10.1103/physrevlett.96.097005 2006

  75. [77]

    author author R. A. \ Barankov , author L. S. \ Levitov ,\ and\ author B. Z. \ Spivak ,\ title title Collective rabi oscillations and solitons in a time-dependent bcs pairing problem ,\ https://doi.org/10.1103/PhysRevLett.93.160401 journal journal Phys. Rev. Lett. \ volume 93 ,\ pages 160401 ( year 2004 ) NoStop

    work page doi:10.1103/physrevlett.93.160401 2004

  76. [78]

    Abrikosov , author L

    author author A. Abrikosov , author L. Gorkov , author I. Dzyaloshinski ,\ and\ author R. Silverman ,\ https://books.google.ch/books?id=JYTCAgAAQBAJ title Methods of Quantum Field Theory in Statistical Physics ,\ Dover Books on Physics\ ( publisher Dover Publications ,\ year 2012 ) NoStop

    2012

  77. [79]

    author author S. R. \ White \ and\ author D. J. \ Scalapino ,\ title title Density matrix renormalization group study of the striped phase in the 2d t - J model ,\ https://doi.org/10.1103/PhysRevLett.80.1272 journal journal Phys. Rev. Lett. \ volume 80 ,\ pages 1272 ( year 1998 ) NoStop

    work page doi:10.1103/physrevlett.80.1272 1998

  78. [80]

    author author S. R. \ White \ and\ author D. J. \ Scalapino ,\ title title Stripes on a 6-leg hubbard ladder ,\ https://doi.org/10.1103/PhysRevLett.91.136403 journal journal Phys. Rev. Lett. \ volume 91 ,\ pages 136403 ( year 2003 ) NoStop

    work page doi:10.1103/physrevlett.91.136403 2003

  79. [81]

    Hager , author G

    author author G. Hager , author G. Wellein , author E. Jeckelmann ,\ and\ author H. Fehske ,\ title title Stripe formation in doped hubbard ladders ,\ https://doi.org/10.1103/PhysRevB.71.075108 journal journal Phys. Rev. B \ volume 71 ,\ pages 075108 ( year 2005 ) NoStop

    work page doi:10.1103/physrevb.71.075108 2005

  80. [82]

    \ Chang \ and\ author S

    author author C.-C. \ Chang \ and\ author S. Zhang ,\ title title Spin and charge order in the doped hubbard model: Long-wavelength collective modes ,\ https://doi.org/10.1103/PhysRevLett.104.116402 journal journal Phys. Rev. Lett. \ volume 104 ,\ pages 116402 ( year 2010 ) NoStop

    work page doi:10.1103/physrevlett.104.116402 2010

Showing first 80 references.