Pith. sign in

REVIEW 3 major objections 5 minor 43 references

Spinon Singlet in Quantum Colored String: Origin of $d$-Wave Pairing in a Partially-Filled Stripe

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read Spinon singlet is the origin of d-wave pairing in stripe phases

desk verdict A genuinely new sign-rule derivation for d-wave PPC from a spinon-singlet string picture, with real DMRG backing; the main soft spot is the truncated Hilbert space controlling the mechanism claim. read the letter →

arxiv 2412.04379 v1 pith:GRXAWZQX submitted 2024-12-05 cond-mat.str-el cond-mat.supr-con

Jia-Long Wang , Shi-Jie Hu , Xue-Feng Zhang This is my paper · ORCID
classification cond-mat.str-elcond-mat.supr-con
keywords spinonsingletquantumcoloredstringd-wavepairingstripephasepair-paircorrelationst-JmodelHubbardcupratesuperconductivity
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 claims that the microscopic mechanism of d-wave pairing in a partially-filled stripe is the formation of a spinon singlet from two spinons with opposite chiralities in a quantum colored string. This mechanism is derived from an effective model built bottom-up from the t-Jz Hamiltonian, and is verified by matching pair-pair correlation patterns across the t-J, t-J-α, and Hubbard models using large-scale DMRG calculations. If correct, it explains why d-wave pairing appears specifically where charge stripes fluctuate, and why Cooper pairs accumulate at the hole-rich stripes rather than uniformly.

What carries the argument

The central object is the two-spinon quantum colored string (QCS): a fluctuating one-dimensional string of color quasi-particles—spinons, holons, and dual-holes—embedded in a π-phase-shifted antiferromagnetic background, described by an effective Hamiltonian $H_{\rm CS}^{e}$ that combines a diagonal confinement energy proportional to $|\Gamma_z|$ with off-diagonal hopping and fluctuation terms. The load-bearing identity is the sign rule for the ground-state expansion coefficients: a two-spinon basis with chirality sequence $\chi_1\chi_2$ has ${\rm Sgn}_s = -1$ for $\Uparrow\Downarrow$ and $+1$ for $\Downarrow\Uparrow$ (up to the global sign convention), and the long-distance pair-pair correlation between two bonds carries sign ${\rm Sgn}_s\,{\rm Sgn}_{s'}\,{\rm Sgn}_\Delta$, where ${\rm Sgn}_\Delta$ accounts for spin exchange. Because the spinon singlet configurations have the lowest potential energy and dominate the wavefunction, this sign product yields positive correlations for same-oriented bonds and negative correlations between x- and y-bonds.

What would settle it

Compute the pair-pair correlation function G_{b,b'} directly from the full DMRG ground state |ψ_D⟩ for the t-Jz model on an 11×6 cylinder at J = 0.6, without projecting onto the truncated QCS space, and compare the sign between a distant y-bond and x-bond; if that sign is positive or the d-wave pattern disappears, the spinon-singlet mechanism's central claim is falsified. Alternatively, increasing the truncation bound |Γ_z| from 5 to 8 and checking whether the leading basis contributions' signs change would test whether the sign rule is a truncation artifact.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that doping a fully-filled stripe with two electrons creates a two-spinon quantum colored string whose ground state is dominated by spinon singlet configurations with chirality sequences $\chi_1\chi_2 = \Uparrow\Downarrow$ and $\Downarrow\Uparrow$. The sign of the wavefunction expansion coefficient of such a basis is determined by the chirality order, and the pair-pair correlation function between distant bonds obeys a sign product rule. This rule forces positive correlations between same-oriented bonds (y-y and x-x) and negative correlations between distant x-bonds and y-bonds—the defining signature of d-wave pairing. The same negative sign between x- and y-bonds is observed by DMRG for the t-Jz model, the t-J-α model with α from 0 to 1, and the Hubbard model, and the pattern persists for a half-filled stripe on a cylinder of circumference Ly = 8.

Load-bearing premise

The central assumption is that the truncated QCS Hilbert space (|Γ_z| ≤ 5) faithfully represents the low-energy physics, even though at J = 0.6 the DMRG wavefunction projected onto this space retains only 44.3% of its weight, with renormalized fidelity 90.8%, so the omitted components could in principle carry opposing sign correlations.

Editorial extensions

If this is right

  • The mechanism implies that d-wave pairing is generated locally along the fluctuating string, so Cooper pairs should be concentrated around the hole-rich stripe rather than distributed uniformly.
  • Enhancing antiferromagnetic xy-exchange stabilizes the spinon singlet and strengthens the d-wave pattern, as confirmed by DMRG for increasing α.
  • The same spinon-singlet sign rule is conjectured to extend to multi-stripe configurations and to a Luttinger-liquid description of the half-filled stripe.
  • The 4×4 hole-checkerboard structure seen by STM in underdoped cuprates may correspond to two holons placed above and below a spinon singlet, connecting the mechanism to local pairing observations.

Reading between the lines

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

  • If the sign rule is robust, the same mechanism should predict a sensitive dependence of d-wave pairing on chirality-breaking perturbations, such as a magnetic field or staggered flux, which could be tested in DMRG or cold-atom simulators.
  • A direct test would be to compute the pair-pair correlation directly from the full DMRG wavefunction without projection onto the truncated QCS space; if the negative x-y sign flips, the truncation is the source of the pattern.
  • The mechanism may apply to other geometries where quantum strings form, such as doped ladders or two-leg stripes, and could be probed by measuring the muon-spin-rotation or neutron-scattering signatures of local singlet formation.
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 / 5 minor

Summary. This paper constructs an effective quantum colored string model (QCSM), Eq. (2), for the t-J_z model and obtains its ground-state wavefunction by exact diagonalization in a truncated Hilbert space Ω with |Γ_z|≤5. The authors identify two-spinon configurations with opposite chiralities as a spinon singlet and derive sign rules for the pair-pair correlation function G_{b,b'} between bonds: positive between two y-bonds and between two x-bonds, and negative between an x-bond and a y-bond, which is the d-wave pairing pattern. The predicted PPC patterns are compared with DMRG results for the t-J_z, t-J-α, and Hubbard models. The paper concludes that the spinon singlet is the microscopic origin of d-wave superconductivity in a fluctuating, partially-filled stripe.

Significance. If the mechanism is correct, this work offers a concrete microscopic picture for d-wave pairing in striped superconductors, connecting stripe geometry with Cooper-pair correlations in a way that goes beyond the RVB paradigm. The paper has clear strengths: the renormalized fidelity FR≈96% at J=1 and ≈90.8% at J=0.6 demonstrates good sign control of the projected wavefunction; the qualitative reproduction of the d-wave PPC pattern across t-J_α and Hubbard models by independent DMRG is nontrivial and supportive; and the explicit identification of the contributing basis states gives a falsifiable sign rule that can be tested in future calculations. The central burden is that the sign-rule derivation is performed inside a truncated Hilbert space that at J=0.6 contains only 44.3% of the DMRG weight, so the causal claim about the spinon singlet needs additional quantitative control over the omitted component.

major comments (3)
  1. [Appendix B and 'd-wave pairing'] The central mechanism is established inside the truncated space Ω. At J=0.6, the value used throughout the main text, the DMRG weight in Ω is only W≈44.3%, with renormalized fidelity FR≈90.8%. The sign enumeration of the basis pairs in Figs. 3 and 4 is restricted to Ω, and no decomposition of the full DMRG G_{b,b'} into inside-Ω and outside-Ω contributions is given. The reported FR only controls the relative phase of the projected component; it does not control the sign of the omitted 55.7% of the DMRG wavefunction, which could in principle reverse the x-y PPC sign. Please provide a direct check, for example by computing G_{b,b'} from the DMRG wavefunction projected onto Ω and comparing with the full DMRG result, or by evaluating the contribution from the complement, or by demonstrating convergence of the sign with respect to the cutoff |Γ_z|. Without such a check, the statement that the spinon singlet is the origin of the observed d-wave pattern is not fully controlled.
  2. ['d-wave pairing'] The derivation of negative G_{b,b'} for x-bonds and y-bonds is based on a small number of representative basis pairs, with the text stating that four basis pairs contribute over 86% to G_{b,b'} for the color-shaded bonds. The paper claims that all contributing pairs can be systematically identified, but no exhaustive enumeration, weight table, or general algebraic sign rule is provided. Since G_{b,b'} is a signed sum over many basis pairs, a sign rule established for 86% of the weight of one bond pair does not by itself establish the sign for all long-distance pairs. I ask for an exhaustive classification of the contributing basis pairs for representative target bonds, including their signs and weights, or an explicit proof of the sign rule for arbitrary long-distance x-y pairs.
  3. [Conclusion and Outlook / 'Four-spinon QCS'] The abstract and conclusion make a causal claim that the spinon singlet is the origin of d-wave pairing in t-J-α and Hubbard models. The QCSM, however, is derived from the t-J_z model, and the evidence for α>0 and for the Hubbard model is only the qualitative similarity of PPC patterns; the text itself states that the effective theory has not yet fully captured the t-J model. Unless the spinon-singlet content of the actual t-J/Hubbard ground states is quantified (for example, by projecting those DMRG wavefunctions onto a colored-string basis or by measuring the weight of the two-spinon singlet component), the across-models 'origin' statement is stronger than the evidence presented. This does not weaken the t-J_z result, but it requires either additional data or a more cautious phrasing.
minor comments (5)
  1. [Conclusion and Outlook] The text contains several typos: 'emergency' should be 'emergence', 'spionon' should be 'spinon', and 'dule-hole-spinon' in the caption of Fig. 4 should be 'dual-hole-spinon'.
  2. ['d-wave pairing'] The word 'violet' in the sentence 'their interference violet the simple behavior of Sgn0' should be 'violate'.
  3. [Appendix A] The abbreviation 'PCC' is used ('the PCC between two x-bonds'), but the paper consistently uses 'PPC' for the pair-pair correlation function; please unify the terminology.
  4. [Appendix C, Table II] The labels '2/3 hole-filled stripe' and '1/2 hole-filled stripe' are confusing because the main text uses '2/3 hole-doped' and 'half-filled' for the same objects; please make the filling notation consistent throughout.
  5. ['d-wave pairing'] The notation 'Sgn Δ = ∓' is introduced without a prior definition; define explicitly what 'Sgn' of an operator or a basis state means, and state the convention used to fix the overall sign of the wavefunction.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the d-wave PPC sign is an emergent ED output validated against independent DMRG, not imposed by construction.

full rationale

The central derivation is not circular. The QCSM (Eq. 2) is taken from the authors' prior Ref. [34], which is a self-citation, but the present paper does not rely on it as an unverified premise: Eq. (2) is explicitly constructed as a bottom-up effective theory from the t-Jz model, and its ground state is obtained by exact diagonalization in the truncated space Ω. The resulting signs and the pair-pair correlation pattern are outputs of the ED calculation, not fitted inputs. The spinon-singlet sign rule (Sgn_s = ∓1 for opposite chiralities) is read off the ED coefficients in Fig. 3(a), and the negative x-y PPC follows from the derived product Sgn_s Sgn_s' Sgn_Δ, not from imposing a d-wave pattern. Independent DMRG calculations for the t-Jz, t-J-α, and Hubbard models reproduce the same negative x-y PPC (Figs. 1, 2, 6, and 7), providing external anchoring outside the fitted or truncated model. The reported QCS-space fidelity (W≈44.3%, F_R≈90.8% at J=0.6, App. B) is a truncation-robustness limitation and a possible correctness concern, but it is not circularity: the truncated wavefunction is not adjusted to match the target PPC, and the DMRG comparison is an independent check. The self-citation to Ref. [34] supplies the effective model, not the d-wave conclusion, so it does not make the derivation circular.

Assumptions & free parameters 0 free parameters · 4 assumptions · 3 invented entities

The central mechanism rests on the QCS effective model and on the leading-order sign enumeration. The model itself is the authors' own construction (Ref. 34), benchmarked to DMRG, so the main ledger entries are domain assumptions about truncation, basis dominance, and the absence of interference rather than fitted constants. No free parameters were introduced to force the d-wave result; J=0.6 and U=8, t'=-0.2 are standard model inputs.

assumptions (4)
  • domain assumption The QCS representation with CQPs and ESF Gamma_z, truncated at |Gamma_z| <= 5, captures the physically relevant low-energy subspace of the t-Jz stripe ground state.
    Invoked in 'Two-Spinon QCS'; App. B shows the DMRG weight inside the truncated space is W = 44.3% at J = 0.6, so the claim depends on the omitted weight not changing the sign structure.
  • domain assumption Only two-spinon and three-spinon bases need be kept in the PPC sign analysis, with two-spinon bases dominating.
    Used in 'd-wave pairing'; based on accumulated weights in Fig. 3(a) and the assertion that four basis pairs contribute over 86% to G_b,b'. This is leading-order enumeration, not exhaustive proof.
  • domain assumption The long-distance sign of the matrix element entering G_b,b' is given by the product of the basis sign, the source basis sign, and the spin-exchange sign, with no other interference terms.
    Central to the sign rule in 'd-wave pairing'; the paper itself notes that when a spinon singlet is close to a dual hole, interference can change the simple behavior, so the long-distance claim relies on locality.
  • domain assumption Negative finite-cylinder PPC between distant x-bonds and y-bonds is a sufficient hallmark of d-wave pairing.
    Used to identify d-wave in all DMRG figures; no d-wave form factor extraction or off-diagonal long-range order check is provided.
invented entities (3)
  • Color quasi-particles (spinons, holons, dual-holes) with chirality labels c = r,g,b and chi = up/down
    purpose: Decompose electron configurations into QCS basis states used to derive the PPC sign rule.
    Constructed in 'Effective theory' from the electron-string mapping; no direct experimental observable is attached outside the model. Falsifiability comes only through the derived PPC pattern matched to DMRG in the same model family.
  • Spinon singlet as the pairing object in a partially-filled stripe
    purpose: Central claimed origin of the d-wave PPC sign pattern.
    Identified in the QCS ground state via chirality-opposite spinon bases; its existence is inferred from the model, not independently measured. The paper's 'Luttinger liquid of spinon singlets' is labeled a conjecture.
  • Quantum colored string (QCS) with pi-phase shift and ESF Gamma_z
    purpose: Geometrical representation of the hole-rich stripe; the effective Hamiltonian Eq. (2) acts on QCS states.
    Introduced by the same authors in Ref. [34] and treated here as the foundation; its validity is benchmarked by fidelity to DMRG rather than by independent experiment.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Spinon Singlet in Quantum Colored String: Origin of $d$-Wave Pairing in a Partially-Filled Stripe." pith.science (2026). https://pith.science/paper/GRXAWZQX

@misc{pith2026241204379,
  author       = {Pith},
  title        = {Pith review of: Spinon Singlet in Quantum Colored String: Origin of $d$-Wave Pairing in a Partially-Filled Stripe},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GRXAWZQX}},
  note         = {Machine review of arXiv:2412.04379}
}
abstract

Although both experimental observations and numerical simulations have reached a consensus that the stripe phase is intertwined with superconductivity in cuprates, the microscopic mechanism behind $d$-wave pairing in the presence of stripes remains unclear. Using the effective theory of quantum colored strings, we derive the wavefunction in Fock space. Our results show that two spinons with opposite chiralities tend to pair into a spinon singlet, which in turn facilitates the formation of negative pair-pair correlations between distant $x$-bonds and $y$-bonds, a hallmark of the $d$-wave pairing pattern. The same pair-pair correlation pattern is observed across various models, as confirmed by large-scale density matrix renormalization group calculations. Based on these results, we conclude that the spinon singlet is the origin of $d$-wave superconductivity in a fluctuating, partially-filled stripe, and this mechanism may also extend to multi-stripe configurations.

Figures

Figures reproduced from arXiv: 2412.04379 by the authors.

Figure 1
Figure 1. FIG. 1. (a) A schematic diagram shows the mapping from an elec [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The PPC function, starting from a target [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) Distributions of accumulated weights (AW) in [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (5 more)
Figure 5
Figure 5. Figure 5: FIG. 5. (a) The PPC function [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The PPC function, starting from a target [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The PPC function, starting from a target [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Schematic diagrams illustrate the processes of moving a [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Comparison of (a,c) the amplitudes and (b,d) signs between [PITH_FULL_IMAGE:figures/full_fig_p007_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

43 extracted references · 37 canonical work pages

  1. [34]

    Quantum colored strings in the hole-doped $t$-$J_z$ model

    Jia-Long Wang, Shi-Jie Hu, and Xue-Feng Zhang, “Quan- tum colored strings in the hole-doped t- jz model,” (2024), arXiv:2406.01980 [cond-mat.str-el]

    work page Pith review arXiv 2024

  2. [1]

    Theory of su- perconductivity,

    J. Bardeen, L. N. Cooper, and J. R. Schrie ffer, “Theory of su- perconductivity,” Phys. Rev.108, 1175–1204 (1957)

    1957

  3. [2]

    Correlated electrons in high-temperature super- conductors,

    Elbio Dagotto, “Correlated electrons in high-temperature super- conductors,” Rev. Mod. Phys.66, 763 (1994)

    work page 1994

  4. [3]

    Magnetic, transport, and optical properties of monolayer cop- per oxides,

    M. A. Kastner, R. J. Birgeneau, G. Shirane, and Y . Endoh, “Magnetic, transport, and optical properties of monolayer cop- per oxides,” Reviews of Modern Physics 70, 897–928 (1998)

    work page 1998

  5. [4]

    Doping a mott insulator: Physics of high-temperature superconductiv- ity,

    Patrick A. Lee, Naoto Nagaosa, and Xiao-Gang Wen, “Doping a mott insulator: Physics of high-temperature superconductiv- ity,” Rev. Mod. Phys.78, 17 (2006)

    work page 2006

  6. [5]

    From quantum matter to high-temperature super- conductivity in copper oxides,

    B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, and J. Zaanen, “From quantum matter to high-temperature super- conductivity in copper oxides,” Nature (London) 518, 179–186 (2015)

    work page 2015

  7. [6]

    When superconductivity crosses over: From bcs to bec,

    Qijin Chen, Zhiqiang Wang, Rufus Boyack, Shuolong Yang, and K. Levin, “When superconductivity crosses over: From bcs to bec,” Rev. Mod. Phys.96, 025002 (2024)

    work page 2024

  8. [7]

    Detection of a Cooper-pair density wave in Bi2Sr2CaCu2O8+x,

    M. H. Hamidian, S. D. Edkins, Sang Hyun Joo, A. Kostin, H. Eisaki, S. Uchida, M. J. Lawler, E. A. Kim, A. P. MacKen- zie, K. Fujita, Jinho Lee, and J. C. S´eamus Davis, “Detection of a Cooper-pair density wave in Bi2Sr2CaCu2O8+x,” Nature (Lon- don) 532, 343–347 (2016), arXiv:1511.08124 [cond-mat.supr- con]

    arXiv 2016

Show all 43 references
  1. [8]

    Pre-formed Cooper pairs in copper oxides and LaAlO 3—SrTiO3 heterostructures,

    Ivan Bo ˇzovi´c and Jeremy Levy, “Pre-formed Cooper pairs in copper oxides and LaAlO 3—SrTiO3 heterostructures,” Nature Physics 16, 712–717 (2020)

    2020

  2. [9]

    The emergence of global phase coherence from local pairing in underdoped cuprates,

    Shusen Ye, Changwei Zou, Hongtao Yan, Yu Ji, Miao Xu, Ze- hao Dong, Yiwen Chen, Xingjiang Zhou, and Yayu Wang, “The emergence of global phase coherence from local pairing in underdoped cuprates,” Nature Physics 19, 1301–1307 (2023), arXiv:2306.05926 [cond-mat.supr-con]

    2023 arXiv

  3. [10]

    One-dimensional nature of the magnetic fluctuations in YBa 2Cu3O6.6,

    H. A. Mook, Pengcheng Dai, F. Dogan, and R. D. Hunt, “One-dimensional nature of the magnetic fluctuations in YBa 2Cu3O6.6,” Nature (London) 404, 729–731 (2000), arXiv:cond-mat/0004362 [cond-mat.supr-con]

    2000 arXiv

  4. [11]

    Imaging the energy gap modulations of the cuprate pair-density-wave state,

    Zengyi Du, Hui Li, Sang Hyun Joo, Elizabeth P. Donoway, Jinho Lee, J. C. S ´eamus Davis, Genda Gu, Peter D. Johnson, and Kazuhiro Fujita, “Imaging the energy gap modulations of the cuprate pair-density-wave state,” Nature (London)580, 65– 70 (2020), arXiv:2109.14033 [cond-mat....

    2020 arXiv

  5. [12]

    Colloquium: Theory of intertwined orders in high temperature superconductors,

    Eduardo Fradkin, Steven A. Kivelson, and John M. Tranquada, “Colloquium: Theory of intertwined orders in high temperature superconductors,” Rev. Mod. Phys.87, 457–482 (2015)

    2015

  6. [13]

    Spins, stripes, and superconductivity in hole-doped cuprates,

    John M. Tranquada, “Spins, stripes, and superconductivity in hole-doped cuprates,” in AIP Conference Proceedings (AIP, 2013)

    2013

  7. [14]

    Stripe order in the underdoped region of the two-dimensional hubbard model,

    B. Zheng, Chia-Min Chung, Philippe Corboz, Georg Ehlers, Mingpu Qin, Reinhard M. Noack, Hao Shi, Steven R. White, Shiwei Zhang, and Garnet Kin-Lic Chan, “Stripe order in the underdoped region of the two-dimensional hubbard model,” Science 358, 1155 – 1160 (2016)

    2016

  8. [15]

    Tangent space approach for thermal tensor net- work simulations of the 2d hubbard model,

    Qiaoyi Li, Yuan Gao, Yuan-Yao He, Yang Qi, Bin-Bin Chen, and Wei Li, “Tangent space approach for thermal tensor net- work simulations of the 2d hubbard model,” Phys. Rev. Lett. 130, 226502 (2023)

    2023

  9. [16]

    Stripes, antiferromag- netism, and the pseudogap in the doped hubbard model at finite temperature,

    Alexander Wietek, Yuan-Yao He, Steven R. White, Antoine Georges, and E. Miles Stoudenmire, “Stripes, antiferromag- netism, and the pseudogap in the doped hubbard model at finite temperature,” Phys. Rev. X11, 031007 (2021)

    2021

  10. [17]

    Coexistence of superconductivity with partially filled stripes in the hubbard model,

    Hao Xu, Chia-Min Chung, Mingpu Qin, Ulrich Schollw ¨ock, Steven R. White, and Shiwei Zhang, “Coexistence of superconductivity with partially filled stripes in the hubbard model,” Science 384, eadh7691 (2024), https://www.science.org/doi/pdf/10.1126/science.adh7691

    2024 doi

  11. [18]

    Temperature Dependence of Spin and Charge Orders in the Doped Two-Dimensional Hubbard Model,

    Bo Xiao, Yuan-Yao He, Antoine Georges, and Shiwei Zhang, “Temperature Dependence of Spin and Charge Orders in the Doped Two-Dimensional Hubbard Model,” Physical Review X 13, 011007 (2023), arXiv:2202.11741 [cond-mat.str-el]

    2023 arXiv

  12. [19]

    Energetics of domain walls in the 2d t- j model,

    Steven R. White and D. J. Scalapino, “Energetics of domain walls in the 2d t- j model,” Phys. Rev. Lett. 81, 3227–3230 (1998)

    1998

  13. [20]

    Robust d-wave superconductivity in the square-lattice t- j model,

    Shoushu Gong, W. Zhu, and D. N. Sheng, “Robust d-wave superconductivity in the square-lattice t- j model,” Physical Re- view Letters 127, 097003 (2021)

    2021

  14. [21]

    Plaquette versus ordinary d- wave pairing in the t ′ -hubbard model on a width-4 cylinder,

    Chia-Min Chung, Mingpu Qin, Shiwei Zhang, Ulrich Schollw¨ock, and Steven R. White (The Simons Collaboration on the Many-Electron Problem), “Plaquette versus ordinary d- wave pairing in the t ′ -hubbard model on a width-4 cylinder,” Phys. Rev. B 102, 041106 (2020)

    2020

  15. [22]

    Emergent superconductivity and competing charge or- ders in hole-doped square-lattice t- j model,

    Xin Lu, Feng Chen, W. Zhu, D. N. Sheng, and Shou-Shu Gong, “Emergent superconductivity and competing charge or- ders in hole-doped square-lattice t- j model,” Phys. Rev. Lett. 132, 066002 (2024)

    2024

  16. [23]

    Ground-state phase diagram and superconductivity of the doped hubbard model on six-leg square cylinders,

    Yi-Fan Jiang, Thomas P. Devereaux, and Hong-Chen Jiang, “Ground-state phase diagram and superconductivity of the doped hubbard model on six-leg square cylinders,” Phys. Rev. 6 B 109, 085121 (2024)

    2024

  17. [24]

    Superconductiv- ity in the doped hubbard model and its interplay with next- nearest hopping t’,

    Hong-Chen Jiang and Thomas P. Devereaux, “Superconductiv- ity in the doped hubbard model and its interplay with next- nearest hopping t’,” Science 365, 1424–1428 (2019)

    2019

  18. [25]

    Ground state phase diagram of the doped hub- bard model on the four-leg cylinder,

    Yi-Fan Jiang, Jan Zaanen, Thomas P. Devereaux, and Hong- Chen Jiang, “Ground state phase diagram of the doped hub- bard model on the four-leg cylinder,” Phys. Rev. Res.2, 033073 (2020)

    2020

  19. [26]

    d-wave superconduc- tivity, pseudogap, and the phase diagram oft-t′- j model at finite temperature,

    Dai-Wei Qu, Bin-Bin Chen, Xin Lu, Qiaoyi Li, Shou-Shu Gong, Yang Qi, Wei Li, and Gang Su, “ d-wave superconduc- tivity, pseudogap, and the phase diagram oft-t′- j model at finite temperature,” (2023), arXiv:2211.06322 [cond-mat.str-el]

    2023 arXiv

  20. [27]

    Fragmented cooper pair condensation in striped superconductors,

    Alexander Wietek, “Fragmented cooper pair condensation in striped superconductors,” Physical Review Letters129, 177001 (2022)

    2022

  21. [28]

    The resonating valence bond state in La2CuO4 and superconductivity,

    P. W. Anderson, “The resonating valence bond state in La2CuO4 and superconductivity,” Science 235, 1196–1198 (1987)

    1987

  22. [29]

    Direct evidence for Cooper pairing without a spectral gap in a disor- dered superconductor above Tc,

    Koen M. Bastiaans, Damianos Chatzopoulos, Jian-Feng Ge, Doohee Cho, Willem O. Tromp, Jan M. van Ruitenbeek, Mark H. Fischer, Pieter J. de Visser, David J. Thoen, Eduard F. C. Driessen, Teunis M. Klapwijk, and Milan P. Allan, “Direct evidence for Cooper pairing without a spectr...

    2021 arXiv

  23. [30]

    Origin and fate of the pseudogap in the doped Hub- bard model,

    Fedor ˇSimkovic, Riccardo Rossi, Antoine Georges, and Michel Ferrero, “Origin and fate of the pseudogap in the doped Hub- bard model,” Science 385, eade9194 (2024), arXiv:2209.09237 [cond-mat.str-el]

    2024 arXiv

  24. [31]

    Order out of disorder in a gas of elastic quantum strings in 2 + 1 dimensions,

    J. Zaanen, “Order out of disorder in a gas of elastic quantum strings in 2 + 1 dimensions,” Phys. Rev. Lett. 84, 753–756 (2000)

    2000

  25. [32]

    The geometric order of stripes and Luttinger liq- uids,

    J. Zaanen, O. Y . Osman, H. V . Kruis, Z. Nussinov, and J. Tworzydlo, “The geometric order of stripes and Luttinger liq- uids,” Philosophical Magazine, Part B 81, 1485–1531 (2001), arXiv:cond-mat/0102103 [cond-mat.str-el]

    2001 arXiv

  26. [33]

    Quantum-fluctuation-induced collisions and subsequent excitation gap of an elastic string between walls,

    Yoshihiro Nishiyama, “Quantum-fluctuation-induced collisions and subsequent excitation gap of an elastic string between walls,” Phys. Rev. B66, 184501 (2002)

    2002

  27. [35]

    Ferromagnetism in a narrow, almost half- filled s band,

    Yosuke Nagaoka, “Ferromagnetism in a narrow, almost half- filled s band,” Phys. Rev.147, 392–405 (1966)

    1966

  28. [36]

    Extension of nagaoka’s theorem on the large-u hubbard model,

    Hal Tasaki, “Extension of nagaoka’s theorem on the large-u hubbard model,” Phys. Rev. B40, 9192–9193 (1989)

    1989

  29. [37]

    Strings in strongly correlated electron systems,

    P. Fulde and F. Pollmann, “Strings in strongly correlated electron systems,” Annalen der Physik 520, 441–449 (2008), arXiv:0711.2129 [cond-mat.str-el]

    2008 arXiv

  30. [38]

    Chiral edge states and fractional charge separation in a system of interacting bosons on a kagome lattice,

    Xue-Feng Zhang and Sebastian Eggert, “Chiral edge states and fractional charge separation in a system of interacting bosons on a kagome lattice,” Phys. Rev. Lett.111, 147201 (2013)

    2013

  31. [39]

    Quantum dynamics of topological strings in a frustrated Ising antiferromagnet,

    Zheng Zhou, Changle Liu, Zheng Yan, Yan Chen, and Xue- Feng Zhang, “Quantum dynamics of topological strings in a frustrated Ising antiferromagnet,” npj Quantum Materials 7, 60 (2022), arXiv:2010.01750 [cond-mat.str-el]

    2022 arXiv

  32. [40]

    Quantum strings in quan- tum spin ice,

    Yuan Wan and Oleg Tchernyshyov, “Quantum strings in quan- tum spin ice,” Phys. Rev. Lett.108, 247210 (2012)

    2012

  33. [41]

    Quantum domain walls induce incommensurate super- solid phase on the anisotropic triangular lattice,

    Xue-Feng Zhang, Shijie Hu, Axel Pelster, and Sebastian Eg- gert, “Quantum domain walls induce incommensurate super- solid phase on the anisotropic triangular lattice,” Phys. Rev. Lett. 117, 193201 (2016)

    2016

  34. [42]

    Charge stripe manipulation of superconducting pairing sym- metry transition,

    Chao Chen, Peigeng Zhong, Xuelei Sui, Runyu Ma, Ying Liang, Shijie Hu, Tianxing Ma, Hai-Qing Lin, and Bing Huang, (a) (b) (c) (d) (e) (f) FIG. 6. The PPC function, starting from a target x-bond (black line), is calculated using: (a) QCSM (2), (b-e) DMRG in the t-J- α model (1)...

    2024

  35. [43]

    Phase string effect in a doped antiferromagnet,

    D. N. Sheng, Y . C. Chen, and Z. Y . Weng, “Phase string effect in a doped antiferromagnet,” Phys. Rev. Lett. 77, 5102–5105 (1996). APPENDIX A. PPC function starting from a target x-bond The d-wave pairing pattern means that the PPCs between x-bonds and y-bonds are negative, w...

    1996

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.