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REVIEW 3 major objections 5 minor 72 references

Numerical study of bi-layer two-orbital model for La$_{3}$Ni$_{2}$O$_{7}$ on a plaquette ladder

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper argues that a bilayer two-orbital model of the nickelate superconductor La3Ni2O7, solved on a plaquette ladder, shows simultaneous quasi-long-range spin, charge, and pairing correlations when the inter-layer antiferromagnetic…

desk verdict Careful DMRG ladder study with new spin/charge/pairing correlations, but the PDW claim is underdetermined by the single-bond analysis. read the letter →

arxiv 2411.13399 v1 pith:YUQNGRZN submitted 2024-11-20 cond-mat.str-el cond-mat.supr-con

Yang Shen , Jiale Huang , Xiangjian Qian , Guang-Ming Zhang , Mingpu Qin This is my paper · ORCID
classification cond-mat.str-elcond-mat.supr-con
keywords nickelatesuperconductivityLa3Ni2O7bilayertwo-orbitalmodelplaquetteladderdensitymatrixrenormalizationgrouppairwavechargeantiferromagneticcorrelation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that the recently discovered 80 K superconductor La3Ni2O7 can be described by a bilayer model with two nickel orbitals, and that on a minimal two-dimensional-like plaquette ladder this model develops simultaneous quasi-long-range order in spin, charge, and pairing channels. The central new signal is a period-2 sign oscillation in the pair-pair correlation for both the 3dx2-y2 and 3dz2 orbitals, which the authors interpret as possible evidence of a pair density wave rather than uniform s-wave or d-wave superconductivity. If correct, superconductivity in this material would arise from inter-layer 3dz2 singlets that become phase-coherent through the 3dx2-y2 planes, while an orbital-selective charge density wave and Neel-type antiferromagnetism coexist with the pairing tendency. This matters because it narrows the competing microscopic mechanisms for an unconventional superconductor whose pairing symmetry is still unsettled.

What carries the argument

The central object is the bilayer two-orbital Hamiltonian of Eq. (1), which couples 3dx2-y2 orbitals within each layer through hopping tx2-y2 and hybridization tx2-y2,z2, and couples the two layers through 3dz2 hopping tz2 and a local antiferromagnetic exchange J between the inter-layer 3dz2 spins, with double occupancy of 3dz2 forbidden. The plaquette ladder geometry, length Lx = 32, is the minimum setup that retains two-dimensional characteristics while remaining tractable for DMRG. The load-bearing quantity is the equal-time spin-singlet pair-pair correlation D(r) measured from a reference inter-layer bond at x = 4, whose power-law envelope and period-2 sign oscillation carry the evidence for quasi-long-range pairing and possible pair density wave order.

What would settle it

Compute the pair-pair correlation D(r) with reference bonds placed at several different x positions, or with the 3dx2-y2 charge density wave artificially suppressed; if the period-2 sign pattern depends on the reference bond or disappears when the CDW is removed, the pair density wave interpretation would be falsified. A second check is to repeat the calculation on longer ladders or with periodic boundary conditions to see whether the oscillation and the fitted exponents Ksc are stable.

Watch

Extended reading notes

Core claim

On a plaquette ladder version of the bilayer two-orbital Hamiltonian, with inter-layer 3dz2 antiferromagnetic exchange J=0.5 (in units of the inter-layer hopping tz2), a Hubbard repulsion U=8 on the 3dx2-y2 orbitals, and fillings 1/16 hole doping for 3dz2 and 9/16 for 3dx2-y2, large-scale DMRG finds quasi-long-range correlations in all three channels. The 3dx2-y2 orbital develops a charge density wave with wavelength 2 lattice units and fitted exponent Kc=0.97(9), while the 3dz2 orbital shows no comparable charge order, indicating orbital-selective charge ordering. Both orbitals show Neel-type spin density waves with power-law exponents Ks=0.44(1) for 3dx2-y2 and Ks=0.31(2) for 3dz2. The spin-singlet pair-pair correlations decay algebraically with exponent Ksc=2.6(3) for the dz2-dz2 channel and Ksc=1.3(3) and 2.2(1) for horizontal and vertical dx2-y2 bonds, and in every channel the sign oscillates with period 2 along the ladder direction. The authors describe this pattern as D(r) ~ $r^{{-Ksc}}$ cos(Q·r + $\theta$) with Q = pi, and conclude that the system hosts possible pair density wave order.

Load-bearing premise

The conclusion that the sign oscillation indicates a pair density wave rests on treating that oscillation as an intrinsic property of the pairing correlation, rather than as an artifact of the single reference bond or of the coexisting period-2 charge order.

Editorial extensions

If this is right

  • If the period-2 sign oscillation in D(r) is intrinsic, the pairing state is not the s±-wave or d-wave state predicted by earlier Fermi-surface studies, but a pair density wave with ordering vector Q = pi along the ladder.
  • The coexistence of quasi-long-range spin, charge, and pairing correlations suggests that superconductivity in La3Ni2O7 emerges from a correlated state with intertwined orders, not from a simple weak-coupling mechanism.
  • Since the pairing exponents Ksc place the pairing susceptibility in the divergent regime for both orbital channels, the plaquette ladder may be a valid precursor for true long-range superconducting order in the two-dimensional limit.
  • The orbital-selective charge order on 3dx2-y2, absent on 3dz2, predicts that charge modulation experiments on the high-pressure phase should see predominantly in-plane dx2-y2 character rather than dz2 character.
  • The decrease of Tc at pressures above 18 GPa can be explained within this model by the increase of J reducing the hole doping of the 3dz2 orbital and shrinking the gamma pocket.

Reading between the lines

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

  • The period-2 oscillation in D(r) could partly be an artifact of the single reference bond at x = 4 or of the coexisting period-2 charge density wave on 3dx2-y2; the paper does not independently check this, so a calculation with multiple reference bonds or with the CDW suppressed would test whether the pairing oscillation is intrinsic.
  • The ferromagnetic spin alignment along the y-direction is explicitly noted by the authors as a possible artifact of the narrow ladder width; a wider ladder or a true two-dimensional calculation could change the magnetic structure from Neel-type to a different ordering vector.
  • If the pair density wave interpretation survives, it would connect La3Ni2O7 to the broader family of unconventional superconductors in which pair density wave order is a known competitor or companion of uniform superconductivity.
  • A direct experimental signature would be a spatially modulated superconducting gap or a field-induced pair density wave response in the high-pressure phase, which could be sought in future scanning tunneling or Josephson junction measurements.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript reports large-scale DMRG simulations of a bilayer two-orbital model for La3Ni2O7 on a plaquette ladder with Lx=32, using bond dimensions up to m=40000 and truncation-error extrapolation. The model has the dz2 orbital near half-filling and the dx2-y2 orbital near quarter-filling, with a substantial interlayer antiferromagnetic exchange J=0.5 for the dz2 orbitals. The authors find an orbital-selective period-2 CDW on the dx2-y2 orbital, Neel-type SDW correlations on both orbitals, and algebraic pair-pair correlations whose signs oscillate with period 2; they interpret the oscillation as evidence of a possible pair density wave. They note that the study is limited to a quasi-one-dimensional plaquette ladder and that the persistence of these orders in two dimensions requires further work.

Significance. The numerical effort is substantial and the DMRG protocol, including truncation-error extrapolation, is appropriate and strengthens confidence in the raw correlation data. If the central claims hold, the model provides a concrete example in which orbital-selective charge order, antiferromagnetic correlations, and pairing fluctuations coexist in a geometry with two-dimensional character, which is of direct relevance to current discussions of La3Ni2O7. The comparisons with the earlier one-dimensional study and with alternative pairing scenarios are useful. However, the pair-density-wave interpretation is not yet supported by the data as presented, and the correlation-exponent fits need systematic validation before the quasi-long-range conclusions can be fully trusted.

major comments (3)
  1. [Superconducting correlations, Fig. 4 and Eq. (3)] The period-2 sign oscillation of D(r) is presented as evidence for a pair density wave, but the correlation is computed with a single reference interlayer bond at x=4 in a ground state that also has a robust period-2 CDW on the dx2-y2 orbital (Fig. 2). A coexisting 2kF charge modulation can imprint an alternating sublattice phase onto equal-time pair-pair correlations even when the intrinsic pairing is uniform, and the inter-orbital hybridization tx2-y2,z2 can transmit the dx2-y2 CDW into the dz2 pairing channel. To distinguish an intrinsic Q=pi pairing modulation from a reference-bond or CDW artifact, the authors should compute D(r) with a reference bond on the other sublattice (e.g., x=5) or decompose the correlation into Q=0 and Q=pi components. Without such a control, the observed oscillation does not by itself establish a pair density wave.
  2. [Charge and spin density distributions, Fig. 2 inset] The CDW exponent Kc=0.97(9) is obtained by fitting 'solely the early segment of extrapolated envelope data', with no systematic fit-range scan or convergence criterion reported. Because the quasi-long-range CDW claim and the interpretation of the pairing correlations both rely on envelope amplitudes, the authors should show the stability of Kc, Ks, and Ksc with respect to the fitted distance window and justify the chosen segment. At present the quoted exponents are not robustly established.
  3. [Superconducting correlations, paragraph on Ksc and chi_sc] The text states that for both channels the pairing is 'close to or within the region with divergent pairing susceptibility', but the reported exponents are Ksc=2.6(3) for dz2-dz2, Ksc=1.3(3) for horizontal dx2-y2 bonds, and Ksc=2.2(1) for vertical dx2-y2 bonds. With chi_sc ~ T^{-(2-Ksc)}, only the horizontal dx2-y2 channel is clearly in the divergent regime, while the dz2-dz2 and vertical dx2-y2 channels have Ksc>2 and are not divergent. The statement should be made channel by channel, with the uncertainty in Ksc propagated, and the claim softened accordingly. This is load-bearing because the pairing instability is a central result.
minor comments (5)
  1. [Introduction] 'attracted a lot of intention' should read 'attracted a lot of attention'.
  2. [Discussion and Conclusion] There are several typographical errors: 'plaqutte' should be 'plaquette', 'encricles' in the Fig. 1 caption should be 'encircles', and 'atmoic sites' should be 'atomic sites'.
  3. [References] Reference [55] and reference [68] appear to be the same arXiv preprint (arXiv:2402.10485), with [55] missing a year; the duplicate should be removed or consolidated. In the conclusion, the reference list contains '56, 56' and should be corrected.
  4. [Superconducting correlations] The causal statement that the interlayer AFM super-exchange is the pairing driving force is not tested within this manuscript, because only J=0.5 is considered; a comparison with smaller J or J=0 would substantiate this mechanism claim.
  5. [Discussion and Conclusion] The phrase 'possible pair density wave' is used repeatedly, but the discussion would benefit from stating explicitly what additional calculation or observable would confirm or falsify the PDW interpretation in this ladder geometry.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the model is an input, and the correlation functions are independent DMRG outputs; the PDW interpretation is underdetermined but not circular.

full rationale

The paper studies a previously proposed bilayer two-orbital Hamiltonian (Eq. 1) taken from the authors' own prior work [35]. This is self-reference, but not circular in the derivation chain: the DMRG ground-state charge, spin, and pairing correlations are computed from the Hamiltonian, not derived from the conclusions of [35]. The parameters (tx2-y2=0.8, tx2-y2,z2=0.4, U=8, J=0.5, Delta_mu=1.2) are fixed inputs chosen to reach target fillings; the extracted exponents Kc, Ks, and Ksc are measured outputs of the simulation. No parameter is fitted to the target correlation and then renamed a prediction. The self-citation to [35] motivates the J term as the pairing-driving exchange, but the numerical result is not forced by that citation: the calculation could in principle have produced short-range correlations or no sign oscillation. The PDW interpretation rests on the observed period-2 sign oscillation in D(r), which is an output, not an input. The concern that the oscillation may be an artifact of the single reference bond at x=4 or of the coexisting period-2 CDW is an underdetermination/scientific-interpretation issue, not circularity. Accordingly, no circular step satisfies the requirement of exhibiting a reduction of a prediction to its own input.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central results rest on a model whose parameters are inputs from prior DFT or author work, and on the assumption that a width-2 ladder captures 2D physics. The most fragile added element is the interpretation of the period-2 sign oscillation as PDW-like; no new particle or field is introduced.

free parameters (6)
  • J (inter-layer AFM exchange for dz2) = 0.5
    Set to the 'large J' regime; the central pairing and PDW-like results depend on this choice, and no scan over J is reported.
  • U (Hubbard repulsion on dx2-y2) = 8
    Chosen Hubbard interaction, not derived; standard for nickelate model studies.
  • tx2-y2 (intra-layer dx2-y2 hopping) = 0.8
    Taken from DFT-based parameter estimates; not fit in this paper.
  • tx2-y2,z2 (intra-layer hybridization) = 0.4
    Same parameter set; the sign pattern comes from orbital symmetry.
  • Delta_mu (chemical potential difference) = 1.2
    Tuned to target 1/16 hole doping on dz2 and 9/16 filling on dx2-y2; the resulting densities are then analyzed.
  • hm (pinning magnetic field) = 0.5
    Artificial symmetry-breaking field applied at one edge site to extract spin density; it shapes the measured magnetic profile.
assumptions (5)
  • domain assumption The bilayer two-orbital Hamiltonian Eq. (1) is a valid low-energy description of pressurized La3Ni2O7, with neglected hoppings being small.
    The model is taken from the authors' prior work [35] and parameter estimates [32]; the paper does not derive it from ab initio calculations.
  • ad hoc to paper The local constraint nd < 2 on the dz2 orbital (no double occupancy) is imposed.
    Introduced in the model to enforce the Mott-insulating character of dz2; it is an input that affects the spin and pairing physics.
  • domain assumption A width-2 plaquette ladder with Lx=32 is a reliable proxy for studying the two-dimensional instability of the bilayer model.
    The paper explicitly states this is a minimum setup with 2D characteristic, and the authors acknowledge the geometry may produce artifacts such as ferromagnetic correlations along y.
  • domain assumption Power-law fits of correlation envelopes on a finite ladder with truncation-error extrapolation give the asymptotic exponents.
    The analysis treats the finite-size DMRG results as representative of the thermodynamic limit; only truncation error, not finite-size scaling, is extrapolated.
  • ad hoc to paper The Q=pi sign oscillation in the pairing correlation is an intrinsic pairing symmetry rather than a consequence of the reference bond or the coexisting CDW.
    This assumption underlies the 'possible pair density wave' claim; no independent check distinguishes PDW from CDW-induced modulation.

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Cite this review

Pith. "Pith review of Numerical study of bi-layer two-orbital model for La$_{3}$Ni$_{2}$O$_{7}$ on a plaquette ladder." pith.science (2026). https://pith.science/paper/YUQNGRZN

@misc{pith2026241113399,
  author       = {Pith},
  title        = {Pith review of: Numerical study of bi-layer two-orbital model for La$_3$Ni$_2$O$_7$ on a plaquette ladder},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YUQNGRZN}},
  note         = {Machine review of arXiv:2411.13399}
}
abstract

The recently discovered high-$T_c$ superconductivity in La$_{3}$Ni$_{2}$O$_{7}$ with $T_c \approx 80K$ provides another intriguing platform to explore the microscopic mechanism of unconventional superconductivity. In this work, we study a previously proposed bi-layer two-orbital model Hamiltonian for La$_{3}$Ni$_{2}$O$_{7}$ [Y. Shen, et al, Chinese Physics Letters 40, 127401 (2023)] on a plaquette ladder, which is a minimum setup with two-dimensional characteristic. We employ large-scale Density Matrix Renormalization Group calculations to accurately determine the ground state of the model. We determine the density, magnetic structure, and the pairing property of the model. We find that with large effective inter-layer anti-ferromagnetic exchange for the 3$d_{z^2}$ orbital, both spin, charge, and pairing correlation display quasi-long-range behavior, which could be viewed as a precursor of possible true long-range order in the two dimensional limit. Interestingly, sign oscillation for the pairing correlation are observed for both the 3$d_{x^2-y^2}$ and 3$d_{z^2}$ orbitals, indicating the presence of possible pair density wave in the system. Even though we only study the model on a quasi one-dimensional plaquette ladder geometry due to the computational difficulty, the results on the spin, charge, and pairing correlation provide valuable insight in the clarification of the properties of La$_{3}$Ni$_{2}$O$_{7}$ in the future.

Figures

Figures reproduced from arXiv: 2411.13399 by the authors.

Figure 1
Figure 1. FIG. 1: (a) Crystal structure and (b) schematic illustra [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The charge density distribution of the (a) 3 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The spin density distribution of the (a) 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The spin-singlet pair-pair correlation functions for [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

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Works this paper leans on

72 extracted references · 36 canonical work pages

  1. [35]

    Kaneko, H

    T. Kaneko, H. Sakakibara, M. Ochi, and K. Kuroki, Physical Review B 109, 045154 (2024)

    work page 2024

  2. [1]

    J. G. Bednorz and K. A. M¨ uller, Zeitschrift f¨ ur Physik B Condensed Matter 64, 189 (1986)

    1986

  3. [2]

    We set the 3dz2 bond at x = 4 (the dashed oval in Fig

    (3) Different pairing channels are calculate by setting bond i (formed by site (i, 1) and (i, 2)) and bond j (formed by site (j, 1) and (j, 2)) to different orbitals. We set the 3dz2 bond at x = 4 (the dashed oval in Fig. 4 (a)) as the refer- ence bond in the calculation of the pair-pair correlations to make sure the pairing correlation results are least ...

    work page

  4. [3]

    Kamihara, T

    Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, Journal of the American Chemical Society 130, 3296 (2008)

    2008

  5. [4]

    Kamihara, H

    Y. Kamihara, H. Hiramatsu, M. Hirano, R. Kawamura, H. Yanagi, T. Kamiya, and H. Hosono, Journal of the American Chemical Society 128, 10012 (2006)

    work page 2006

  6. [5]

    V. I. Anisimov, D. Bukhvalov, and T. M. Rice, Phys. Rev. B 59, 7901 (1999), URL https://link.aps.org/ doi/10.1103/PhysRevB.59.7901

    doi:10.1103/physrevb.59.7901 1999

  7. [6]

    D. Li, K. Lee, B. Y. Wang, M. Osada, S. Crossley, H. R. Lee, Y. Cui, Y. Hikita, and H. Y. Hwang, Nature 572, 624 (2019)

    work page 2019

  8. [7]

    Osada, B

    M. Osada, B. Y. Wang, B. H. Goodge, K. Lee, H. Yoon, K. Sakuma, D. Li, M. Miura, L. F. Kourkoutis, and H. Y. 6 Hwang, Nano letters 20, 5735 (2020)

    work page 2020

Show all 72 references
  1. [8]

    S. Zeng, C. Li, L. E. Chow, Y. Cao, Z. Zhang, C. S. Tang, X. Yin, Z. S. Lim, J. Hu, P. Yang, et al., Science advances 8, eabl9927 (2022)

    2022

  2. [9]

    Osada, B

    M. Osada, B. Y. Wang, B. H. Goodge, S. P. Harvey, K. Lee, D. Li, L. F. Kourkoutis, and H. Y. Hwang, Ad- vanced Materials 33, 2104083 (2021)

    2021

  3. [10]

    Lee and W

    K.-W. Lee and W. E. Pickett, Phys. Rev. B 70, 165109 (2004), URL https://link.aps.org/doi/10. 1103/PhysRevB.70.165109

    2004

  4. [11]

    Nomura and R

    Y. Nomura and R. Arita, Reports on Progress in Physics 85, 052501 (2022)

    2022

  5. [12]

    B. H. Goodge, D. Li, K. Lee, M. Osada, B. Y. Wang, G. A. Sawatzky, H. Y. Hwang, and L. F. Kourkoutis, Proceedings of the National Academy of Sciences 118, e2007683118 (2021), https://www.pnas.org/doi/pdf/10.1073/pnas.2007683118, URL https://www.pnas.org/doi/abs/10.1073/pnas. 2...

    2021 doi

  6. [13]

    Zhang, Y.-f

    G.-M. Zhang, Y.-f. Yang, and F.-C. Zhang, Phys. Rev. B 101, 020501 (2020), URL https://link.aps.org/doi/ 10.1103/PhysRevB.101.020501

    2020 doi

  7. [14]

    Hayward and M

    M. Hayward and M. Rosseinsky, Solid State Sci- ences 5, 839 (2003), ISSN 1293-2558, international Conference on Inorganic Materials 2002, URL https://www.sciencedirect.com/science/article/ pii/S1293255803001110

    2003

  8. [15]

    H. Sun, M. Huo, X. Hu, J. Li, Z. Liu, Y. Han, L. Tang, Z. Mao, P. Yang, B. Wang, et al., Nature621, 493 (2023)

    2023

  9. [16]

    Hou, P.-T

    J. Hou, P.-T. Yang, Z.-Y. Liu, J.-Y. Li, P.-F. Shan, L. Ma, G. Wang, N.-N. Wang, H.-Z. Guo, J.-P. Sun, et al., Chinese Physics Letters 40, 117302 (2023), URL https://dx.doi.org/10.1088/0256-307X/40/11/ 117302

    2023 doi

  10. [17]

    Zhang, D

    Y. Zhang, D. Su, Y. Huang, Z. Shan, H. Sun, M. Huo, K. Ye, J. Zhang, Z. Yang, Y. Xu, et al., Nature Physics pp. 1–5 (2024)

    2024

  11. [18]

    Zhang, C

    M. Zhang, C. Pei, Q. Wang, Y. Zhao, C. Li, W. Cao, S. Zhu, J. Wu, and Y. Qi, Journal of Materials Sci- ence and Technology 185, 147 (2024), ISSN 1005- 0302, URL https://www.sciencedirect.com/science/ article/pii/S1005030223009829

    2024

  12. [19]

    G. Wang, N. N. Wang, X. L. Shen, J. Hou, L. Ma, L. F. Shi, Z. A. Ren, Y. D. Gu, H. M. Ma, P. T. Yang, et al., Phys. Rev. X 14, 011040 (2024), URL https://link. aps.org/doi/10.1103/PhysRevX.14.011040

    2024 doi

  13. [20]

    Wang, H.-H

    M. Wang, H.-H. Wen, T. Wu, D.-X. Yao, and T. Xiang, Chinese Physics Letters 41, 077402 (2024), URL https: //dx.doi.org/10.1088/0256-307X/41/7/077402

    2024 doi

  14. [21]

    Y. Zhu, D. Peng, E. Zhang, B. Pan, X. Chen, L. Chen, H. Ren, F. Liu, Y. Hao, N. Li, et al., Nature 631, 531 (2024)

    2024

  15. [22]

    J.-X. Wang, Z. Ouyang, R.-Q. He, and Z.-Y. Lu, Phys. Rev. B 109, 165140 (2024), URL https://link.aps. org/doi/10.1103/PhysRevB.109.165140

    2024 doi

  16. [23]

    Sakakibara, M

    H. Sakakibara, M. Ochi, H. Nagata, Y. Ueki, H. Sakurai, R. Matsumoto, K. Terashima, K. Hi- rose, H. Ohta, M. Kato, et al., Phys. Rev. B 109, 144511 (2024), URL https://link.aps.org/doi/10. 1103/PhysRevB.109.144511

    2024

  17. [24]

    Li, Y.-J

    Q. Li, Y.-J. Zhang, Z.-N. Xiang, Y. Zhang, X. Zhu, and H.-H. Wen, Chinese Physics Letters 41, 017401 (2024)

    2024

  18. [25]

    Li, C.-Q

    J. Li, C.-Q. Chen, C. Huang, Y. Han, M. Huo, X. Huang, P. Ma, Z. Qiu, J. Chen, X. Hu, et al., Science China Physics, Mechanics & Astronomy 67, 117403 (2024)

    2024

  19. [26]

    Zhang, C

    M. Zhang, C. Pei, X. Du, W. Hu, Y. Cao, Q. Wang, J. Wu, Y. Li, H. Liu, C. Wen, et al., arXiv preprint arXiv:2311.07423 (2023)

    2023 arXiv

  20. [28]

    S. N. Abadi, K.-J. Xu, E. G. Lomeli, P. Puphal, M. Isobe, Y. Zhong, A. V. Fedorov, S.-K. Mo, M. Hashimoto, D.-H. Lu, et al., arXiv preprint arXiv:2402.07143 (2024)

    2024

  21. [29]

    J. Yang, H. Sun, X. Hu, Y. Xie, T. Miao, H. Luo, H. Chen, B. Liang, W. Zhu, G. Qu, et al., Nature Com- munications 15, 4373 (2024)

    2024

  22. [30]

    Zhang, L.-F

    Y. Zhang, L.-F. Lin, A. Moreo, T. A. Maier, and E. Dagotto, Nature Communications 15, 2470 (2024)

    2024

  23. [31]

    Zhang, L.-F

    Y. Zhang, L.-F. Lin, A. Moreo, and E. Dagotto, Phys. Rev. B 108, L180510 (2023), URL https://link.aps. org/doi/10.1103/PhysRevB.108.L180510

    2023 doi

  24. [32]

    Lechermann, J

    F. Lechermann, J. Gondolf, S. B¨ otzel, and I. M. Eremin, Phys. Rev. B 108, L201121 (2023), URL https://link. aps.org/doi/10.1103/PhysRevB.108.L201121

    2023 doi

  25. [33]

    Pardo and W

    V. Pardo and W. E. Pickett, Phys. Rev. B 83, 245128 (2011), URL https://link.aps.org/doi/10. 1103/PhysRevB.83.245128

    2011

  26. [34]

    Y. Gu, C. Le, Z. Yang, X. Wu, and J. Hu, arXiv preprint arXiv:2306.07275 (2023)

    2023 arXiv

  27. [36]

    Y. Shen, M. Qin, and G.-M. Zhang, Chinese Physics Let- ters 40, 127401 (2023)

    2023

  28. [37]

    Z. Luo, X. Hu, M. Wang, W. W´ u, and D.-X. Yao, Phys. Rev. Lett. 131, 126001 (2023), URL https://link.aps. org/doi/10.1103/PhysRevLett.131.126001

    2023 doi

  29. [38]

    C. Lu, Z. Pan, F. Yang, and C. Wu, Phys. Rev. Lett. 132, 146002 (2024), URL https://link.aps.org/doi/ 10.1103/PhysRevLett.132.146002

    2024 doi

  30. [39]

    Cao and Y.-f

    Y. Cao and Y.-f. Yang, Phys. Rev. B 109, L081105 (2024), URL https://link.aps.org/doi/10.1103/ PhysRevB.109.L081105

    2024

  31. [40]

    Ouyang, J.-M

    Z. Ouyang, J.-M. Wang, J.-X. Wang, R.-Q. He, L. Huang, and Z.-Y. Lu, Phys. Rev. B 109, 115114 (2024), URL https://link.aps.org/doi/10. 1103/PhysRevB.109.115114

    2024

  32. [41]

    Oh and Y.-H

    H. Oh and Y.-H. Zhang, Phys. Rev. B 108, 174511 (2023), URL https://link.aps.org/doi/10. 1103/PhysRevB.108.174511

    2023

  33. [42]

    Fan, J.-F

    Z. Fan, J.-F. Zhang, B. Zhan, D. Lv, X.-Y. Jiang, B. Normand, and T. Xiang, Phys. Rev. B 110, 024514 (2024), URL https://link.aps.org/doi/10. 1103/PhysRevB.110.024514

    2024

  34. [43]

    Qu, D.-W

    X.-Z. Qu, D.-W. Qu, J. Chen, C. Wu, F. Yang, W. Li, and G. Su, Phys. Rev. Lett. 132, 036502 (2024), URL https://link.aps.org/doi/10.1103/PhysRevLett. 132.036502

    2024 doi

  35. [44]

    Q.-G. Yang, D. Wang, and Q.-H. Wang, Phys. Rev. B 108, L140505 (2023), URL https://link.aps.org/doi/ 10.1103/PhysRevB.108.L140505

    2023 doi

  36. [45]

    Xue and F

    J.-R. Xue and F. Wang, Chinese Physics Letters 41, 057403 (2024), URL https://dx.doi.org/10.1088/ 0256-307X/41/5/057403

    2024

  37. [46]

    W. W´ u, Z. Luo, D.-X. Yao, and M. Wang, arXiv preprint arXiv:2307.05662 (2023)

    2023 arXiv

  38. [47]

    Z. Liu, M. Huo, J. Li, Q. Li, Y. Liu, Y. Dai, X. Zhou, 7 J. Hao, Y. Lu, M. Wang, et al., Nature Communications 15, 7570 (2024)

    2024

  39. [48]

    J. Zhan, Y. Gu, X. Wu, and J. Hu, arXiv preprint arXiv:2404.03638 (2024)

    2024 arXiv

  40. [49]

    Ouyang, M

    Z. Ouyang, M. Gao, and Z.-Y. Lu, npj Quantum Mate- rials 9, 80 (2024)

    2024

  41. [50]

    Christiansson, F

    V. Christiansson, F. Petocchi, and P. Werner, Phys. Rev. Lett. 131, 206501 (2023), URL https://link.aps.org/ doi/10.1103/PhysRevLett.131.206501

    2023 doi

  42. [51]

    D. A. Shilenko and I. V. Leonov, Phys. Rev. B 108, 125105 (2023), URL https://link.aps.org/doi/10. 1103/PhysRevB.108.125105

    2023

  43. [52]

    X. Chen, P. Jiang, J. Li, Z. Zhong, and Y. Lu, arXiv preprint arXiv:2307.07154 (2023)

    2023 arXiv

  44. [53]

    K. Chen, X. Liu, J. Jiao, M. Zou, C. Jiang, X. Li, Y. Luo, Q. Wu, N. Zhang, Y. Guo, et al., Phys. Rev. Lett. 132, 256503 (2024), URL https://link.aps.org/ doi/10.1103/PhysRevLett.132.256503

    2024 doi

  45. [54]

    Z. Dan, Y. Zhou, M. Huo, Y. Wang, L. Nie, M. Wang, T. Wu, and X. Chen, arXiv preprint arXiv:2402.03952 (2024)

    2024 arXiv

  46. [55]

    K.-J. Zhou, X. Chen, J. Choi, Z. Jiang, J. Mei, K. Jiang, J. Li, S. Agrestini, M. Garcia-Fernandez, H. Sun, et al. (2024)

    2024

  47. [57]

    Z. Luo, B. Lv, M. Wang, W. W´ u, and D.-X. Yao, npj Quantum Materials 9, 61 (2024)

    2024

  48. [58]

    Liu, J.-W

    Y.-B. Liu, J.-W. Mei, F. Ye, W.-Q. Chen, and F. Yang, Phys. Rev. Lett. 131, 236002 (2023), URL https:// link.aps.org/doi/10.1103/PhysRevLett.131.236002

    2023 doi

  49. [59]

    Yang, G.-M

    Y.-f. Yang, G.-M. Zhang, and F.-C. Zhang, Phys. Rev. B 108, L201108 (2023), URL https://link.aps.org/doi/ 10.1103/PhysRevB.108.L201108

    2023 doi

  50. [60]

    H. Liu, C. Xia, S. Zhou, and H. Chen, arXiv preprint arXiv:2311.07316 (2023)

    2023 arXiv

  51. [61]

    Jiang, Z

    K. Jiang, Z. Wang, and F.-C. Zhang, Chinese Physics Letters 41, 017402 (2024), URL https://dx.doi.org/ 10.1088/0256-307X/41/1/017402

    2024 doi

  52. [62]

    S. R. White, Phys. Rev. Lett. 69, 2863 (1992), URL https://link.aps.org/doi/10.1103/PhysRevLett.69. 2863

    1992 doi

  53. [63]

    S. R. White, Phys. Rev. B 48, 10345 (1993), URL https: //link.aps.org/doi/10.1103/PhysRevB.48.10345

    1993 doi

  54. [64]

    Giamarchi, Quantum physics in one dimension , vol

    T. Giamarchi, Quantum physics in one dimension , vol. 121 (Clarendon press, 2003)

    2003

  55. [65]

    Lu, D.-W

    X. Lu, D.-W. Qu, Y. Qi, W. Li, and S.-S. Gong, Phys. Rev. B 107, 125114 (2023), URL https://link.aps. org/doi/10.1103/PhysRevB.107.125114

    2023 doi

  56. [66]

    D.-K. Seo, W. Liang, M.-H. Whangbo, Z. Zhang, and M. Greenblatt, Inorganic chemistry 35, 6396 (1996)

    1996

  57. [67]

    S. R. White and A. L. Chernyshev, Phys. Rev. Lett. 99, 127004 (2007), URL https://link.aps.org/doi/ 10.1103/PhysRevLett.99.127004

    2007 doi

  58. [68]

    X. Chen, J. Choi, Z. Jiang, J. Mei, K. Jiang, J. Li, S. Agrestini, M. Garcia-Fernandez, H. Sun, X. Huang, et al., Nature Communications 15, 9597 (2024)

    2024

  59. [69]

    Khasanov, T

    R. Khasanov, T. J. Hicken, D. J. Gawryluk, L. P. Sorel, S. B¨ otzel, F. Lechermann, I. M. Eremin, H. Luetkens, and Z. Guguchia, arXiv preprint arXiv:2402.10485 (2024)

    2024 arXiv

  60. [70]

    Y. Wang, K. Jiang, Z. Wang, F.-C. Zhang, and J. Hu, arXiv preprint arXiv:2401.15097 (2024)

    2024 arXiv

  61. [71]

    Shen, G.-M

    Y. Shen, G.-M. Zhang, and M. Qin, Phys. Rev. B 108, 165113 (2023), URL https://link.aps.org/doi/ 10.1103/PhysRevB.108.165113

    2023 doi

  62. [72]

    D. F. Agterberg, J. S. Davis, S. D. Edkins, E. Fradkin, D. J. Van Harlingen, S. A. Kivelson, P. A. Lee, L. Radz- ihovsky, J. M. Tranquada, and Y. Wang, Annual Review of Condensed Matter Physics 11, 231 (2020)

    2020

  63. [73]

    J. Li, P. Ma, H. Zhang, X. Huang, C. Huang, M. Huo, D. Hu, Z. Dong, C. He, J. Liao, et al., Preprint at https://arxiv. org/abs/2404 11369 (2024)

    2024

  64. [74]

    Our in-house DMRG code is developed with TensorKit package at https://github.com/Jutho/TensorKit.jl

Pith tools

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