Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

Intermediate bond-order-wave phase and nature of the order-to-order transition in the one-dimensional Hubbard-Holstein model

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 The paper claims that the one-dimensional, half-filled Hubbard-Holstein model hosts a narrow bond-order-wave phase between spin-density-wave and charge-density-wave order, with a second-order transition between the two charge orders at…

desk verdict A careful exact-QMC study that makes a plausible case for a new BOW phase in the Hubbard-Holstein model, but the central thermodynamic-limit claim rests on unconverged bond observables; worth refereeing, not yet proven. read the letter →

arxiv 2412.13263 v1 pith:SWHLMVWZ submitted 2024-12-17 cond-mat.str-el

Manuel Weber This is my paper · ORCID
classification cond-mat.str-el
keywords Hubbard-Holsteinmodelbond-orderwavecharge-densityspin-densityLuther-Emeryliquidretardedinteractiondirected-loopquantumMonteCarlophasetransition
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 one-dimensional, half-filled Hubbard-Holstein model, which combines local electron repulsion with coupling to lattice vibrations, hosts a narrow bond-order-wave phase between spin-density-wave and charge-density-wave order at strong coupling. The claim matters because this intermediate region had been debated as an extended metallic Luther-Emery liquid, and the model is a standard testbed for how electron-electron and electron-phonon interactions compete. Using an exact directed-loop quantum Monte Carlo method for retarded interactions on chains of up to 642 sites, the paper finds that once the extended metallic regime disappears, the spin- and charge-gapped region actually contains bond order, with the bond correlation ratio and bond susceptibility growing with system size. It further argues that the transition from bond order to charge order is second order and is governed by a gapless-charge Luther-Emery fixed point at moderate repulsion, turning first order only at stronger repulsion. If correct, this shows that a purely local, frequency-dependent interaction can stabilize bond order that the weak-coupling Peierls picture misses.

What carries the argument

The load-bearing object is the frequency-dependent Hubbard interaction $U(\omega)=U-\lambda W/[1-(\omega/\omega_0)^2]$, obtained by integrating out the phonons; because its strength and sign vary with frequency, one local density-density interaction can simultaneously favour spin and charge order, and their competition stabilizes bond order. The numerical machinery is an exact directed-loop quantum Monte Carlo algorithm for retarded interactions, which permits system sizes of several hundred sites. The finite-size diagnostics are the Luttinger parameters $K_{\rho/\sigma}(L)$ extracted from the long-wavelength structure factor, the correlation ratios $R_{\rho/b}(L)$ at the ordering vector $q=\pi$, and the charge and bond susceptibilities $\chi_{\rho/b}(q=\pi)/L$; the bond correlation ratio and bond susceptibility carry the BOW signal.

What would settle it

Extend the bond correlation ratio $R_b(L)$ and the bond susceptibility $\chi_b(q=\pi)/L$ at $U/t=6.0$ and $\lambda\approx 1.555$ to $L\gtrsim 1000$ using a method that can reach those sizes, such as density-matrix renormalization group or cluster quantum Monte Carlo. If $R_b(L)$ saturates well below one or $\chi_b(q=\pi)/L$ turns over instead of growing linearly with $L$, the intermediate regime is not a true bond-order-wave phase.

Watch

Extended reading notes

Core claim

The central discovery is that the intermediate spin- and charge-gapped region of the 1D Hubbard-Holstein model at strong coupling is not merely the tail of a metallic Luther-Emery phase but a distinct bond-ordered state. The evidence is the finite-size growth of the bond correlation ratio $R_b(L)$ and the bond susceptibility $\chi_b(q=\pi)/L$ for system sizes up to $L=642$, together with exponentially decaying charge correlations, while the charge correlation ratio $R_\rho(L)$ scales to one only on the CDW side of the transition. The paper interprets the bond-order-to-charge-order boundary as a continuous quantum phase transition described by a Luther-Emery fixed point with Luttinger parameter $K_\rho<1$, consistent with a one-dimensional deconfined quantum critical point, and finds that this transition becomes first order at larger $U$. The driving mechanism is the frequency dependence of the same local interaction $U(\omega)=U-\lambda W/[1-(\omega/\omega_0)^2]$ that arises after integrating out the phonons.

Load-bearing premise

The central claim rests on the assumption that the growing bond-order signal, which has not converged at the largest simulated size of 642 sites, saturates into true long-range order in the limit of an infinitely long chain rather than turning over into a crossover.

Editorial extensions

If this is right

  • At strong coupling, the phase diagram contains three ordered phases, spin-density-wave (SDW), bond-order-wave (BOW), and charge-density-wave (CDW) order, with BOW confined to a narrow parameter window between spin and charge order.
  • The BOW-CDW transition is continuous at moderate electron repulsion and can be viewed as a one-dimensional deconfined quantum critical point; it becomes first order at stronger repulsion.
  • Bond order can arise from a purely local electron-phonon coupling, so the standard weak-coupling Peierls classification of phonon-driven order is incomplete.
  • Previous numerical studies on smaller chains likely could not resolve the BOW phase because its signatures are weak and require system sizes of hundreds of sites.
  • The BOW phase at strong coupling connects naturally to the weak-coupling Luther-Emery regime, so the BOW phase boundary likely emerges from the tip of the extended metallic region.

Reading between the lines

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

  • Inference: varying the phonon frequency $\omega_0$ should move or reshape the BOW pocket, since the retardation range controls the effective competition; the paper leaves this as an open question.
  • Inference: if the BOW-CDW transition is truly a deconfined quantum critical point in one dimension, it should display emergent critical properties that could be tested by measuring both order parameters near $\lambda_{c2}$.
  • Inference: a calculation on chains longer than 642 sites, for example with tensor-network methods, could settle the thermodynamic-limit question directly by checking whether $R_b(L)$ continues to rise toward one.
  • Inference: the same frequency-dependent screening mechanism might stabilize bond order in multi-orbital or layered materials where the effective Hubbard interaction is dynamically screened, suggesting a search strategy for bond-order-wave materials.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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 studies the half-filled one-dimensional Hubbard-Holstein model at phonon frequency ω0/t=1 using an exact directed-loop quantum Monte Carlo method for retarded interactions. For U/t=6.0, it identifies three regimes as a function of electron-phonon coupling λ: an SDW phase, a narrow intermediate phase with both spin and charge gaps, and a CDW phase. The intermediate phase is attributed to bond-order-wave (BOW) order, and the BOW-CDW transition is argued to be second-order at moderate U, described by a Luther-Emery-liquid fixed point with Kρ≈0.59, turning first-order for U/t≳7.5. The phase diagram is extended to U/t=5.0–8.0, with the BOW regime disappearing for U/t≳8.0.

Significance. If the claims hold, this is a significant result: it would establish a BOW phase in the Hubbard-Holstein model induced by the frequency dependence of a purely local interaction, and it would sharpen the analogy to the extended Hubbard model while connecting to ideas of deconfined criticality in one dimension. The work uses an exact method, reaches system sizes up to L=642, and checks multiple observables (Kσ/ρ, correlation ratios, susceptibilities, and real-space correlation functions), which are genuine strengths. However, the direct finite-size evidence for BOW order is not yet converged, so the central claim is not fully established in the present version.

major comments (3)
  1. [Results, Fig. 2(d) and Fig. 3(d)] The existence of the BOW phase is inferred primarily from the increase of Rb(L) for L≳82 and the overshoot of χb(q=π)/L, but the paper states that systems are too small to observe convergence of χb(q=π)/L to its asymptotic behavior, and Rb(L) remains far below 1 at the largest sizes. Since no finite-size extrapolation of Rb(L) or χb(q=π)/L to L→∞ is provided, the data are also compatible with a crossover or with an LEL regime that has not yet flowed to the disordered limit. Please provide a controlled extrapolation (e.g., crossing analysis of Rb, scaling collapse, or a direct estimate of the BOW order parameter) to establish thermodynamic-limit BOW order.
  2. [Results, Fig. 2(b) inset and Fig. 4(e)] The second-order BOW-CDW scenario relies on the Kρ(L) peak height converging to a nonzero value, but the extrapolation uses a power-law fit whose functional form is acknowledged to be unknown, and the correlation-function exponent at the same point drifts between Kρ≈0.59 and 0.50 (Fig. S4(c)). Given the very narrow window λc2−λc1≈0.038 at U/t=6, the LEL fixed-point interpretation needs a more robust demonstration that the peak height remains finite in the thermodynamic limit rather than decaying to zero, which would indicate a first-order transition.
  3. [Results, Fig. 2(c)] The claim of a second-order BOW-CDW transition is based on a crossing of Rρ(L) that drifts slowly with L. A drifting crossing is also expected for a weak first-order transition, and no scaling collapse or critical-exponent analysis is shown. Since the order of the transition is a central claim, additional evidence (e.g., Binder cumulant, correlation-length scaling, or energy-gap behavior across λc2) is needed.
minor comments (5)
  1. [Discussion and Fig. S5] The statement that the intermediate phase extends to much weaker U is based on Kρ/σ(L) alone, because the paper notes that even L=642 is too small for a direct Rb(L) signature at U/t<5.0; this extrapolation should be labeled as more speculative than the strong-coupling result.
  2. [Figs. 2–4] No error bars are visible in the main figures; please state whether statistical errors are smaller than the symbol size or add representative error bars.
  3. [Fig. 3] The red guide-to-eye lines use Kρ≈0.45, which differs from the extrapolated Kρ≈0.59 reported in Fig. 2(b); the text attributes this to crossover effects, but the figure caption should clarify that these lines are guides and not fits.
  4. [Acknowledgments] The acknowledgments switch between first-person singular ('I am very grateful') and plural ('The authors gratefully acknowledge'); please make the wording consistent.
  5. [Fig. 2(d)] The legend in Fig. 2(d) lists system sizes up to L=322, while the text and Fig. 4 use L=642; please clarify whether L=642 data are omitted from Fig. 2 for clarity or add them.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the BOW-phase claim is derived from independent QMC observables with no fitted parameter enforcing the target result.

full rationale

The paper's central claims are the existence of a narrow BOW phase between SDW and CDW order and a second-order BOW–CDW transition at moderate U. These are supported by direct, independently defined observables: Kσ(L) dropping below one marks the spin gap opening, Kρ(L) peaking at λc2 marks the charge-gap closing point, and Rb(L) increasing with L identifies bond order. None of these quantities is defined in terms of the claimed phase diagram, and no parameter is fitted to reproduce the BOW phase. The self-cited directed-loop QMC method [63] is an exact, parameter-free algorithm whose stated assumptions (exact integration over phonons and diagrammatic sampling of the fermionic action) do not include the target result; it is real external evidence rather than a circular premise. The LEL fixed-point interpretation at the BOW–CDW boundary is imported from an external Gaussian/bosonization theory [11], not from the authors' own prior work. The acknowledged finite-size limitations—Rb(L) still far from one and the comment that 'systems are too small to observe convergence to the asymptotic behavior'—are a correctness and extrapolation risk, not a circularity: the observables are not constructed from the conclusion. Similarly, the power-law extrapolation of Kρ(L) to 0.59(1) is explicitly admitted to have no known exact fitting function and is used only as a guide to the eye. Therefore no load-bearing step reduces by construction to its inputs.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The model parameters U, λ, ω0 are inputs from the Hamiltonian and are not fitted. The central BOW phase claim does not require adjusting any parameter. However, the quantitative assignment of a second-order BOW-CDW transition uses an extrapolated Kρ value with an unknown fitting form, and the extension of the BOW phase to weak coupling is inferred from Luttinger parameter behavior rather than from direct bond order. There are no new postulated particles or forces.

free parameters (2)
  • Asymptotic Luttinger parameter Kρ at the BOW-CDW boundary (U/t=6.0) = 0.59(1)
    Extrapolated from finite-size Kρ(L) with a power-law fit [11]; used to identify a gapless LEL fixed point. The exact fitting function is unknown, and correlation-function fits give a lower exponent (~0.50).
  • Guide-to-eye exponent for charge/bond correlations in Fig. 3 = 0.45
    Hand-chosen value used to compare power-law decays at λ=1.5561; not a fit, but illustrates size effects in the critical regime.
assumptions (4)
  • domain assumption Luttinger-liquid finite-size estimate Kρ/σ(L) = π Sρ/σ(q1)/q1 accurately reflects the true Luttinger parameters in the thermodynamic limit for the Hubbard-Holstein model, including when coupled to phonons.
    Used in Eq. (4) and throughout to identify phases; the SM notes that finite-size corrections are strong when coupled to phonons (Ref [56]) and that Kρ(L) is hard to extrapolate.
  • domain assumption The BOW-CDW transition can be described by an LEL fixed point with gapless charge and gapped spin excitations, with Kρ < 1, based on bosonization of the extended Hubbard model.
    Used in the Discussion to interpret the peak in Kρ(L) and power-law charge correlations as a second-order transition. The analogy to the extended Hubbard model [11] is assumed to hold.
  • standard math The directed-loop QMC method for retarded interactions is exact and its O(Lβ log β) scaling is as stated in Ref [63].
    The correctness of the numerical method is taken from Ref [63] by the same author; no independent verification in this paper.
  • domain assumption The SDW-BOW transition is a BKT transition with an exponentially small spin gap, making it unresolvable at accessible system sizes.
    Invoked to justify the failure to directly observe the spin gap near λc1 and to use Kσ(L) dropping below one as the phase boundary estimate, as in Ref [9].

how reviews work

0 comments
Cite this review

Pith. "Pith review of Intermediate bond-order-wave phase and nature of the order-to-order transition in the one-dimensional Hubbard-Holstein model." pith.science (2026). https://pith.science/paper/SWHLMVWZ

@misc{pith2026241213263,
  author       = {Pith},
  title        = {Pith review of: Intermediate bond-order-wave phase and nature of the order-to-order transition in the one-dimensional Hubbard-Holstein model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SWHLMVWZ}},
  note         = {Machine review of arXiv:2412.13263}
}
read the original abstract

The Hubbard-Holstein model is one of the central models that describe the competition between electron-electron and electron-phonon interactions. In one dimension and at half-filling, the interplay between an electronic spin-density wave and a phonon-driven charge-density wave is considered to stabilize an intermediate Luther-Emery liquid. Here we show that, once the extended metallic regime disappears, a narrow bond-order-wave phase emerges. We use an exact directed-loop quantum Monte Carlo method for retarded interactions to simulate system sizes of several hundred sites, necessary to observe the weak signatures of this novel phase. Our results suggest a second-order quantum phase transition between the two charge orders that only turns first-order at strong coupling. Our findings are reminiscent of the extended Hubbard model, but they are driven by competing frequency dependencies of the same dynamically-screened Hubbard interaction.

Figures

Figures reproduced from arXiv: 2412.13263 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic phase diagram of the 1D Hubbard-Holstein [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Finite-size dependence of the (a) spin and (b) charge [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Finite-size dependence of the (a)–(c) spin and (d)–(f) [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Competing Dirac masses in one dimension: Symmetry-enhanced pseudo-first-order transition and deconfined criticality

    cond-mat.str-el 2025-09 conditional novelty 7.0 of 10

    Competing site and bond electron-phonon couplings in one dimension produce a transition that looks first-order with an emergent U(1) symmetry on finite scales, then becomes a narrow coexistence phase, and quantum phon...

Reference graph

Works this paper leans on

90 extracted references · 75 canonical work pages · cited by 1 Pith paper

  1. [1]

    Sipos, A

    B. Sipos, A. F. Kusmartseva, A. Akrap, H. Berger, L. Forr´ o, and E. Tutiˇ s, From Mott state to supercon- ductivity in 1T-TaS2, Nature Materials 7, 960 (2008)

    2008

  2. [2]

    Keimer, S

    B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, and J. Zaanen, From quantum matter to high-temperature superconductivity in copper oxides, Nature 518, 179 (2015)

    2015

  3. [3]

    Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken, J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T. Taniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo- Herrero, Correlated insulator behaviour at half-filling in magic-angle graphene superlattices, Nature 556, 80 (2018)

    2018

  4. [4]

    Y. Wang, H. Wu, G. T. McCandless, J. Y. Chan, and M. N. Ali, Quantum states and intertwining phases in kagome materials, Nature Reviews Physics 5, 635 (2023)

    2023

  5. [5]

    Gruner, Density Waves in Solids(CRC Press, Boca Raton, 1994)

    G. Gruner, Density Waves in Solids(CRC Press, Boca Raton, 1994)

    1994

  6. [6]

    Pouget, The Peierls instability and charge density wave in one-dimensional electronic conductors, Comptes Rendus Physique 17, 332 (2016)

    J.-P. Pouget, The Peierls instability and charge density wave in one-dimensional electronic conductors, Comptes Rendus Physique 17, 332 (2016)

    2016

  7. [7]

    Nakamura, Mechanism of CDW-SDW Transition in One Dimension, Journal of the Physical Society of Japan 68, 3123 (1999)

    M. Nakamura, Mechanism of CDW-SDW Transition in One Dimension, Journal of the Physical Society of Japan 68, 3123 (1999)

    1999

  8. [8]

    Nakamura, Tricritical behavior in the extended Hub- bard chains, Phys

    M. Nakamura, Tricritical behavior in the extended Hub- bard chains, Phys. Rev. B 61, 16377 (2000)

    2000

Show all 90 references
  1. [9]

    Sengupta, A

    P. Sengupta, A. W. Sandvik, and D. K. Campbell, Bond- order-wave phase and quantum phase transitions in the one-dimensional extended Hubbard model, Phys. Rev. B 65, 155113 (2002)

    2002

  2. [10]

    Ground-State Phase Diagram of a Half-Filled One-Dimensional Extended Hubbard Model

    A. W. Sandvik, P. Sengupta, and D. K. Campbell, Com- ment on “Ground-State Phase Diagram of a Half-Filled One-Dimensional Extended Hubbard Model”, Phys. Rev. Lett. 91, 089701 (2003)

    2003

  3. [11]

    A. W. Sandvik, L. Balents, and D. K. Campbell, Ground State Phases of the Half-Filled One-Dimensional Ex- tended Hubbard Model, Phys. Rev. Lett. 92, 236401 (2004)

    2004

  4. [12]

    Tsuchiizu and A

    M. Tsuchiizu and A. Furusaki, Phase Diagram of the One-Dimensional Extended Hubbard Model at Half Fill- ing, Phys. Rev. Lett. 88, 056402 (2002)

    2002

  5. [13]

    Tsuchiizu and A

    M. Tsuchiizu and A. Furusaki, Ground-state phase di- agram of the one-dimensional half-filled extended Hub- bard model, Phys. Rev. B 69, 035103 (2004)

    2004

  6. [14]

    Y. Z. Zhang, Dimerization in a Half-Filled One- Dimensional Extended Hubbard Model, Phys. Rev. Lett. 92, 246404 (2004)

    2004

  7. [15]

    Ejima and S

    S. Ejima and S. Nishimoto, Phase Diagram of the One- Dimensional Half-Filled Extended Hubbard Model, Phys. Rev. Lett. 99, 216403 (2007)

    2007

  8. [16]

    Tam, S.-W

    K.-M. Tam, S.-W. Tsai, and D. K. Campbell, Functional Renormalization Group Analysis of the Half-Filled One- Dimensional Extended Hubbard Model, Phys. Rev. Lett. 96, 036408 (2006)

    2006

  9. [17]

    Peierls, Quantum theory of solids(Oxford University Press, Oxford, 1955)

    R. Peierls, Quantum theory of solids(Oxford University Press, Oxford, 1955)

    1955

  10. [18]

    Holstein, Studies of polaron motion: Part I

    T. Holstein, Studies of polaron motion: Part I. The molecular-crystal model, Annals of Physics 8, 325 (1959)

    1959

  11. [19]

    J. E. Hirsch and E. Fradkin, Phase diagram of one- dimensional electron-phonon systems. II. The molecular- crystal model, Phys. Rev. B 27, 4302 (1983)

    1983

  12. [20]

    W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in Polyacetylene, Phys. Rev. Lett. 42, 1698 (1979)

    1979

  13. [21]

    Fradkin and J

    E. Fradkin and J. E. Hirsch, Phase diagram of one-dimensional electron-phonon systems. I. The Su- Schrieffer-Heeger model, Phys. Rev. B 27, 1680 (1983)

    1983

  14. [22]

    Weber, F

    M. Weber, F. Parisen Toldin, and M. Hohenadler, Com- peting orders and unconventional criticality in the Su- Schrieffer-Heeger model, Phys. Rev. Research 2, 023013 (2020)

    2020

  15. [23]

    Beyl, Hybrid Quantum Monte Carlo for Con- densed Matter Models, PhD thesis, Universit¨ at W¨ urzburg (2020)

    S. Beyl, Hybrid Quantum Monte Carlo for Con- densed Matter Models, PhD thesis, Universit¨ at W¨ urzburg (2020)

    2020

  16. [24]

    Cai, Z.-X

    X. Cai, Z.-X. Li, and H. Yao, Antiferromagnetism In- duced by Bond Su-Schrieffer-Heeger Electron-Phonon Coupling: A Quantum Monte Carlo Study, Phys. Rev. Lett. 127, 247203 (2021)

    2021

  17. [25]

    G¨ otz, S

    A. G¨ otz, S. Beyl, M. Hohenadler, and F. F. Assaad, Valence-bond solid to antiferromagnet transition in the two-dimensional Su-Schrieffer-Heeger model by Langevin dynamics, Phys. Rev. B 105, 085151 (2022)

    2022

  18. [26]

    Jeckelmann, C

    E. Jeckelmann, C. Zhang, and S. R. White, Metal- insulator transition in the one-dimensional Holstein model at half filling, Phys. Rev. B 60, 7950 (1999)

    1999

  19. [27]

    Zhong and H.-B

    J. Zhong and H.-B. Sch¨ uttler, Polaronic anharmonicity in the Holstein-Hubbard model, Phys. Rev. Lett. 69, 1600 (1992)

    1992

  20. [28]

    Berger, P

    E. Berger, P. Val´ aˇ sek, and W. von der Linden, Two- dimensional Hubbard-Holstein model, Phys. Rev. B 52, 4806 (1995). 6

    1995

  21. [29]

    J. K. Freericks and M. Jarrell, Competition between Electron-Phonon Attraction and Weak Coulomb Repul- sion, Phys. Rev. Lett. 75, 2570 (1995)

    1995

  22. [30]

    Hotta and Y

    T. Hotta and Y. Takada, Unconventional superconduc- tivity in the Hubbard-Holstein model, Physica B: Con- densed Matter 230-232, 1037 (1997)

    1997

  23. [31]

    Fehske, A

    H. Fehske, A. P. Kampf, M. Sekania, and G. Wellein, Nature of the Peierls- to Mott-insulator transition in 1D, The European Physical Journal B - Condensed Matter and Complex Systems 31, 11 (2003)

    2003

  24. [32]

    Fehske, G

    H. Fehske, G. Wellein, G. Hager, A. Weiße, and A. R. Bishop, Quantum lattice dynamical effects on single- particle excitations in one-dimensional Mott and Peierls insulators, Phys. Rev. B 69, 165115 (2004)

    2004

  25. [33]

    Koller, D

    W. Koller, D. Meyer, Y. Ono, and A. C. Hewson, First- and second-order phase transitions in the Holstein- Hubbard model, EPL (Europhysics Letters) 66, 559 (2004)

    2004

  26. [34]

    Honerkamp, H

    C. Honerkamp, H. C. Fu, and D.-H. Lee, Phonons and d-wave pairing in the two-dimensional Hubbard model, Phys. Rev. B 75, 014503 (2007)

    2007

  27. [35]

    Bauer, Competing interactions and symmetry break- ing in the Hubbard-Holstein model, EPL (Europhysics Letters) 90, 27002 (2010)

    J. Bauer, Competing interactions and symmetry break- ing in the Hubbard-Holstein model, EPL (Europhysics Letters) 90, 27002 (2010)

    2010

  28. [36]

    Bauer and A

    J. Bauer and A. C. Hewson, Competition between anti- ferromagnetic and charge order in the Hubbard-Holstein model, Phys. Rev. B 81, 235113 (2010)

    2010

  29. [37]

    E. A. Nowadnick, S. Johnston, B. Moritz, R. T. Scalet- tar, and T. P. Devereaux, Competition Between Antifer- romagnetic and Charge-Density-Wave Order in the Half- Filled Hubbard-Holstein Model, Phys. Rev. Lett. 109, 246404 (2012)

    2012

  30. [38]

    Johnston, E

    S. Johnston, E. A. Nowadnick, Y. F. Kung, B. Moritz, R. T. Scalettar, and T. P. Devereaux, Determinant quan- tum Monte Carlo study of the two-dimensional single- band Hubbard-Holstein model, Phys. Rev. B 87, 235133 (2013)

    2013

  31. [39]

    Murakami, P

    Y. Murakami, P. Werner, N. Tsuji, and H. Aoki, Or- dered phases in the Holstein-Hubbard model: Interplay of strong Coulomb interaction and electron-phonon cou- pling, Phys. Rev. B 88, 125126 (2013)

    2013

  32. [40]

    Wang, W.-S

    D. Wang, W.-S. Wang, and Q.-H. Wang, Phonon en- hancement of electronic order and negative isotope effect in the Hubbard-Holstein model on a square lattice, Phys. Rev. B 92, 195102 (2015)

    2015

  33. [41]

    Weber, F

    M. Weber, F. F. Assaad, and M. Hohenadler, Phonon spectral function of the one-dimensional Holstein- Hubbard model, Phys. Rev. B 91, 235150 (2015)

    2015

  34. [42]

    Ohgoe and M

    T. Ohgoe and M. Imada, Competition among Super- conducting, Antiferromagnetic, and Charge Orders with Intervention by Phase Separation in the 2D Holstein- Hubbard Model, Phys. Rev. Lett. 119, 197001 (2017)

    2017

  35. [43]

    Karakuzu, L

    S. Karakuzu, L. F. Tocchio, S. Sorella, and F. Becca, Superconductivity, charge-density waves, antiferromag- netism, and phase separation in the Hubbard-Holstein model, Phys. Rev. B 96, 205145 (2017)

    2017

  36. [44]

    N. C. Costa, K. Seki, S. Yunoki, and S. Sorella, Phase di- agram of the two-dimensional Hubbard-Holstein model, Communications Physics 3, 80 (2020)

    2020

  37. [45]

    Y. Wang, I. Esterlis, T. Shi, J. I. Cirac, and E. Dem- ler, Zero-temperature phases of the two-dimensional Hubbard-Holstein model: A non-Gaussian exact diago- nalization study, Phys. Rev. Res. 2, 043258 (2020)

    2020

  38. [46]

    Z. Han, S. A. Kivelson, and H. Yao, Strong Coupling Limit of the Holstein-Hubbard Model, Phys. Rev. Lett. 125, 167001 (2020)

    2020

  39. [47]

    N. K. Yirga, K.-M. Tam, and D. K. Campbell, Phonon- induced instabilities in correlated electron Hamiltonians, Phys. Rev. B 107, 235120 (2023)

    2023

  40. [48]

    Takada and A

    Y. Takada and A. Chatterjee, Possibility of a metal- lic phase in the charge-density-wave–spin-density-wave crossover region in the one-dimensional Hubbard- Holstein model at half filling, Phys. Rev. B 67, 081102 (2003)

    2003

  41. [49]

    R. T. Clay and R. P. Hardikar, Intermediate Phase of the One Dimensional Half-Filled Hubbard-Holstein Model, Phys. Rev. Lett. 95, 096401 (2005)

    2005

  42. [50]

    Tezuka, R

    M. Tezuka, R. Arita, and H. Aoki, Density-Matrix Renormalization Group Study of Pairing when Electron- Electron and Electron-Phonon Interactions Coexist: Ef- fect of the Electronic Band Structure, Phys. Rev. Lett. 95, 226401 (2005)

    2005

  43. [51]

    R. P. Hardikar and R. T. Clay, Phase diagram of the one- dimensional Hubbard-Holstein model at half and quarter filling, Phys. Rev. B 75, 245103 (2007)

    2007

  44. [52]

    Tezuka, R

    M. Tezuka, R. Arita, and H. Aoki, Phase diagram for the one-dimensional Hubbard-Holstein model: A density- matrix renormalization group study, Phys. Rev. B 76, 155114 (2007)

    2007

  45. [53]

    Tam, S.-W

    K.-M. Tam, S.-W. Tsai, D. K. Campbell, and A. H. Cas- tro Neto, Retardation effects in the Holstein-Hubbard chain at half filling, Phys. Rev. B 75, 161103 (2007)

    2007

  46. [54]

    Fehske, G

    H. Fehske, G. Hager, and E. Jeckelmann, Metallicity in the half-filled Holstein-Hubbard model, Europhysics Let- ters 84, 57001 (2008)

    2008

  47. [55]

    Ejima and H

    S. Ejima and H. Fehske, DMRG analysis of the SDW- CDW crossover region in the 1D half-filled Hubbard- Holstein model, in Journal of Physics Conference Se- ries, Journal of Physics Conference Series, Vol. 200 (IOP,

  48. [56]

    Tam, S.-W

    K.-M. Tam, S.-W. Tsai, and D. K. Campbell, Validity of the Tomonaga Luttinger liquid relations for the one- dimensional Holstein model, Phys. Rev. B 84, 165123 (2011)

    2011

  49. [57]

    Hohenadler and F

    M. Hohenadler and F. F. Assaad, Excitation spectra and spin gap of the half-filled Holstein-Hubbard model, Phys. Rev. B 87, 075149 (2013)

    2013

  50. [58]

    Bakrim and C

    H. Bakrim and C. Bourbonnais, Nature of ground states in one-dimensional electron-phonon Hubbard models at half filling, Phys. Rev. B 91, 085114 (2015)

    2015

  51. [59]

    Greitemann, S

    J. Greitemann, S. Hesselmann, S. Wessel, F. F. Assaad, and M. Hohenadler, Finite-size effects in Luther-Emery phases of Holstein and Hubbard models, Phys. Rev. B 92, 245132 (2015)

    2015

  52. [60]

    S. Zhao, Z. Han, S. A. Kivelson, and I. Esterlis, One- dimensional Holstein model revisited, Phys. Rev. B 107, 075142 (2023)

    2023

  53. [61]

    Luther and V

    A. Luther and V. J. Emery, Backward Scattering in the One-Dimensional Electron Gas, Phys. Rev. Lett. 33, 589 (1974)

    1974

  54. [62]

    Fehske, G

    H. Fehske, G. Wellein, A. Weiße, F. G¨ ohmann, H. B¨ uttner, and A. Bishop, Peierls-insulator Mott- insulator transition in 1D, Physica B: Condensed Matter 312-313, 562 (2002)

    2002

  55. [63]

    Weber, F

    M. Weber, F. F. Assaad, and M. Hohenadler, Directed- Loop Quantum Monte Carlo Method for Retarded Inter- actions, Phys. Rev. Lett. 119, 097401 (2017)

    2017

  56. [64]

    Roberts, S

    B. Roberts, S. Jiang, and O. I. Motrunich, Deconfined 7 quantum critical point in one dimension, Phys. Rev. B 99, 165143 (2019)

    2019

  57. [65]

    Mudry, A

    C. Mudry, A. Furusaki, T. Morimoto, and T. Hikihara, Quantum phase transitions beyond Landau-Ginzburg theory in one-dimensional space revisited, Phys. Rev. B 99, 205153 (2019)

    2019

  58. [66]

    Aryasetiawan, M

    F. Aryasetiawan, M. Imada, A. Georges, G. Kotliar, S. Biermann, and A. I. Lichtenstein, Frequency- dependent local interactions and low-energy effective models from electronic structure calculations, Phys. Rev. B 70, 195104 (2004)

    2004

  59. [67]

    Aryasetiawan, K

    F. Aryasetiawan, K. Karlsson, O. Jepsen, and U. Sch¨ onberger, Calculations of Hubbard U from first- principles, Phys. Rev. B 74, 125106 (2006)

    2006

  60. [68]

    Werner and A

    P. Werner and A. J. Millis, Dynamical Screening in Cor- related Electron Materials, Phys. Rev. Lett. 104, 146401 (2010)

    2010

  61. [69]

    Werner and M

    P. Werner and M. Casula, Dynamical screening in cor- related electron systems—from lattice models to realis- tic materials, Journal of Physics: Condensed Matter 28, 383001 (2016)

    2016

  62. [70]

    A. W. Sandvik, Stochastic series expansion method with operator-loop update, Phys. Rev. B 59, R14157 (1999)

    1999

  63. [71]

    O. F. Sylju ˚ asen and A. W. Sandvik, Quantum Monte Carlo with directed loops, Phys. Rev. E 66, 046701 (2002)

    2002

  64. [72]

    A. W. Sandvik, A generalization of Handscomb’s quan- tum Monte Carlo scheme-application to the 1D Hubbard model, Journal of Physics A: Mathematical and General 25, 3667 (1992)

    1992

  65. [73]

    A. W. Sandvik, R. R. P. Singh, and D. K. Campbell, Quantum Monte Carlo in the interaction representation: Application to a spin-Peierls model, Phys. Rev. B 56, 14510 (1997)

    1997

  66. [74]

    R. T. Clay, A. W. Sandvik, and D. K. Campbell, Possible exotic phases in the one-dimensional extended Hubbard model, Phys. Rev. B 59, 4665 (1999)

    1999

  67. [75]

    Eggert, I

    S. Eggert, I. Affleck, and M. Takahashi, Susceptibility of the spin 1/2 Heisenberg antiferromagnetic chain, Phys. Rev. Lett. 73, 332 (1994)

    1994

  68. [76]

    See Supplemental Material for additional results

  69. [77]

    Hohenadler, Interplay of Site and Bond Electron- Phonon Coupling in One Dimension, Phys

    M. Hohenadler, Interplay of Site and Bond Electron- Phonon Coupling in One Dimension, Phys. Rev. Lett. 117, 206404 (2016)

    2016

  70. [78]

    Giamarchi, Quantum Physics in One Dimension(Ox- ford University Press, 2003)

    T. Giamarchi, Quantum Physics in One Dimension(Ox- ford University Press, 2003)

    2003

  71. [79]

    Hofmeier, J

    D. Hofmeier, J. Willsher, U. F. P. Seifert, and J. Knolle, Spin-Peierls instability of deconfined quantum critical points, Phys. Rev. B 110, 125130 (2024)

    2024

  72. [80]

    Fabrizio, A

    M. Fabrizio, A. O. Gogolin, and A. A. Nersesyan, From Band Insulator to Mott Insulator in One Dimension, Phys. Rev. Lett. 83, 2014 (1999)

    1999

  73. [81]

    Fabrizio, A

    M. Fabrizio, A. Gogolin, and A. Nersesyan, Critical prop- erties of the double-frequency sine-Gordon model with applications, Nuclear Physics B 580, 647 (2000)

    2000

  74. [82]

    Wilkens and R

    T. Wilkens and R. M. Martin, Quantum Monte Carlo study of the one-dimensional ionic Hubbard model, Phys. Rev. B 63, 235108 (2001)

    2001

  75. [83]

    M. E. Torio, A. A. Aligia, and H. A. Ceccatto, Phase diagram of the Hubbard chain with two atoms per cell, Phys. Rev. B 64, 121105 (2001)

    2001

  76. [84]

    A. P. Kampf, M. Sekania, G. I. Japaridze, and P. Brune, Nature of the insulating phases in the half-filled ionic Hubbard model, Journal of Physics Condensed Matter 15, 5895 (2003)

    2003

  77. [85]

    Y. Z. Zhang, C. Q. Wu, and H. Q. Lin, Inducement of bond-order wave due to electron correlation in one di- mension, Phys. Rev. B 67, 205109 (2003)

    2003

  78. [86]

    C. D. Batista and A. A. Aligia, Exact Bond Ordered Ground State for the Transition between the Band and the Mott Insulator, Phys. Rev. Lett. 92, 246405 (2004)

    2004

  79. [87]

    S. R. Manmana, V. Meden, R. M. Noack, and K. Sch¨ onhammer, Quantum critical behavior of the one- dimensional ionic Hubbard model, Phys. Rev. B 70, 155115 (2004)

    2004

  80. [88]

    Otsuka and M

    H. Otsuka and M. Nakamura, Ground-state phase dia- gram of the one-dimensional Hubbard model with an al- ternating chemical potential, Phys. Rev. B 71, 155105 (2005)

    2005

  81. [89]

    Tincani, R

    L. Tincani, R. M. Noack, and D. Baeriswyl, Critical prop- erties of the band-insulator-to-Mott-insulator transition in the strong-coupling limit of the ionic Hubbard model, Phys. Rev. B 79, 165109 (2009)

    2009

  82. [90]

    8 Supplemental Material for Intermediate bond-order-wave phase and nature of the order-to-order transition in the one-dimensional Hubbard-Holstein model S1

    https://www.nhr-verein.de/unsere-partner . 8 Supplemental Material for Intermediate bond-order-wave phase and nature of the order-to-order transition in the one-dimensional Hubbard-Holstein model S1. SIGNA TURES OF THE SDW–BOW–CDW TRANSITIONS IN DIFFERENT OBSER V ABLES A commo...

Pith tools

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