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Stripes in the extended $t-t^\prime$ Hubbard model: A Variational Monte Carlo analysis

by Vito Marino, Federico Becca, Luca F. Tocchio

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Submission summary

Authors (as registered SciPost users): Vito Marino · Luca Fausto Tocchio
Submission information
Preprint Link:  (pdf)
Date accepted: 2022-05-19
Date submitted: 2022-04-21 19:09
Submitted by: Tocchio, Luca Fausto
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
Approaches: Theoretical, Computational


By using variational quantum Monte Carlo techniques, we investigate the instauration of stripes (i.e., charge and spin inhomogeneities) in the Hubbard model on the square lattice at hole doping $\delta=1/8$, with both nearest- ($t$) and next-nearest-neighbor hopping ($t^\prime$). Stripes with different wavelengths $\lambda$ (denoting the periodicity of the charge inhomogeneity) and character (bond- or site-centered) are stabilized for sufficiently large values of the electron-electron interaction $U/t$. The general trend is that $\lambda$ increases going from negative to positive values of $t^\prime/t$ and decreases by increasing $U/t$. In particular, the $\lambda=8$ stripe obtained for $t^\prime=0$ and $U/t=8$ [L.F. Tocchio, A. Montorsi, and F. Becca, SciPost Phys. {\bf 7}, 21 (2019)] shrinks to $\lambda=6$ for $U/t\gtrsim 10$. For $t^\prime/t<0$, the stripe with $\lambda=5$ is found to be remarkably stable, while for $t^\prime/t>0$, stripes with wavelength $\lambda=12$ and $\lambda=16$ are also obtained. In all these cases, pair-pair correlations are highly suppressed with respect to the uniform state (obtained for large values of $|t^\prime/t|$), suggesting that striped states are not superconducting at $\delta=1/8$.

Published as SciPost Phys. 12, 180 (2022)

Author comments upon resubmission

Dear Editor,

we resubmit the manuscript, following the referees' comments and suggestions.

Your sincerely,
Luca F. Tocchio, on behalf of all the authors

List of changes

-) We included a brief description of the pair-density-wave scenario in the introduction, following a suggestion of the first referee.
-) We added in the introduction a comparison with the VMC work of the new Ref. 37, following a suggestion of the second referee.
-) We updated the variational wave function in the Method section, in order to describe also site-centered stripes.
-) We added a comment in the Method section, noticing that a superconductive state with the "antiphase" pairing, that can represent a pair density wave state, is never the optimal wave function.
-) We added a column to Table II, following a suggestion of the first referee.
-) We made the lines in the figures thicker, following a suggestion of the first referee.
-) We added a comment in the Results section, on the negligible difference between bond-centered and site-centered stripes when the wavelength is even, following a question of the second referee.
-) We updated the phase diagram, where now the (site-centered) stripe with wavelength 5 is present when t'/t<0. The inclusion of stripes with odd wavelength follows a question of the second referee.
-) We updated the phase diagram at U/t=12 and positive t'/t, where now the stripe with wavelength 12 is the optimal solution at t'/t=0.15 and t'/t=0.20.
-) We added two extra references, that is the new references 18 and 34.

Reports on this Submission

Anonymous Report 2 on 2022-5-6 (Invited Report)


I had a look at the replies and the modified paper.
The authors have responded to all concerns raised by the referee.
As I said in the previous report, the present work is one of the important pieces of recent intensive numerical investigation of the Hubbard model.
With additional calculations, the quality of the paper has improved.
I recommend that this paper be accepted for publication in SciPost.

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Anonymous Report 1 on 2022-4-29 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2111.04623v3, delivered 2022-04-29, doi: 10.21468/SciPost.Report.5005


The authors have substantially revised their manuscript and extended it with additional results. I find the new part on the comparison between site and bond centered stripes very interesting. The authors have also checked the variational energy of a pure PDW state. (In this context it would have been interesting to check also the energy of a PDW state coexisting with CDW and SDW orders, i.e. without setting \Delta_c= \Delta_s=0). In any case, the authors have addressed all points I raised in my previous report, and I can thus recommend this manuscript to be published in SciPost Physics.

There is one remaining small point which is not clear to me: on p5 the authors write: "Stripes with odd wavelength can only be site-centered". I do not see why they cannot be bond-centered (with a structure like odUdoodUdoodUdoodUdo… with U: large up spin, d: smaller down spin, o: a hole with a tiny up spin). I don't think this is energetically favored at 1/8 doping (maybe close in energy as in the lambda=6 case, or slightly higher as in the lambda=4 case), but it would be good to rephrase the statement (or explain why odd wavelength stripes cannot be bond-centered).

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