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Metallic and insulating stripes and their relation with superconductivity in the doped Hubbard model

by Luca F. Tocchio, Arianna Montorsi, Federico Becca

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Submission summary

Authors (as registered SciPost users): Luca Fausto Tocchio
Submission information
Preprint Link:  (pdf)
Date accepted: 2019-08-01
Date submitted: 2019-07-16 02:00
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


The dualism between superconductivity and charge/spin modulations (the so-called stripes) dominates the phase diagram of many strongly-correlated systems. A prominent example is given by the Hubbard model, where these phases compete and possibly coexist in a wide regime of electron dopings for both weak and strong couplings. Here, we investigate this antagonism within a variational approach that is based upon Jastrow-Slater wave functions, including backflow correlations, which can be treated within a quantum Monte Carlo procedure. We focus on clusters having a ladder geometry with $M$ legs (with $M$ ranging from $2$ to $10$) and a relatively large number of rungs, thus allowing us a detailed analysis in terms of the stripe length. We find that stripe order with periodicity $\lambda=8$ in the charge and $2\lambda=16$ in the spin can be stabilized at doping $\delta=1/8$. Here, there are no sizable superconducting correlations and the ground state has an insulating character. A similar situation, with $\lambda=6$, appears at $\delta=1/6$. Instead, for smaller values of dopings, stripes can be still stabilized, but they are weakly metallic at $\delta=1/12$ and metallic with strong superconducting correlations at $\delta=1/10$, as well as for intermediate (incommensurate) dopings. Remarkably, we observe that spin modulation plays a major role in stripe formation, since it is crucial to obtain a stable striped state upon optimization. The relevance of our calculations for previous density-matrix renormalization group results and for the two-dimensional case is also discussed.

Author comments upon resubmission

Dear Editor,

We would like to resubmit the manuscript ``Metallic and insulating stripes and their relation with superconductivity in the doped
Hubbard model'' by L.F. Tocchio, A. Montorsi, and F. Becca to SciPost Physics.

We thank the Referees for their positive reports and for strongly recommending the paper for publication after some clarifications
have been provided. We submitted a reply to both the Referee reports, where we answer the questions raised by them; the revised version of the manuscript includes
all the pertinent modifications.

We also provide a point-to-point list of all the changes.

Sincerely yours,
L.F. Tocchio, A. Montorsi, and F. Becca

List of changes

List of modifications:

1) We changed Figs. 2, 3, 4, 5, 6, and 7 (and/or their captions) according to the suggestion of Referee 1 (points 1 and 2)
and Referee 2 (points 6 and 7);
2) We modified the Conclusions, according to the comments of Referee 1 (points 3 and 4);
3) We changed the "Variational Monte Carlo method" section, where the static structure factor is introduced, following the point 2 raised by Referee 2;
4) We added a few comments in the section "Results" in order to clarify the points 3 and 4 raised by Referee 2.
5) We added two extra references on the t-J model in the "Introduction" section and a related comment on the nature of the pairing term in the variational
state in the "Variational Monte Carlo method" section.

Published as SciPost Phys. 7, 021 (2019)

Reports on this Submission

Anonymous Report 2 on 2019-7-30 (Invited Report)


The authors have adequately addressed all of the points raised in my referee report and in the report of referee 1 and, in addition, have included all of the suggested improvements to the text and figures from both referees.

I recommend that this improved manuscript now be published in this form on SciPost.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 1 on 2019-7-17 (Invited Report)


The authors have revised and improved their paper according to all points of criticisms and I can thus recommend publication of this interesting article in SciPost in its present form.

  • validity: high
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: excellent

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