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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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|Authors (as registered SciPost users):||Vito Marino · Luca Fausto Tocchio|
|Preprint Link:||https://arxiv.org/abs/2111.04623v2 (pdf)|
|Date submitted:||2021-12-01 11:41|
|Submitted by:||Tocchio, Luca Fausto|
|Submitted to:||SciPost Physics|
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) 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. 7, 21 (2019)] shrinks to $\lambda=6$ for $U/t\gtrsim 10$. The latter value of the stripe wavelength is found to be stable in a large region of the phase diagram, within the $(t^\prime/t,U/t)$ plane. For $U/t=8$ and positive values of $t^\prime/t$, stripes with wavelength $\lambda=12$ and $\lambda=16$ are also obtained, while, for $U/t=12$ and negative $t^\prime/t$, the state with $\lambda=4$ can be stabilized. 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 $\delta=1/8$.
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- Cite as: Anonymous, Report on arXiv:2111.04623v2, delivered 2022-01-04, doi: 10.21468/SciPost.Report.4136
In this paper the authors study the extended doped Hubbard model for U/t = 8 and U/t=12 at hole doping 1/8 for different values of the next-nearest neighbor hopping t' with variational Monte Carlo, based on fermionic wave functions including a density-density Jastrow factor and backflow correlations. Their auxiliary non-interacting Hamiltonian includes charge and spin modulations of a bond-centered stripe of wavelength lambda as well as an antiferromagnetic and pairing contribution, besides the regular kinetic term. Their main results include that lambda increases with increasing t'/t and decreasing U/t (in agreement with previous studies), that the stripes are not superconducting, and that uniform states are stabilized at sufficiently large values of |t'/t|. A lambda=4 stripe is found over a certain negative t'/t range for U/t=12, whereas for U/t=8 it is absent.
This work, which builds upon a previous work by two of the authors for t'=0, provides an interesting contribution to the ongoing efforts in understanding the phase diagram of the Hubbard model and the competition between uniform and stripe states. It provides further support for the shift in stripe period as a function of t'/t and U/t, and of the existence of a uniform superconducting state at large values of |t'/t|. Whether stripes coexist with superconductivity (in particular away from one hole per unit length) is still controversial, thus the absence of superconductivity in all stripes is an important and interesting result.
While I believe this work will eventually be suitable for publication in SciPost, it has some weak points, listed below, which should be addressed and improved first.
(1) It seems the authors have missed the fact that an extensive VMC study on the t-t' Hubbard model was already performed in [K. Ido, T. Ohgoe and M. Imada, PRB 97, 045138 (2018)] (however, without including backflow correlations). The authors cite this paper in their introduction (Ref. ), however, as a QMC calculation, and not as a VMC study. It would be important to compare their findings with these previous results in their discussion.
(2) The authors consider bond-centered stripes in their study. While there is a rather strong consensus that the lambda=8 stripe at 1/8 doping is bond-centered, this is not necessarily true for other stripe periods (or other dopings); in previous works there is rather a tendency that site-centered stripes are stabilized. Have the authors checked that site-centered stripes have a higher variational energy than bond-centered stripes? Or are the energies similar? It would be important to verify and discuss this point.
(3) It is not clear to me whether the authors have also considered odd values of lambda, since they only present results for even values. Have the authors checked the variational energies of odd lambda values and found that they are higher? Or was there a particular reason to restrict the study to even values? This point should be discussed/mentioned in the paper.
- Cite as: Anonymous, Report on arXiv:2111.04623v2, delivered 2021-12-30, doi: 10.21468/SciPost.Report.4116
Hubbard model is one of the fundamental Hamiltonians for strongly-correlated systems. It is also one of the minimal Hamiltonians for cuprate high-Tc superconductors. Although the form of the Hubbard model is simple, its ground state property is still highly controversial.
Recent rapid improvement in computational resources and numerical techniques has enabled intensive research on the ground-state property of the Hubbard model. In particular, stripe states gather much attention as a candidate for the ground state.
In this paper, the authors investigate the stability of stripe states in the Hubbard model on the square lattice at 1/8 hole doping. The authors apply the variational Monte Carlo method using Jastrow and backflow correlation factors.
First, the authors study the U dependence at t’=0. Stripe states are stabilized for U>4, and a period-8 (period-6) stripe state is realized in the intermediate U (large U) region. Superconductivity is suppressed in the stripe states.
Then, the authors investigate t’ dependence for U=8 and 12. For U=8, they obtain period-6, 8, 12, 16 stripes with the period increasing as t’ increases. For U=12, period-4, 6, 8 states are stabilized. Again the period increases with increasing t’. The superconducting correlation function is suppressed in the stripe states as in the cases of t’=0.
The paper is clearly written, and the present work is one of the important pieces of recent intensive numerical investigation of the Hubbard model. Thus, I recommend that this paper be accepted for publication in SciPost.
Below, I list several comments.
1. The authors employ the backflow correlation factor, which is an off-diagonal correlation factor in Fock space. Therefore, it is expected to describe nontrivial correlation effects. Then, it would be interesting to investigate the effects of the backflow factor. In what kind of solution does the backflow factor become important? In other words, how much is the energy gain due to the backflow correlation factor? Is there a significant difference in energy gain between uniform and stripe states?
2. Is it meaningful to put optimized Delta_x and Delta_y values in Figs. 5 and 7? Are the uniform states superconducting in the entire region (the authors briefly discuss superconductivity in Fig. 9, but the parameter region is limited.)?
3. For general readers, I recommend the authors put discussions on the relevance to the pair-density-wave (PDW) scenario in the cuprate phase diagram.
4. In Table II, it would be helpful to show the energy difference between uniform and stripe states.
5. Throughout the paper, I recommend the authors make the lines thicker and the colors darker in the figures. The symbols are not very visible when printed.