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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
This is not the current version.
|As Contributors:||Luca Fausto Tocchio|
|Submitted by:||Tocchio, Luca Fausto|
|Submitted to:||SciPost Physics|
|Domain(s):||Theor. & Comp.|
|Subject area:||Condensed Matter Physics - 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 are also discussed.
Submission & Refereeing History
Reports on this Submission
Anonymous Report 2 on 2019-6-14 Invited Report
In this manuscript, the authors describe variational Monte Carlo
calculations on the two-dimensional Hubbard model treated in ladder
geometry (actually, toroidal geometry with the x- spatial direction
taken to be much larger than the transverse y-direction). The authors
confirm and build on previous numerical results (in particular, those
in Ref. ) with an independent method. These calculations find
that completely filled and thus insulating stripe configurations
dominate at moderate to strong Coulomb interaction strength and 1/8
doping away 1/2 filling. In addiition the authors' more extensive
treatment of the effect of changing the band filling on the stripe
configurations and pairing strength adds to the picture and makes
important progress towards determining the ground-state phase diagram
as a function of doping for the two-dimensional Hubbard model, as
depicted by the authors in Fig. 8. In particular, that filled stripes
and thus non-superconducting behavior is dominant at particular
commensurate band fillings, but that strong superconducting
correlations can occur at more generic fillings, and that the stripe
order melts at sufficiently large doping is an important result.
Aside from a few details mentioned below, I feel that the work in the
has been very carefully carried out and that the results are very
clearly depicted and described. Thus, I strongly recommend that this
work be accepted for publication on SciPost Physics.
I do, however, also have the following detailed comments, most of
which are relatively minor and which I hope would improve the
readability of the manuscript and the support for the calculation
1) The authors have treated lattices (primarily) with L_y = 2, 6,
10. Is there a particular reason that L_y = 4, 8 were left out? In
particular, L_y = 4 might be useful to compare directly with other
work such as that in Ref. .
2) I'm not convinced that the low-q behavior of the charge structure
factor is such a definitive probe of the metallicity; it
esssentially just probes the behavior of the integral of the
charge-charge correlation function. It cannot distinguish between
normal metallicity and superconductivity. A more transport-oriented
measure such as the Drude weight or the electric susceptibility
would be better. In addition, for the low-q behavior of the
structure factor, it would be good to carry out a finite-size
scaling by studying the behavior of the lowest non-zero q point as
a function of L_x.
3) I think that some details of the authors description of the
variational energies, Fig. 2 and Table 1, can be tightened up. It
seems to me that the energies quoted must be intensive, i.e.,
energy per site. If so, this should be stated explicitly.
In Table 1, how are the errors in the energies estimated? (It is a
little suspect that they are all exactly the same on an absolute
scale, for all states and fillings.
Also, can the variational energies be compared to other variational
calculations, such as the DMRG calculations in Ref. ? (Here
considerations of the effect of boundary conditions as well applied
extrapolations would presumably complicate the matter.)
4) In the variational state, it is not completely clear to me what
effect allowing/turning on the pairing correlation have on the
results. On the one hand, we have statements that the metallicity
and pairing are necessarily suppressed in the filled stripe
states. On the other, we have statements (e.g., on p. 10, last
paragraph) that various pairing correlations can be induced
without significantly changing the energy. Presumably the latter
statement is only a description of the initial variational state,
which is then modified by the Jastrow factors, Monte Carlo
optimization, and back flow. But a more clear statement could be
made about the instrinsic robustness of the pairing in the
5) I feel that the analysis of the pairing correlations, Fig. 7, is
a little too casual. In particular, what is important is the
exponent of the power-law decay at moderate to large distance,
assuming that they can be sufficiently accurately calculated at
these distances. For the stripe states, the pairing is always
suppressed, even at short distance, but this can be viewed as a
prefactor that multiplies the asymptotics. The authors should
comment more directly and quantitatively on the asymptotics, if
possible, or describe why the asymptotics cannot be reliably
extracted if not.
6) In Fig. 2, it would be better to label the x axis with 1/10, 1/6,
1/2, rather than decimal notation and/or list the treated M-values
7) It might be useful to remind the reader of the treated optimal
stripe wavelengths in the legend or caption of Figs. 5, 6, 7, and
Table 1 to make interpretation easier for the reader, even though I
realize that the optimal stripe wavelengths are mentioned multiple
times in the text.
Anonymous Report 1 on 2019-6-12 Invited Report
In this paper the authors study the doped Hubbard model by means of the variational Monte Carlo (VMC) approach based on Jastrow-Slater wave functions, improved by the inclusion of backflow correlations. They find an insulating stripe ordered state with a period 8 in the charge order at 1/8 doping, in agreement with other numerical methods from other recent works. The period 8 stripe is also the lowest energy state at smaller doping (1/10 and 1/12), whereas at doping 1/6 they find an insulating period 6 stripe. In between these dopings the stripes exhibit strong superconducting correlations. The authors report that the spin modulation is crucial in order to obtain the stripes in this work; whereas without the spin modulation uniform states are favored (which is the reason why uniform states have been predicted in previous works based on VMC). They also find that stripes with in-phase pairing are lower in energy than the ones with anti-phase pairing, and that uniform states become favorable for dopings larger than 0.2.
Understanding the ground state phase diagram of the 2D Hubbard model has been a major and central challenge since many decades. While 10 years ago there was a huge discrepancy between the results obtained with various numerical approaches, the situation has changed in recent years and there is now a growing consensus, at least regarding certain aspects of the phase diagram. With this work the authors provide an important contribution in this context, in particular they strengthen the recent result that the ground state of the doped 2D Hubbard model in the strongly correlated regime is a stripe state, and not a uniform d-wave superconducting state, which has been an open and very controversial question for many years. Having additional support from VMC regarding this question is another milestone towards a full solution of the 2D Hubbard model.
Besides this main result, the authors provide other interesting results, e.g. that the period 8 stripe is also the lowest energy state at smaller dopings, and the nature of the stripes (insulating/metallic/superconducting) for different dopings. It is also very interesting to finally understand, why stripes have not been predicted in previous VMC studies (because in previous works the spin order has not been included in the variational optimization). The paper is also well written and the results are presented in a clear way.
For these reasons I can definitely recommend publication of this interesting article in SciPost. I only have minor points for improvement and questions attached below.
Questions and comments:
1. The arrangement of the figures could be improved; there are sometimes several pages between a figure and the text describing it. A possible solution could be to make the figures more compact, e.g. putting the two panels in Figs. 3, 5, 6 next to each other rather than on top of each other, and then combine more figures on one page.
2. The caption of Fig. 4 is a bit cumbersome to read with the listing of all plot symbols. Putting a legend in each figure panel would make it more readable and more compact.
3. I did not quite understand the following statement in the conclusions part: "A weakly metallic behavior with no superconducting correlations may be related to the proximity to the phase separation region that, within the variational Monte Carlo accuracy, is present close to half filling."
Could the authors explain in more detail what they mean by this statement?
4. Is it in principle possible to further improve the results and check the stability of the stripe order using additional Lanczos steps and combination with fixed-node Monte Carlo? (I am not requesting this for this work, but just to know if there would be further room for improvement).