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DMRG study of strongly interacting $\mathbb{Z}_2$ flatbands: a toy model inspired by twisted bilayer graphene

by P. Myles Eugenio, Ceren B. Dağ

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Paul Eugenio
Submission information
Preprint Link: scipost_202009_00013v1  (pdf)
Date accepted: 2020-10-28
Date submitted: 2020-09-16 17:02
Submitted by: Eugenio, Paul
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Quantum Physics
Approaches: Theoretical, Computational


Strong interactions between electrons occupying bands of opposite (or like) topological quantum numbers (Chern$=\pm1$), and with flat dispersion, are studied by using lowest Landau level (LLL) wavefunctions. More precisely, we determine the ground states for two scenarios at half-filling: (i) LLL's with opposite sign of magnetic field, and therefore opposite Chern number; and (ii) LLL's with the same magnetic field. In the first scenario -- which we argue to be a toy model inspired by the chirally symmetric continuum model for twisted bilayer graphene -- the opposite Chern LLL's are Kramer pairs, and thus there exists time-reversal symmetry ($\mathbb{Z}_2$). Turning on repulsive interactions drives the system to spontaneously break time-reversal symmetry -- a quantum anomalous Hall state described by one particle per LLL orbital, either all positive Chern $|++\cdots+>$ or all negative $|--\cdots->$. If instead, interactions are taken between electrons of like-Chern number, the ground state is an $SU(2)$ ferromagnet, with total spin pointing along an arbitrary direction, as with the $\nu=1$ spin-$\frac{1}{2}$ quantum Hall ferromagnet. The ground states and some of their excitations for both of these scenarios are argued analytically, and further complimented by density matrix renormalization group (DMRG) and exact diagonalization.

Published as SciPost Phys. Core 3, 015 (2020)

Author comments upon resubmission

“The authors provided a simple model to capture the essential physics of two strongly interacting flat bands for the cases that the bands have identical (spin-1/2 QH) or opposite (Z2) Chern number. By studying the model numerically with a cylinder geometry, it was found that the gap to the first excited states remains finite for the Z2 case and it is significantly shrunk for the spin-1/2 QH case. Although the obtained results are rather obvious, the used numerical approaches would be useful information for future studies of quantum Hall system. Furthermore, the discussion on the nature of the excited states, especially a connection to 1D Ising-type spin physics, is interesting. Therefore, the contents of this manuscript seem to be enough for the publication in SciPost.”

Our reply: We thank the referee for recommending our Manuscript for publication in SciPost.

“However, I think that the discussion about numerical results is insufficient and ambiguous descriptions are seen in the presentation. Hence, I cannot recommend the publication in the present form. Then, I would like to request the improvement of the manuscript with addressing the following questions/suggestions.”

Our reply: We thank the referee for pointing out possible improvements for our Manuscript. We implemented the suggestions provided by the referee carefully and answered their questions in the following.

“- Is the number of LLL orbitals counted like N=LxLy/(2π) ? If yes, how can one take N=4,6,8, …, as shown in FIG.3 and FIG.4? Namely, what is the relation between N and Lx?”

Our reply: N is the number of orbitals in the lowest Landau level, which is proportional to the size of the 1D discrete chain after dimensional reduction technique is applied to the 2D problem in continuum. Thus N could be any positive integer number. The size of the 2D system in the x-axis, namely Lx, can be calculated from N=LxLy/(2π) for a given Ly. We briefly noted this in page 4 in Section III.

“- What is the definition of gap? Is it ΔE=E1−E0? Also, does the single-particle gap provide the same result in the Z2 case or not? It would be a useful information for DMRG calculation.”

Our reply: The gap is defined as the energy difference between the many body ground and first excited states. We note this in the captions of Figs. 3 and 5 as well as the main text in Section IV. In our revised Manuscript, we study the comparison between the energies of the first excited states and the lowest-lying spin-flip state which is a product state. For small Ly circumference values, the first excited state is indeed a product state where the spin-flip excitation is observed at the edge of the 1D chain. As Ly increases, an entangled state at the edge of the 1D chain becomes energetically more favorable compared to the edge spin-flip product state. Fig. 4 in our revised Manuscript shows this change in the nature of the first excited states with respect to Ly for orbital numbers 6 to 8 performed with exact diagonalization. This observation is intuitive and can be explained via the localization of the Wannier wavefunctions. For a fixed magnetic length l_B, as Ly increases, the distance between the centers of the wavefunctions decreases, thus giving rise to more significant overlaps between nearest neighbor Wannier wavefunctions. As a result of this decrease in localization, the first excited states become extended at the edge of the 1D chain. In Section IV, we discuss this observation and give an example of an extended first excited state based on ED data.

"- It is unclear how the parameters ls, lB are chosen in FIG.3 and FIG.4. For example, the used choice of lB=1 corresponds to B=1. Does it fulfill the condition that the gap between Landau levels is much larger than the interaction strength?"

Our reply: l_B, the magnetic length is set to 1 for all computations in the Manuscript. This choice sets the scale for the rest of the parameters in the problem. One can always scale the interaction strength accordingly to fulfill the condition that higher Landau levels are ignored thanks to exact flatbands of Landau levels. Scaling the interaction strength, and thus the Hamiltonian, since the interaction term is the only term in the Hamiltonian, only scales the absolute energies of the states. In the revised Manuscript, we plotted the gap of the Z2 Hamiltonian for different l_s, the range of Coulomb interaction (Fig. 10 in Supplement D). We find that the gap shrinks with increasing ls, and consequently the gap crashes for the Z2 system with screening length ls=3 l_B, resulting in a ground state with charge accumulation at the edges and center. However the gap is still visible beyond ls=1l_B, as seen in data for ls=1.5. We conclude that while a stable gap -- between the fully polarized ground states and the edgelike excited states -- is persistent for different interaction ranges, the nature of the ground state completely changes after some interaction range to a more fluidlike state with charge accumulation at the edges from a uniform one particle per site incompressible state. Nonetheless, the fully polarized states remain eigenstates for ls=3l_B, albeit at a higher energy. For these reasons, it is likely the change in the nature of the ground state due to increasing the screening length is a consequence of the open boundary conditions. iDMRG or a tunable confining potential, the latter which could be tuned to increase the energy of the edge states, are potential ways to regulate this change. Investigation of whether there is indeed a critical screening length between two different phases, and how it responds to a confining potential, is beyond the scope of our paper. Thus we leave this as a future study.

"- In FIG.3, why the gap is larger for larger Ly, even though the gap between Landau levels typically depends only on B?"

Our reply: We thank the referee for this important question. The calculations produce an additional factor of Ly as given in Supplement A due to normalization of the wavefunctions. We replotted our data with the rescaling performed correctly. Figs. 3 shows the updated gap values for Z2 Hamiltonian for both contact and screened Coulomb interactions as a function of total orbital number N. In both cases, we observe that the gap approaches the same value for different Ly as N increases. The results for spin half Hamiltonian are also updated with the correct scaling, which can be seen in Figs. 5. Here the main conclusion does not change: the gap decreases as N increases for all Ly with a similar trend.

“-Related to the above question, the data only for two circumferences may be not enough to confirm the saturation of the gap in the thermodynamic limit. If possible, it would be better to add one more data for another Ly even if it is smaller than Ly=15”

Our reply: As suggested by the referee, we computed another two sets of Ly value, which is Ly=10 and Ly=8, for Z2 Hamiltonian with screened Coulomb interaction. All our data exhibits the same trend, suggesting a gap in thermodynamic limit. For spin half Hamiltonian we computed another three sets of Ly values. In addition to Ly=20 and Ly=15, now we also have Ly=8,10 and 12. From the consistent behavior observed from these five data sets, the gap significantly shrinks as the orbital number increases in the spin half QH system.

“- In FIG.3, one may naively think that the gap for the short-range contact-like interaction is larger than that for the screened Coulomb interaction but the result is opposite. Is the limit ls→0
of Vsc quantitatively connected to Vδ?”

Our reply: We thank the referee for this interesting question. Indeed the limit ls goes to zero of Vsc is quantitatively connected to Vdelta. We derived the relation between them in Supplement D under two assumptions, see Eq. 25. The two are equal to each other up to a coefficient which explains why the gap for contactlike interaction is smaller than the gap for screened Coulomb interaction with ls=1l_B. It should be understood from Fig 10 (supplement of updated Manuscript) that the energy cost of a “spin-flip” at the edge grows with shrinking screening length, and therefore the gap is largest in the limit ls→0.

“- Is the caption of FIG.4 correct? What does "plotted as (ΔE)−1 mean?
-The horizontal axis of FIG.4 should start from 1/N=0”
Our reply: We fixed these typos.

"If possible, it would be better to discuss the gap in the thermodynamic limit."
Our reply: We extended the discussion of our results for both Hamiltonians in Section IV and discussed the presence or absence of a gap in the thermodynamic limit. The related discussion for Z2 and spin-half QH Hamiltonians are at pages 6 and 7, respectively.

“- Which parameter value is used in FIG.5 and FIG.6?”
Our reply: Fig. 6a and b in revised Manuscript are plotted for Z2 Hamiltonian with contact-like interaction. In the revised manuscript, we noted this both in the caption and the main text. Additionally in the revised manuscript, we looked for inverse participation ratio greater than 0.99, to ensure that we are counting only the product states.

“- In FIG.5 and FIG. 6 the data for Ly=2 seems to deviate from the systematic size dependence. Is a periodic boundary indeed applied in the y-direction for Ly=2 namely, is the hopping taken twice?”
Our reply: Periodic boundary conditions are taken around the cylinder. As discussed in the main text and previous replies, the localization of the LLL’s is controlled by the ratio Ly/l_B. Since we fix l_B=1, scaling the cylinder circumference has the effect of controlling the localization. In the limit of an infinitely thin cylinder, we find that the LLL’s are significantly separated, and consequently the entire spectrum (determined by using ED) becomes composed of product states, reminiscent of the Ising Hamiltonian with no external fields (classical limit). As Ly increases, so does the overlap of the LLL wavefunctions, and consequently states composed of entangled orbitals appear around Ly \sim 2\pi. This behavior was discussed in Section IV in detail for the first excited state, which is a product state until Ly\sim 2\pi, and then is replaced by an entangled state for larger Ly (Fig. 4). The product state which is the first excited state for Ly<2\pi, still remains an eigenstate independent of Ly, albeit at a higher energy. Thus the total number of product excited states grows with N regardless of Ly (Fig 6a) but their fraction compared to the number of entangled states shrinks with larger Ly (Fig 6b). Due to this reason, we do not expect that Ly=2 data should have a similar trend with L=8 data. For Ly=2, the increase in the product states as a function of N is faster compared to Ly=8, however the decrease in its r value (ratio of product states to the dimension of Hilbert space) is significantly slower than Ly=8. We noted this observation at page 8, Section V.

"- In FIG.7, it would be helpful for readers to put numbers in the color bar to estimate how strongly a spin is localized."
Our reply: We have made the requested change. Please see the updated figure.

"- The accuracy of DMRG should be written, even if the discarded weight is negligible."
Our reply: We have included the limits on the bond dimension and the cutoff for DMRG of Z2 and spin-half in the captions of Fig 3 and 5, respectively.

List of changes

We defined and motivated all parameters as clearly as possible in the revised Manuscript. We expanded the DMRG results section (Sec. IV) and the section on excited states (Sec. V), in addition to updating our Figures that plot the gap from ground state to the first excited states in Z2 Hamiltonian with the correct Ly scaling. As the referee suggested, we discussed the gap in the thermodynamic limit for both models. We improved the presentation of our Manuscript and the captions of the figures as much as possible. We fixed typos and updated Supplement sections A and B; additionally added new Supplement sections C and D. We improved our understanding of the excited states in Z2 Hamiltonian. We thank the referee for their helpful comments which helped us to improve our Manuscript significantly. The revised Manuscript is also uploaded on Arxiv

Reports on this Submission

Anonymous Report 2 on 2020-10-20 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202009_00013v1, delivered 2020-10-20, doi: 10.21468/SciPost.Report.2104


1. hot topic 2. presents unbiased numerical results


1. overall findings are in line with previous analytical and numerical investigations: limited novelty of results


In the manuscript two interacting flat topological bands are considered in the situation where the Chern numbers of the two bands are either equal or opposite. Analytical considerations are combined with numerical DMRG results for two different forms of the interaction Hamiltonian. The authors find that for opposite Chern number a quantum anomalous Hall state forms, for equal Chern number a ferromagnet and present their results in quite some detail. The overall findings are in line with previous analytical and numerical investigations, for instance Ref. 16 and 17 in the manuscript in the context of twisted bilayer graphene, with differences in the detailed assumptions on interactions, (lack of) kinetic terms and numerical approaches. I therefore recommend publication in SciPost Core.

  • validity: good
  • significance: good
  • originality: ok
  • clarity: good
  • formatting: good
  • grammar: excellent

Author:  Paul Eugenio  on 2020-10-27  [id 1019]

(in reply to Report 2 on 2020-10-20)

We are thankful to the referee for their feedback on our manuscript. While it is true that our results verify both the analytic mean field theory (Ref 16) and previous numerical results (Ref 17) for the anomalous Hall ferromagnetism in twisted bilayer graphene (TBG), and have done so without introducing a bandwidth, such a perspective does not underscore the full potential that our simple model (Eqn 9) provides.

Indeed, because of its simplicity, our model has allowed us to study the problem of strongly interacting Chern bands for systems with much larger cylinder circumferences than Ref 17: Ly=8 to 20 magnetic lengths compared to their Ly=3 unit cells. And this being done with a matrix product state (MPS) bond dimension many orders of magnitude smaller than Ref 17 (see Fig 3 of the Arxiv version of their paper). In fact, despite being in the limit where the ratio of bandwidth-to-interaction-strength is zero, i.e the extremely strong coupling limit, the bond dimension of our ground states remains 1; as opposed to Ref 17, where the bond dimension grows with the inverse of that ratio. This is a consequence of the hybrid momentum-real space representation of the Landau-gauged wavefunctions, used in the construction of our model. Such a representation provided us an advantage in representing the anomalous Hall, in comparison to the tight-binding model used in Ref 17. Moreover, it has allowed us to study the nature of the lowest energy excitations, leading us directly to a (so far as we know) undiscovered connection between a two-dimensional strongly interacting topologically non-trivial system of electrons and one-dimensional spin-Ising physics bearing that same topology.

In this way, what the referee refers to as a “lack of kinetic terms” -- which is more correctly a lack of bandwidth, as kinetic energy is implicit in the construction of Eqn 9 -- appears to be an advantage in the limit of strong interactions. If we were to introduce a periodic potential, and therefore induce a bandwidth, our bond dimension could only grow, opposite and hence complementary to the bond dimension scaling of Ref 17. Nonetheless, our model successfully captures the spontaneously broken anomalous ferromagnetic ground state without introducing a finite bandwidth, highlighting the importance of the topology in the formation of it. Introducing a small but finite bandwidth would only shrink the gap, and thus we expect the ground states to remain fully polarized, assuming the bandwidth is not too large.

These explanations are briefly mentioned at the end of the Introduction in our revised Manuscript (page 3).

Beyond the clear numerical advantage afforded by our toy model Eqn 9, its ability to concisely capture the non-trivial topology makes it a valuable platform for the multiple burgeoning topics in topological many-body physics, including the question of the possibility of the fractional quantum Hall in the presence of time-reversal symmetry. As outlined in our conclusion, if a tunable potential were to be introduced which biases one set of lowest Landau levels over their time-reversal partners, a phase diagram could be mapped in the space of that tunable potential. In the limit where one lowest Landau level is prohibitively expensive to occupy -- defined when the strength of the tunable potential is much larger than the interaction strength -- our model Eqn 9 would reduce to a well-studied quantum Hall Hamiltonian (see Eqn 2 in Ref 50, Arxiv). It is therefore guaranteed to exhibit fractional quantum Hall physics at various electron densities (e.g \nu=1/3) in this limit. And it has been shown previously (see Ref 47-51) that DMRG is capable of capturing those states. Understanding how these states evolve as one changes the tunable potential could provide invaluable insight into the nature of the fractional quantum Hall in Moire systems.

Additionally, it has been argued (by Ref 24) that degenerate Landau levels with zero bandwidth and time-reversal symmetry have the potential to form excitonic Laughlin states, given the intra-valley repulsion is tuned sufficiently larger than the inter-valley repulsion. Since the intra and inter-valley scattering is explicit in Eqn 9, one need only tune their relative strengths in order to enter into a regime where such excitonic states are energetically competitive with the spontaneously valley (Chern) polarized ground states; the latter which we’ve shown here are products states in the Landau gauge.

The potential uses for this model go beyond TBG and equilibrium physics. As shown in a previous work Ref 58 authored by one of us, the degeneracy structure of a spectrum has profound effects in dynamics of information scrambling probed by out-of-time-order correlators (OTOC). Because the ferromagnetic regime of the 1D transverse-field Ising model (TFIM) is dual to a 1D topological superconductor (the Kitaev chain), the presence of edge states results in a topological degeneracy which exists throughout the spectrum, and such a degeneracy inhibits information scrambling of edge operators, even at infinite temperature. How this extends to higher than one dimension has yet to be studied, in part hindered by the growth of the bond dimension in time. We’ve shown in our manuscript that our integer-filled Z2 Hamiltonian on open cylinders likewise exhibits topological degeneracy throughout the spectrum; and because of its connection to 1D Ising spin physics, as well as the presence of low-bond dimension excited states, our model may be one of the best candidates for numerically exploring information scrambling and in general nonequilibrium quantum physics in two dimensions.

Given the potential applications of our simple model in different research subfields, we believe that it is best suited for SciPost Physics. We again thank the referee for their time and comments, which have motivated us to further stress the advantages of our toy model, and we have updated our manuscript to follow.

Anonymous Report 1 on 2020-10-12 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202009_00013v1, delivered 2020-10-12, doi: 10.21468/SciPost.Report.2072


1. The given toy model for strongly interacting opposite-Chern flat bands is simple to understand.
2. A connection between the excited states and 1D Ising-type spin physics is interesting.
3. A useful information about numerical approach to quantum Hall system is provided.
4. The performed numerical calculations and their analyses are reliable.


1. Part of numerical results are rather obvious (even easily expected).


I am satisfied by the author’s reply. By addressing all of my suggestions and comments, the manuscript has been sufficiently improved. The main improvements are the followings:
1. With additional data the finite-size scaling of gap became more convincing.
2. By comparing the energies of a lowest lying product state and a first excited state, a deeper and intuitive insight into the localization of the Wannier wavefunctions was given.
3. The new supplemental information improved the understanding of the excited states in the $\mathbb{Z}_2$ Hamiltonian.
Therefore, I recommend the publication of this manuscript as a SciPost paper in the present form.

Requested changes

I request no further changes.

  • validity: high
  • significance: good
  • originality: ok
  • clarity: high
  • formatting: good
  • grammar: good

Author:  Paul Eugenio  on 2020-10-27  [id 1020]

(in reply to Report 1 on 2020-10-12)

We thank the referee for recommending our Manuscript for SciPost Physics and taking the time to provide us valuable feedback, therefore helping us to significantly improve the paper.

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