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An effective curved space-time geometric theory of generic twist angle graphene with application to a rotating bilayer configuration

by Jia-Zheng Ma, Trinanjan Datta, Dao-Xin Yao

Submission summary

As Contributors: Trinanjan Datta · Jia-Zheng Ma
Preprint link: scipost_202109_00030v1
Date submitted: 2021-09-28 05:23
Submitted by: Ma, Jia-Zheng
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
Approaches: Theoretical, Phenomenological


We propose a new kind of geometric effective theory based on curved space-time single valley Dirac theory with spin connection for twisted bilayer graphene under generic twist angle. This model can reproduce the nearly flat bands with particle-hole symmetry around the first magic angle. The band width is near the former results given by Bistritzer-MacDonald model or density matrix renormalization group. Even more, such geometric formalism allows one to predict the properties of rotating bilayer graphene which cannot be accessed by former theories. As an example, we investigate the Bott index of a rotating bilayer graphene. We relate this to the two-dimensional Thouless pump with quantized charge pumping during one driving period which could be verified by transport measurement.

Current status:
Editor-in-charge assigned

Reports on this Submission

Anonymous Report 4 on 2021-12-29 (Contributed Report)


Please see the attached.


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

Anonymous Report 2 on 2021-12-28 (Invited Report)


The authors of the current manuscript claimed to derive an effective theory for twisted bilayer graphene with arbitrary twisted angles. They also claimed to obtain the theory of rotating bilayer graphene with the twisted angle depending on time.

I found the manuscript has many misleading claims and equations that I will comment on in detail below. Therefore, I do not recommend this paper to publish on SciPost Physics.

1) The first and most confusing equation is the proposed action (4.6a)
-There is no definition of the 4 components fermion field. I guess $\psi=(\psi^{up}_A, \psi^{up}_B,\psi^{down}_A, \psi^{down}_B )$, with A and B are the usual sublattice indices.
-What is the definition of $\gamma^\mu$ in the curved space-time, what is the definition of $\gamma^\mu$ that the authors used in numerical simulations?
-How can one propose the 3+1D Hamiltonian for a purely 2+1D bilayer graphene? In order to have an effective theory as (4.6a), the UV scale (defined lattice size in the z-direction) should be much different from the IR scale (defined by the thickness in the z-direction). But in bi-layer graphene, the thickness and the lattice size in the z-direction are the same. What is the definition of the derivative in the $z$ direction?
-There is no microscopic derivation of (4.6a). The effective theory in strained graphene Refs [52] and [66] comes directly from Tight Binding (TB) Hamiltonian. The effective theory in Ref [15] comes from the expansion about the K point in the momentum space after analyzing the TB Hamiltonian. On the other hand, the action (4.6a) comes from nowhere. How can the authors show that the effect of the general deformation $u^a$ gives this Hamiltonian? If it is a guess, then it is a poor guess.
-If my guess $\psi=(\psi^{up}_A, \psi^{up}_B,\psi^{down}_A, \psi^{down}_B )$ is correct, the upper layer electrons only live on $z=h/2$, the lower layer electrons only live on $z=-h/2$, in the 3+1D action, how come they are coupling with each other? If my guess is incorrect, then what is the definition of $\psi$?
-There is no definition of $\bar{\psi}$, I guess $\bar{\psi}=\psi^\dagger \gamma^0$, then the mass term in (4.6a) only capture the AA tunneling without AB tunneling (if $\psi=(\psi^{up}_A, \psi^{up}_B,\psi^{down}_A, \psi^{down}_B )$). As we know, the AB tunneling is the key to obtaining the flat bands in twist BG, one can look at Ref [13] for detail. So why did the authors of this paper ignore the AB tunneling?
2) What is the meaning of equation (4.5a)? $t_{ij}$ is defined as the tunneling parameter between electrons at site $i$ to site $j$. In (4.5a), $\Sigma^{bc}$ acts on the spinor indices as defined in equation (4.4). So the left-hand and the right-hand sides of the equation (4.5a) are different quantities?
3) The author claimed that their theory can capture large twisted angles, however, in this case, the displacement field $u^a$ varies quickly in space. One needs to consider the coupling between valleys $K$ and $K’$ as in the case of Kukule distortion. In the current manuscript, the authors only consider the effective theory around $K$, which is inconsistent.
4) In the manuscript, there is a very important quantity, which is $\Gamma$ that represents the broadening of the deformation domain wall. The band structure depends heavily on this quantity. In the previous formalism for twisted bilayer graphene, this quantity wasn’t introduced. The authors claimed that this is a phenomenological quantity that depends on photon, impurity, and fluctuation. So at exactly zero temperature, where phonon and fluctuations can be ignored, the band structure can change by an order of magnitude by changing the impurity density? The band structure in the very clean limit (almost no impurities) and the band structure with a small amount of impurity are totally different? I don’t think so.
5) Figure 5 is misleading, the effective theory only works at a small energy limit. In graphene, the expansion around the $K$ point works relatively well up to the energy of about 1eV. But in Fig 5, the author shows the energy scale of order 5-50 eV. This figure makes no sense physically.

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

Anonymous Report 1 on 2021-12-27 (Invited Report)


The authors propose an effective theory for twisted bilayer graphene, consisting of Dirac fermion in curved spacetime. Compared with previous works, it has the following advantages:

(1) can describe the cases of large twist angles where translation symmetry is generically absent,

(2) can incorporate large deformation gradients which is usually not allowed in the continuum description, and

(3) is capable of describing out-of-equilibrium systems such as rotating bilayer graphene.


(1) Important concepts are undefined and the presentations are not clear enough. For example, "deformation singularity" and "domain walls" in section 3 appear without explanations, while the ability to describe the existence of such singularities is one of the key advantages of the current theory over the previous small-deformation-gradient theories.

(2) If I understand correctly, the authors treat one of the two layers as static, which serves as a geometric background for the other layer. This key approximation is not clearly stated in the manuscript and its validity is not discussed either.

(3) It is hard for the readers to connect the results in this paper with the previous theories. For instance, which quantities correspond to the parameters w_0 and w_1 in previous treatments, (or u and w, using the conventions of Ref. [15]) ?

(4) The current theory excludes strains, which are common in experiments and treatable in other theoretical formalisms. Given that the model without strains seems already complicated computational-wise, I doubt it will be an economic description of the general TBG physics.


Based on the strengths and weaknesses described above, I suggest some minor revisions before recommending it for publication.

Requested changes

What were discussed in items (1-3) of the "weakness" section correspond to the requested changes. In addition, here are some optional changes:

(1) It seems to me that equations (4.7)(5.2)(5.3)(5.4) can be largely simplified, by writing for example U_x, U_y just as U_i and allow i=x,y.

(2) Since this theory is capable of dealing with incommensurate twisting angles, I wonder if it would be easy to study the 30-degree case, where the system is a quasicrystal, reproduce the existing results and generate new ones. See for example [] and [].

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

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