Flat Bands and High Chern Numbers in Twisted Multilayer Graphene
Mengxuan Yang
TL;DR
This paper analyzes the electronic properties of twisted multilayer graphene, revealing that flat bands with high Chern numbers occur at specific 'magic' angles, with implications for topological phases.
Contribution
It extends the chiral model to multilayer graphene, showing the equivalence of magic angles with bilayer cases and constructing flat band eigenfunctions with nontrivial topology.
Findings
Magic angles are identical to those in twisted bilayer graphene.
Flat bands have Chern number -n, indicating nontrivial topology.
Band separation at Dirac points varies with tunneling strength.
Abstract
Motivated by recent Physical Review Letters of Wang-Liu and Ledwith-Vishwanath-Khalaf, we study Tarnopolsky-Kruchkov-Vishwanath chiral model of two sheets of -layer Bernal stacked graphene twisted by a small angle using the framework developed by Becker-Embree-Wittsten-Zworski. We show that magic angles of this model are exactly the same as magic angles of chiral twisted bilayer graphene with multiplicity. For small inter-layer tunneling potentials, we compute the band separation at Dirac points as we turning on the tunneling parameter. Flat band eigenfunctions are also constructed using a new theta function argument and this yields a complex line bundle with the Chern number .
| 0.586 | 2.22 | 3.75 | 5.28 | 6.79 | 8.31 | |
|---|---|---|---|---|---|---|
| 0.2345 | 0.0542 | 0.0033 | 0.0022 | 0.0013 |
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Taxonomy
TopicsGraphene research and applications · Topological Materials and Phenomena · Quantum and electron transport phenomena
Flat Bands and High Chern Numbers in Twisted Multilayer Graphene
Mengxuan Yang
Department of Mathematics, University of California, Berkeley, CA 94720, USA.
Abstract.
Motivated by recent Physical Review Letters of Wang–Liu [WL22] and Ledwith–Vishwanath–Khalaf [LVK22], we study Tarnopolsky–Kruchkov–Vishwanath [TKV19] chiral model of two sheets of -layer Bernal stacked graphene twisted by a small angle using the framework developed by Becker–Embree–Wittsten–Zworski [BEWZ22]. We show that magic angles of this model are exactly the same as magic angles of chiral twisted bilayer graphene with multiplicity. For small inter-layer tunneling potentials, we compute the band separation at Dirac points as we turning on the tunneling parameter. Flat band eigenfunctions are also constructed using a new theta function argument and this yields a complex line bundle with the Chern number .
1. Introduction
When two or more sheets of graphene are stacked on top of each other and twisted, it has been observed that at certain angles, namely magic angles, the zero energy band becomes flat and the system becomes superconducting.
In this paper, we consider the chiral model [SJGG12] [TKV19] of two sheets of -layer Bernal stacked graphene twisted by a small angle [WL22]. The twisted multilayer graphene (TMG) Hamiltonian is given by
[TABLE]
where
[TABLE]
with and
[TABLE]
In particular, , and
[TABLE]
In equation (1.2), (resp. ) denotes the tunneling between the top (resp. the bottom) -th layer and -th layer. Without loss of generality we can assume that all ’s are non-zero, since otherwise we have direct sum decomposition of into smaller matrix blocks.
In the corresponding physics model, when two honeycomb lattices are twisted against each another, a periodic honeycomb superlattice, called the moiré lattice, is formed. Bistritzer–MacDonald [BM11] predicted that the symmetries of the periodic moiré lattice in twisted bilayer graphene (TBG) lead to dramatic flattening of the band spectrum. The chiral model of TBG (1.5) was obtained by Tarnopolsky–Kruchkov–Vishwanath [TKV19] by removing certain interaction terms from the operator constructed in [BM11]; it was recently studied in greater mathematical details by the work of Becker et al. [BEWZ22] and Becker–Humbert–Zworski [BHZ22a] [BHZ22b]. Recently, Wang–Liu [WL22] and Ledwith–Vishwanath–Khalaf [LVK22] generalized the chiral TBG model to chiral TMG models which also have the ideal quantum geometry and support high Chern number band,
Remark 1**.**
For , the operator (1.1) reduces to the chiral twisted bilayer (TBG) model
[TABLE]
independent of , see [TKV19] [BEWZ22] [BHZ22a] [BHZ22b] for more details. The dimensionless parameter in (1.1) and (1.5) is essentially the reciprocal of the twisting angle.
Note that the potential (1.4) satisfies the following properties:
[TABLE]
In particular, it is periodic with respect to the lattice with
[TABLE]
The dual lattice of consists of satisfying
[TABLE]
We consider a generalized Floquet condition under the modified translation operator , with (defined in equation (2.3)), i.e., we study the spectrum of satisfying the following boundary conditions:
[TABLE]
The spectrum is symmetric with respect to the origin due to chiral symmetry and we index it as
[TABLE]
see §3.1 for details. The points are called the Dirac points and are typically denoted by and in the physics literature.
For satisfies equation (1.9) of , we define the following set of complex numbers:
[TABLE]
The magic angles are essentially the reciprocals of ’s. And is call simple if and multiplicity of is defined to be the number of ’s () such that .
Our first theorem states that the magic angles of the TMG Hamiltonian coincides with magic angles of the TBG Hamiltonian .
Theorem 1**.**
The set is independent of . More precisely, the multiplicity of each is independent of .
Therefore we can define . This result is essentially contained in the paper [WL22]. A spectral theory characterization of the set (for TBG) was given in [BEWZ22] using a Birman–Schwinger operator whose spectrum is given by .
Our second result studies the role of the tunneling parameter for . When , the operator can be decomposed into direct sums of one operator and operators , where the latter give rise to Dirac cones independent of . The Dirac cones are connected to the band for any . In particular, for , turning on the tunneling parameter such that for results in these Dirac cones separate from the flat band and a band gap is formed. The band separation is studied for :
Theorem 2**.**
Assume is simple. For and at , the first two eigenvalues of are given by and .
Remark 2**.**
The constant is computed in Proposition 5.3. The same result is expect to be true for . We only present the case here for notatinal simplicity purposes. This also gives an explicit result for the band separation mechanism discussed in [WL22].
Using a new theta function argument and the resolvent formula for , for simple, we construct flat band eigenfunctions of . This gives rise to a holomorphic line bundle (see discussions in §6.3 for more details).
Theorem 3**.**
The Chern number of the line bundle L is given by .
This justifies the high Chern number observation made in [WL22] and it relies on a numerical calculation of the integration of the flat band eigenfucntions (see Table 1). In [WKL22], the flat band eigenfunctions are considered as a section of rank holomorphic vector bundle. Our new theta function argument gives an explicit analytic construction and answers a question proposed in [LB22].
We conclude our introduction by discussing its relation to physics:
Integer quantum Hall effect. For two dimensional electron gas in cold temperature, Hall conductance exhibits some plateaus at integer multiples of as the magnetic field varies. It turns out that the Hall conductance at the plateaus is related (via the Berry curvature) to Chern number of the underlying Hermitian line bundles arising from ground state eigenfunctions. By Kubo’s formula, the Hall conductance is given by
[TABLE]
The higher Chern number in the TMG model can yield higher Hall conductance in this case.
Ideal Quantum Geometry and Fractional Chern insulators. Another observation (cf. [WL22]) is that flat band eigenfunctions constructed in (6.9) are holomorphic in . Such flat band is said to satisfy the ideal quantum geometry in physics literature (cf. [CLT*+*15][OM21][MO21] [WCM*+*21]), in the sense that the Berry curvature on the fundamental domain is non-vanishing and proportional to a Fubini–Study metric on induced by the flat band eigenfunctions.
Such flat band was predicted and verified to have Fractional Chern Insulators (FCI) [RB11] [BL13], which are lattice generalizations of the conventional fractional quantum Hall effect in two-dimensional electron gases. For example, FCI can appear at band fillings given by
[TABLE]
with such that . Here (resp. ) corresponds to Abelian states (resp. non-Abelian states) and is even (resp. odd) for bosons (resp. fermions). The stability of FCI are conjectured due to the ideal quantum geometry (cf. [TKV19] [WKL22] [LB22]).
Structure of the paper
In Section 2, we study symmetry groups and irreducible representations. This yields the so-called protected zero eigenstates for all . In Section 3, we recall a generalized Bloch–Floquet theory introduced in [BHZ22a] and give spectral characterizations of magic angles. In Section 4, we prove Theorem 1. Then in Section 5, we prove Theorem 2. In Section 6, we construct the flat band eigenfunctions assuming the simplicity of magic angles and prove Theorem 3.
Acknowledgement
The author is very grateful to Maciej Zworski for introducing him to the field of magic angles as well as for having many helpful discussions and comments on the early manuscript. The author would also like to thank Simon Becker for helpful discussions and providing various numerical supports, and Zhongkai Tao, Jie Wang and Jared Wunsch for helpful discussions related to this project and comments on the manuscript.
2. Hamiltonians of Twisted Multilayer Graphene
In this section, we study symmetries of the Hamiltonian . The symmetry group commutes with the Hamiltonian and split the functional space into irreducible representations. This gives rise to the so-called protected states. We also consider eigenstates with zero eigenvalue for , for which some of them are unprotected in the sense that the corresponding eigenbranches separate from the flat band as we turning on the inter-layer parameter when .
2.1. Symmetries of the Hamiltonian
2.1.1. Translational symmetry
The first identity in (1.6) shows that for ,
[TABLE]
where denotes block diagonal matrices with blocks. Therefore,
[TABLE]
Therefore for the lattice
[TABLE]
and
[TABLE]
we obtain a unitary action of on , or more precisely on , with . We can further extend the action of to or block-diagonally and it yields
[TABLE]
2.1.2. Rotational symmetry
The second identity in (1.6) shows that
[TABLE]
Therefore, applying this to in equation (1.2), we have
[TABLE]
Similarly for , we have
[TABLE]
Hence, for ,
[TABLE]
Therefore, we obtain
[TABLE]
2.1.3. Additional symmetries
We record some additional actions and symmetries involving .
[TABLE]
This implies that the spectrum of is even.
Remark 3**.**
With the notation introduced in (3.6) we also have
[TABLE]
This implies that for each eigenvalue of , there exists another eigenvalue such that .
2.1.4. Group actions on functional spaces
Following [BEWZ22], since , we can combine the two actions into a unitary group action that commutes with :
[TABLE]
Taking a quotient by , we obtain a finite group acting unitarily on and commuting with :
[TABLE]
Restricting to the components, (resp. ) acts on (resp. ) as well and we use the same notation for those actions. With the previous discussions, we have the following
Proposition 2.1**.**
The operator is an unbounded self-adjoint operator with the domain given by . The operator commutes with the unitary action of the group given by (2.11) and
[TABLE]
The same conclusions are valid when is replaced by and by given by (2.12). In that case, the spectrum is discrete.
2.2. Protected states at zero
To show the existence of protected zero eigenstates of the operator , we start with considering all irriducible representations of (resp. ) on (resp. ).
2.2.1. Irreducible representations
Recall as in [BEWZ22, Section 2.2], irreducible unitary representations of are one dimensional and are given by
[TABLE]
Irreducible representations of are one dimensional for (given by – note that , , if and only if ),
[TABLE]
or three dimensional, for :
[TABLE]
The representations are equivalent for in the same orbit of the transpose of , and hence there are only two.
Defining the following functional space:
[TABLE]
There are then 11 irreducible representations: 9 one dimensional and 2 three dimensional. We can decompose into 11 orthogonal subspaces:
[TABLE]
In view of Proposition 2.1 we define
[TABLE]
with and similarly defined. Here we omit the dimensional constant when there is no ambiguity.
2.2.2. Protected zero eigenstates
We start with the case . The kernel of at is given by
[TABLE]
where the set forms the standard basis of . Now we consider the action of on with :
[TABLE]
If for , then are in different irriducible representations of under group action with
[TABLE]
where with . Therefore if and , each of these irriducible representation has a single zero eigenvalue. Since (see (2.10)) commutes with the action of , the spectra of are symmetric with respect to [math] (see Proposition 2.1), it follows that for each as above, has an eigenvalue at [math].
When for some , we have
[TABLE]
since and are always eigenfuctions of and correspondingly, they are protected zero eigenstates naturally. The symmetry of the spectra of (resp. ) implies that and both have two protected eigenvalues at [math]. Since
[TABLE]
we obtained the following result about symmetric protected eigenstates at [math]:
Proposition 2.2**.**
For all and any
[TABLE]
Remark 4**.**
The protected states are also discussed in [WL22] using perturbation theory for infinitesimal. Here we provide a different approach to show the zero eigenstates are protected for all .
2.2.3. Unprotected zero eigenstates and band separation
Now we consider the case , i.e., no tunneling in the upper and lower -layers. We show that in this case, there are additional zero eigenstates of that is unprotected, in the sense that they result in band separations as we turning on the tunneling parameter t (cf. Section 5.).
The operator is decomposed into direct sums of one twisted bilayer operator (see equation (1.3)) and . The kernel of is -dimensional and given by direct sums of and and copies of :
[TABLE]
where
[TABLE]
We can again consider the action of on with .
For simplicity of presentation, we only discuss the case . This yields
[TABLE]
Similar to previous discussion, for , each of these irreducible representation has a single zero eigenvalue, whereas has a double zero eigenvalue. The commutativity of with yields that each have an eigenvalue at [math]. Whereas for from the discussion in the previous subsection shows that the pairs of zero eigenfunctions and of disappears as we turning on the tunneling parameter . This, in fact, is the origin of the Dirac band separation from the flatband, which is discussed in Section 5.
Same argument applies to any , where all the irreducible representations may have more than one zero eigenfunctions. The only protected zero eigenstates are eigenbranchs corresponding to eigenfunctions .
3. Bloch–Floquet theory
In this section we introduce a Bloch–Floquet theory of the Hamiltonian with respect to the translation operator , which gives the band structure of the operator (cf. equation (3.6)) for .
3.1. A generalized Floquet theory approach
We follow [BHZ22a] for the discussion of Floquet theory with respect to the operator with , which has already implicitly appeared in the physics literature (cf. [TKV19]). One of the advantages of this type of Floquet theory is that the set at the flat band, which is nine dimensional (cf. [BHZ22a, Section 3.3]), splits to elements of (see (3.8) below for the definition of ) for nine different . This generalized Floquet theory also simplifies the presentation of band separation mechanism (cf. Section 5).
We first define the inner product on by
[TABLE]
Note that if and , then we have .
The generalized Floquet condition is formulated for as follows:
[TABLE]
where
[TABLE]
We consider the Floquet spectrum (cf. Proposition 3.1 for the discreteness of the spectrum) with Floquet condition (3.1) under the twisted translation operator :
[TABLE]
We define
[TABLE]
which yields . We then consider the operator
[TABLE]
with boundary condition and that gives us equivalent Floquet spectrum:
[TABLE]
Note that also depends on but we omit it here for simplicity. This suggests that we consider as a self-adjoint operator on
[TABLE]
with the domain given by . Note that we sometimes slightly abuse the notation of by restricting to the subspace or of the operator is diagonal.
In other words, we can define a generalized Bloch transform (cf. [TZ23, Chapter 5]):
[TABLE]
which can be checked easily satisfies
[TABLE]
so that . By the commutativity of and we have
[TABLE]
For a fixed , acts on as the operator in (3.6). The modified Bloch transform thus gives rise to bands of eigenfunctions of for .
Remark 5**.**
In , we denote two special points by and by as in physics literature.
3.2. Floquet theory and representations
We further define the spaces (cf. the decomposition (2.15)):
[TABLE]
Note that for
[TABLE]
we have orthogonal decomposition
[TABLE]
Remark 6**.**
Note that if and only if for , .
Comparing the remaining terms in the decomposition (2.15) and (3.10) we obtain
[TABLE]
where being two orbits under action given by
[TABLE]
3.3. Spectral characterization of magic angles
We start by introducing some basic properties of the operator with domain . We first observe that
[TABLE]
since for ,
[TABLE]
where the exponentials form an orthonormal basis of and is the standard basis of . This leads to the following proposition:
Proposition 3.1**.**
The family is a holomorphic family of elliptic Fredholm operators of index [math], and for all , the spectrum of is -periodic:
[TABLE]
Proof.
Since is an elliptic operator in dimension 2, existence of parametrices (cf. [DZ19, Proposition E.32]) implies the Fredholm property (cf. [DZ19, §C.2] for Fredholm operators). In view of (3.12), is invertible for and hence is an operator of index [math]. The same is true for the Fredholm family .
To see (3.13), note that if with , then with for any . ∎
Therefore for each , the operator is an elliptic differential system and hence it has a discrete spectrum that then describes the spectrum of on :
[TABLE]
Note that sometimes we omit and simply write and when there is no ambiguity. To conclude the second identity in (3.14), note that
[TABLE]
Hence, non-zero eigenvalues of on are given by the non-zero singular values of , i.e., eigenvalues of the operator , counted with their multiplicities.
The zero eigenvalue (if exists) of also has the same multiplicity as the zero eigenvalue of , so that eigenvalues are included exactly twice (for ), this follows from the identity
[TABLE]
which follows from Proposition 3.1: the operator is a Fredholm operator of index [math].
Combining with the equation (3.6), The above discussion yields three equivalent characterizations of the existence of a flat band at energy zero:
Proposition 3.2**.**
The following statements are equivalent for :
- (1)
** 2. (2)
E_{0}(\mathbf{k},\alpha,\mathbf{t})=0\text{ for all \mathbf{k}\in{\mathbb{C}}/\Lambda^{*}.}** 3. (3)
.
4. Magic angles of the Twisted multilayer Hamiltonian
The magic angles are defined as the angles ’s at which
[TABLE]
As discussed in Section 3, the Hamiltonian comes from the Floquet theory of and (4.1) means that has a flat band at [math] which is given by the equivalent condition:
[TABLE]
4.1. Magic angles of TBG
We briefly recall several results on flat bands and magic angles of TBG from [BHZ22a].
Let denote the sets of all magic angles of with , then the spectrum of on is given by
[TABLE]
This yields the following proposition:
Proposition 4.1**.**
Suppose and is defined using (3.3) for given by (1.1) for . Then
[TABLE]
In other words, zero energy band is flat if and and if the Bloch eigenvalue is [math] at some , which is the lattice of Dirac points .
The following result is useful when constructing flat band eigenfunctions using the theta function argument when the multiplicity of the flat band is one.
Proposition 4.2**.**
Suppose that . If , then , , has zero of order one at and no other zeros.
In the above proposition, the point is a point of high symmetry (cf. [BEWZ22]). Recall that , so .
4.2. Magic angles of TMG
In this section, we prove the following theorem, which states that magic angles of coincides with magic angles of TBG.
Theorem 4**.**
For any and defined in equation (1.3), we have
[TABLE]
Proof.
By Proposition 2.2 and 3.1, for any both and contains as the spectrum and they are both -periodic for spectrum of ). Therefore we only consider .
We define the following matrices
[TABLE]
[TABLE]
where for . We only need to show that for any fixed is invertible if and only if is invertible for . Following from a direct computation and the identity , we have the following decomposition
[TABLE]
Since , the operator is invertible if and only if is invertible for . ∎
Remark 7**.**
The decomposition (4.5) also concludes that the multiplicity of the flat band of is the same as the multiplicity of the flat band of .
5. Band separation with inter-layer tunneling
We discuss, using the perturbation theory, the band separation mechanism appeared in [WL22] once the tunneling being turned on. We set up a Grushin problem and compute the separation of the band explicitly. For a detailed introduction on the Grushin problem, we refer to [DZ19, Appendix C].
5.1. A Grushin Problem
Following the identity (3.15), eigenvalues of are given by singular values of , i.e., eigenvalues of , counted with multiplicities. For simplicity purposes, we work with the case , the general case is essetially the same. We first introduce the Schur complement formula:
Proposition 5.1**.**
Suppose
[TABLE]
are bounded operators on Banach spaces, then is invertible if and only if is invertible. Moreover, in such case we have
[TABLE]
Proof.
See [DZ19, Appendix C.1]. ∎
Define and . We have
[TABLE]
We consider the operator given by
[TABLE]
We start by considering the zero spectrum of the operator . By equation (2.18) and (2.19) and the discussion of §3.2, are given by
[TABLE]
Here we slightly abuse notations by restricting to the first component because of the direct sum decomposition (2.17). Following the spirit of Remark 6, we define
[TABLE]
These are elements of , or more precisely , for respectively.
5.2. Analysis at K’
To see how zero spectrum changes as we turning on , we consider the following Grushin problem for the perturbed operator at the Dirac point :
[TABLE]
with
[TABLE]
We use to denote the set . For , the Grushin problem (5.5) is invertible with the inverse given by
[TABLE]
with
[TABLE]
where the set is the set of all normalized eigenfunctions of with eigenvalues . Note that by definition of in equation (1.2), the eigenspace of is given by the direct sum of eigenspaces of and . Therefore we can take a complete basis of eigenfunctions with any element is either of the form or , where are eigenfunctions of and are eigenfunctions of . To work with the perturbed operator , we employ the following proposition.
Proposition 5.2**.**
For , the Grushin problem for :
[TABLE]
is invertible. If the inverse is given by , then
[TABLE]
Proof.
This follows directly from [TZ23, Propsition 2.12]. ∎
Proposition 5.2 yields that
[TABLE]
with
[TABLE]
where the inner product is taken in the space . The expression of and the fact that and being zero eigenfunctions of at yields that equation (5.8) vanishes.
Now we compute the second order perturbation term in . Writing and where , we have the following identities
[TABLE]
The operator is then given by the matrix
[TABLE]
while the operator is given by the matrix
[TABLE]
Writing , the above matrix can be rewritten as
[TABLE]
By the direct sum decomposition of , any eigenfunctions is of the form or , the off-diagonal terms in the above matrix vanishes. It thus reduces to a sum of diagonal matrices
[TABLE]
in which and with
[TABLE]
We consider the first term in the expression (5.10). Note that are joint eigenfunctions of and with spectrum
[TABLE]
Therefore,
[TABLE]
Using the fact that and Plancherel theorem, at we obtain the equality
[TABLE]
For , combining equations (5.9) (5.10) and (5.11), the second order perturbation is given by
[TABLE]
recall that is the set of non-zero eigenvalues of with -eigenfunctions . Therefore by Proposition 5.1, 5.2, zero is an eigenvalue for closed to . The zero eigenvalue corresponds to the flat band of .
The eigenvalue of the operator corresponding the Dirac cone of is given by
[TABLE]
which corresponds to the separation of the Dirac band from the flat band. Since this eigenvalue is given by the solution to the equation
[TABLE]
a substitution and a geometric series argument yield the solution .
Combining with the fact that there is a protected states at [math] for any , this proves the following proposition:
Proposition 5.3**.**
For and at , the first two eigenvalues of are given by and , where the constant is given by
[TABLE]
6. Flatband eigenfunctions and Chern numbers
In the section, we construct flat band eigenfunctions of at for using a theta function argument analogous to [BEWZ22] (see also [LTKV20]). The construction also gives an explicit method to compute the Chern number of the flat band of this twisted multilayer model. We start by reviewing some backgrounds in theta functions.
6.1. Review of theta functions
We recall that the theta function is given by
[TABLE]
where
[TABLE]
The function has simple zeros at
[TABLE]
see [Mum83]. If we consider
[TABLE]
we obtain a meromorphic function with simple poles at , simple zeros at , and satisfying Hence, by choosing , we define
[TABLE]
It satisfies
[TABLE]
We then define a function periodic with respect to the lattice :
[TABLE]
6.2. Construction of flat bands
We first transplant the functions constructed above to the lattice and its dual . Recall that in equation (2.2), they are given by
[TABLE]
We introduce change of variables
[TABLE]
in the equation (6.2) (6.4), so that the new function is periodic with respect to the lattice , i.e., we define
[TABLE]
Following from [BHZ22a, Lemma 3.1], is then the fundamental solution of in the sense that
[TABLE]
Therefore the Schwartz kernel of
[TABLE]
is given by , with .
Recall that in [BEWZ22, Propsition 3.4] for simple, a solution to for any is constructed using a theta function argument. In particular, for this implies, by Proposition 4.1, that is a magic angle of . Recall that the magic angles of coincides with the magic angles of . We therefore have a similar construction of all flat band eigenfunctions of .
Recall that by Proposition 4.2, for simple, i.e. flat band has multiplicity one, vanishes at the point . Therefore we have
[TABLE]
Define the function
[TABLE]
with
- (1)
for ; 2. (2)
; 3. (3)
,
where denotes the -th powers of the resolvent and
[TABLE]
It is easy to verify that defined above solves the equation
[TABLE]
for , as the resolvent is invertible in .
For , without loss of generality we consider , otherwise multiplication by reduces it to the zero case. Note that the (Schwartz kernel of) resolvent is meromorphic at with a simple pole, i.e.,
[TABLE]
where is holomorphic in . The function is holomorphic in and has a zero at with order . Therefore, each component is holomorphic in at . In particular, the construction (3) above yields that at
[TABLE]
where . This yields the flat band eigenfunction at provided that the integral appeared in the above equation is non-zero. This is verified numerically for the first five real magic angles, where the multiplicities of the flat band are one (see Table 1).
Remark 8**.**
This agrees with the protected eigenstates result shown using irreducile representations, where is always a protected eigenstate with zero-energy.
6.3. High Chern numbers of the flat band
In this section, assuming the flat band is simple, we show that the flat band eigenfunctions constructed above give rise to a holomorphic line bundle with Chern number . For basics on holomorphic line bundles, Berry connections and Chern numbers, we refer to [TZ23].
We use the notations in §6.1 for simplicity. By equation (6.1) (6.3) and (6.7), we obtain
[TABLE]
where
[TABLE]
Note that
[TABLE]
is a unitary transformation. Here we slightly abuse notations by identifying and with corresponding parts in the rescaled lattice space. Following the standard construction we define
[TABLE]
Therefore, we have the following lemma:
Lemma 6.1**.**
Definition (6.17) gives a holomorphic line bundle over :
[TABLE]
The corresponding family of multipliers is given by .
Proof.
The action of the discrete group , with on the trivial complex line bundle
[TABLE]
(where is defined in (6.9)) is free and proper, and the quotient map is given by . Hence its quotient by that action, , is a smooth complex manifold of dimension 2.
The map satisfies the cocycle conditions with the action on :
[TABLE]
and for . We then have by equation (6.14) and this gives the structure of a complex line bundle over . ∎
The hermitian structure is inherited from and the resulting hermitian structure on of (6.18). In coordinates on , we get
[TABLE]
and this gives a hermitian structure on : from (6.14) we see that
[TABLE]
We then associate a Chern connection and a Berry curvature to the hermitian metric . The Chern number of the line bundle thus can be calculated using the method of multiplier as
[TABLE]
This concludes the proof to Theorem 3.
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