32 magic-angle quantization rule of at bands in twisted bilayer graphene and relationship with the Quantum Hall eect Leonardo A. Navarro-Labastida and Gerardo G. Naumis

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3/2 magic-angle quantization rule of ﬂat bands in twisted bilayer graphene and

relationship with the Quantum Hall eﬀect

Leonardo A. Navarro-Labastida and Gerardo G. Naumis∗

Depto. de Sistemas Complejos, Instituto de F´ısica,

Universidad Nacional Aut´onoma de M´exico (UNAM)

Apdo. Postal 20-364, 01000, CDMX, M´exico.

(Dated: February 2023)

Flat band electronic modes in twisted graphene bilayers are responsible for superconducting and

other highly correlated electron-electron phases. Although some hints were known of a possible

connection between the quantum Hall eﬀect and zero ﬂat band modes, it was not clear how such

connection appears. Here the electronic behavior in twisted bilayer graphene is studied using the

chiral model Hamiltonian. As a result, it is proved that for high-order magic angles, the zero

ﬂat band modes converge into coherent Landau states with a dispersion σ2= 1/3α, where αis a

coupling parameter that incorporates the twist angle and energetic scales. Then it is proved that

the square of the hamiltonian, which is a 2 ×2 matrix operator, turns out to be equivalent in

a ﬁrst approximation to a two-dimensional quantum harmonic oscillator. The interlayer currents

between graphene’s bipartite lattices are identiﬁed with the angular momentum term while the

conﬁnement potential is an eﬀective quadratic potential. By considering the zero mode equation,

the boundary conditions and a scaling argument, a limiting quantization rule for high-order magic

angles is obtained, i.e., αm+1 −αm= 3/2 where mis the order of the angle. From there, an

equipartition and quantization of the kinetic, conﬁnement and angular momentum contributions is

found. All these results are in very good agreement with numerical calculations.

I. INTRODUCTION

In 2018 it was found experimentally that twisted

bilayer graphene (TBG) presents strongly correlated

electron-electron quantum phases leading for example

to unconventional superconductivity and Mott insulator

states [1]. More recently, trilayer twisted graphene

has been found to be the most strongly interacting

correlated material [2, 3]. Such remarkable discoveries

presented a new paradigm in the so-called Moir´e ma-

terials and unveiled the importance of two-dimensional

(2D) materials to understand unconventional supercon-

ductivity in cuprates and heavy fermions systems, as

they share similar quantum phase diagrams [1, 2, 4].

TBG advantages are i) its simplicity, as they are made

from a single chemical element, and ii) they have a high

degree of manipulation that cuprates doesn’t have. In

recent years, there has been a signiﬁcant interest in

these phases of matter from a fundamental point of view

[5–13] but also because they present a lot of possible

electronic applications and quantum computing advan-

tages [13, 14]. There is also an interesting connection

between topological phases, edge states, semimetals,

and fractional quantum Hall eﬀect (FQHE) [15–26]. A

recently paper establishes a connection between heavy

fermion models and TBG [4], opening the prospect of

using heavy fermions physics to the superconducting

physics of TBG and more strongly correlated phases.

The discovery of such phases was proceeded by the

Bistritzer-Mac Donald (BM) theoretical observation that

∗naumis@ﬁsica.unam.mx

twisted bilayer graphene (TBG) develops ﬂat bands at

certain twisting angles which are called magic [27]. BM

considered a continuum Dirac model in which the moir´e

periodicity between layers produces moir´e Bloch´s bands

[27]. The model is continuum in the sense that the inter-

layer potential between Carbon πorbitals is a smooth

function of the spatial separation projected onto the

graphene planes and also the hopping is local and pe-

riodic, allowing to apply the Bloch´s theorem for any

rotation angle. For TBG it was demonstrated that non-

Abelian gauge ﬁelds arise due to the coupling between

layers in the low-energy regime [28, 29].

Flat band modes that arise at magic angles, also known

as zero energy modes, have been investigated in many

recent works [30–39], and in particular, there were hints

in the mathematics for a possible connection with the

quantum Hall eﬀect (QHE) and the lowest Landau level

[30, 33]. There are interesting properties of the zero mode

wave function [33–35], in particular, the connection with

the lowest Landau level reveals that TBG presents topo-

logical phases [33, 40].

Importantly, the wave function is reminiscent of a

quantum hall wave function because is described in

terms of Jacobi theta functions such as in the quantum

hall eﬀect wave function [30–32]. This hidden wave

function is important to understand because leads

to particular localization properties, orbital current,

density wave function distribution, and symmetries of

the pseudo-magnetic gauge ﬁelds. Yet, exactly how

this analogy arises was not clear as no connection

between the quantum harmonic oscillator and the

TBG hamiltonian was ever found. Tarnopolsky et. al.

also found that magic angles were quantized but no

explanation was provided for this fact [30]. Thus there

arXiv:2210.01931v4 [cond-mat.mes-hall] 6 Mar 2023

were two open questions related to the same problem.

The present work shows how these two questions relate

to each other, and also answers them. Moreover, we

ﬁnd that in fact, the zero ﬂat band modes converge into

coherent Landau levels. As we will discuss, this is done

by using boundary layer diﬀerential equations theory

and squaring the Hamiltonian [29, 41], a process that

has also been used in supersymmetry [42–44].

II. CHIRAL TBG AND SQUARED TBG

HAMILTONIANS

The chiral Hamiltonian of twisted bilayer graphene is

a variant of the original Bistritzer-MacDonald Hamil-

tonian in which the AA tunneling is set to zero

[32]. Here we use as basis the wave vectors Φ(r) =

ψ1(r), ψ2(r), χ1(r), χ2(r)Twhere the index 1,2 rep-

resents each graphene layer and ψj(r) and χj(r) are

the Wannier orbitals on each inequivalent site of the

graphene’s unit cell. The chiral Hamiltonian is given by

[30, 31, 45],

H=



0D∗(−r)

D(r) 0 

(1)

where the zero-mode operator is deﬁned as,

D(r) = 

−i¯

∂ αU(r)

αU(−r)−i¯

∂

(2)

and,

D∗(−r) = 

−i∂ αU∗(−r)

αU∗(r)−i∂ 

(3)

with ¯

∂=∂x+i∂y,∂=∂x−i∂y. The potential is,

U(r) = e−iq1·r+eiφe−iq2·r+e−iφ e−iq3·r(4)

where the phase factor is φ= 2π/3 and the vec-

tors are given by q1=kθ(0,−1), q2=kθ(√3

2,1

2),

q3=kθ(−√3

2,1

2), the moir´e modulation vector is kθ=

2kDsin θ

2with kD=4π

3a0is the magnitude of the Dirac

wave vector and a0is the lattice constant of monolayer

graphene. The model contains only the parameter α, de-

ﬁned as α=w1

v0kθwhere w1is the interlayer coupling of

stacking AB/BA with value w1= 110 meV and v0is the

Fermi velocity with value v0=19.81eV

2kD. The operators

∂and ¯

∂are dimensionless as the Hamiltonian Eq. (1)

is written in using units where v0= 1, kθ= 1. The

twist angle only enters in the dimensionless parameter α.

The combinations b1,2=q2,3−q1are the moir´e Bril-

louin zone (mBZ) vectors and also b3=q3−q2. Using

this basis for the reciprocal space lattice, some impor-

tant high symmetry points of the moir´e Brillouin zone

are K= (0,0), K0=−q1, and Γ=q1(see ref. [41]

for a diagram). For further use it is also convenient to

deﬁne a set of unitary vectors q⊥

µperpendicular to the

set qµand given by q⊥

1= (1,0),q⊥

2=−1

2,√3

2,q⊥

−1

2,−√3

2. The moir´e vectors unitary cell are given by

a1,2= (4π/3kθ)(√3/2,1/2). Observe that qµ·a1,2=−φ

for µ= 1,2,3.

In a previous work we showed how, by taking the

square of H, it is possible to write the Hamiltonian as

a 2 ×2 matrix [29, 41],

H2=

−∇2+α2|U(−r)|2αA†(r)

αA(r)−∇2+α2|U(r)|2

(5)

where the squared norm of the potential is an eﬀective

trigonal conﬁnement potential,

|U(r)|2= 3 + 2 cos(b1·r−φ) + 2 cos(b2·r+φ)

+ 2 cos(b3·r+ 2φ)(6)

and the oﬀ-diagonal term is,

A†(r) = −i

µ=1

e−iqµ·r(2q⊥

µ·∇+ 1) (7)

where ∇†=−∇with ∇= (∂x, ∂y) and µ= 1,2,3.

III. ZERO-ENERGY MODES AS COHERENT

LANDAU STATES

Now we investigate the asymptotic limit α→ ∞ by

numerically solving (see appendix C) the Schr¨odinger

equation HΨ(r) = EΨ(r) where Eis the energy. As

the potential is periodic, it satisﬁes Bolch’s theorem, and

thus ψk,j (r) = eik·ruk,j (r) where uk,j (r) has the peri-

odicity of the lattice (see appendix A). The rotational

C3symmetry allows to further simplify the problem (see

appendix B). In Fig. 1 we present the zero mode wave

function, corresponding to E= 0 at the reciprocal space

point k= Γ for the mth magic angles (αm) with m= 8

and m= 9. The electronic maxima of the density form

hexagons which are nearly localized at rµ≈ ±qµ. Such

observation is detailed in Fig. 1. Moreover, the wave-

function for other kpoints follow the same behavior al-

though the Γ point best captures the magic angle behav-

ior [41]. In the limit of αm→ ∞ we have veriﬁed that

in fact, the electron density is almost localized at qµ.

Notice that here we are working with adimensional units

but this suggests a connection with the QHE as solutions

seem self-dual [46], i.e., in real space are similar to those

in reciprocal space with renormalized parameters.

Although there are expressions for the wave-function

[30, 34, 47] at any kpoint that hinted a relationship

with the lowest Landau levels, they depend on the wave

function at the Kpoint, i.e.,

ψk,j (r) = fk(z)ψK,j (r) (8)

where z=x+iy and fk(z) is an analytic function which

satisfy the boundary condition and turns out to be a Ja-

cobi theta function. The form of the ψK,j (r) is not ana-

lytically known. Yet in Figs. 1 and 2 we see numerically

that the electron wave function reaches an asymptotic

limit almost invariant as αm→ ∞. In this limit, the lo-

calization centers for the Γpoint wave function seem to

converge as seen in Figs. 1, 2 and 3. Such wave function

tends to be localized in certain points of space which are

not the stacking points AA, AB, and BA. In that sense,

the solutions are very diﬀerent from the ﬁrst magic an-

gle a fact that was explained elsewhere [41]. As seen in

Fig. 3, for other kpoints diﬀerent from Γthe situation is

quite similar, i.e., the wave functions are more localized

as α→ ∞ and approach the same localization center.

FIG. 1. Numerically obtained zero-mode normalized wave

function localization for some high-order magic angles at

k=Γ. In Panels a) and b) we present Re{ψ1(r)}+Im{ψ2(r)}

(orange curves) and Im{ψ1(r)} − Re{ψ2(r)}(purple curves)

parts of a one layer symmetrized wave function components

ψ±(r) = ψ1(r)∓iµαψ2(r) at the symmetric line (0, y) at

magic angles α8= 11.345 and α9= 12.855 respectively. Pan-

els c) and d), contour plot of the global electronic density

ρ1(r) + ρ2(r) for α8and α9respectively. The vertical line

(yellow line) inside the real-space moire unit cell indicates

the cut along the yaxis used in panels a) and b). The exter-

nal hexagon is the real-space moire unit cell, where the AB

(green), BA (yellow), and AA (red) stacking points are in-

dicated. For higher magic angles, the wave-function density

localizes in 6 high-density points, located at r=±qµ, with

µ= 1,2,3, forming the red spots of maximal density.

To understand how this limiting wave func-

tion arises, let us discuss the zero-mode equation

D(r)ψ1(r), ψ2(r)T= 0 for states in the ﬂat-band.

Although not essential for the analysis, it is easier

to understand the Γ point solution. For this case we

have that due to symmetry, ψ2(r) = iµαψ1(−r) where

µα=±1 depending on the magic angle parity [30].

Therefore, we obtain,

∂ψ1(r) = αµαU(r)ψ1(−r) (9)

∂ψ1(−r) = −αµαU(−r)ψ1(r) (10)

To solve the equation in the limit α→ ∞ we use the

boundary layer theory of diﬀerential equations [48], i.e.,

whenever the gradients are small, we can neglect the

derivative in Eqns. (9)-(10) when compared to the poten-

tial term. Then our solution must satisfy ψj(r)→0. The

solution will be diﬀerent from zero only inside the bound-

ary layer, i.e., whenever ¯

∂ψj(r) is of order αU(±r)ψj(r).

Taken into account the boundary layer we conclude that

the solution must be strongly peaked around certain re-

gions of space. Then is natural to seek the solution

within continuous functions having a peak while keep-

ing the form of Eq. (8). We then propose a coherent

Landau state ansatz for a given layer (and thus suppress

the subindex j),

ψ(z, z∗) = fλ(z)e−1

4σ2|z|2(11)

where fλ(z) is an analytic function [49],

fλ(z) = 1

σ√2πe1

2σ2λ∗ze−1

4σ2λλ∗(12)

The parameter λis the localization center (known as the

guiding coordinates in the QHE problem [50]) and σthe

standard deviation as the electronic density is a Gaus-

sian,

ρ(r) = 1

2πσ2e−|z−λ|2

2σ2(13)

Notice how the Gaussian envelope in Eq. (11) ensures

the boundary layer condition, i.e., the vanishing of the

wave function whenever the gradient is small.

However, still we need to make remarks. As the equa-

tion involves ψ(r) and ψ(−r), the solutions can be writ-

ten as a sum of a symmetrized and antisymmetrized

forms. Therefore, it will be a linear combination of the

symmetrized/antisymmetrized wavefunctions,

ψ±(z, z∗)≈e−1

4σ2|z|21

√2(fλ(z)±f−λ(z)) (14)

provided that σ→ ∞ to avoid overlap between the Gaus-

sians centered at λand −λ. A second reason to neglect

the overlap eﬀect around z= 0 is that U(0) = 0 and

∂ψ±(z, z∗)|z,z∗=0 = 0.

In what follows we will use our ansatz in the zero mode

equation to prove how it satisﬁes the equation and to

obtain σ.

Before doing so, observe that ψ(r) must transform ac-

cording to the C3symmetry group and this can be en-

sured by deﬁning a λ1such that,

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32 magic-angle quantization rule of at bands in twisted bilayer graphene and relationship with the Quantum Hall eect Leonardo A. Navarro-Labastida and Gerardo G. Naumis

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