# Hofstadter butterflies and metal/insulator transitions for moiré
heterostructures

Simon Becker<EMAIL_ADDRESS>Courant Institute of Mathematical Sciences,
Mercer St, New York City, USA.  ,  Lingrui Ge<EMAIL_ADDRESS>University of
California, 410N Rowland Hall, Irvine, USA  and  Jens Wittsten
<EMAIL_ADDRESS>Department of Engineering, University of Borås, SE-501 90
Borås, Sweden

###### Abstract.

We consider a tight-binding model recently introduced by Timmel and Mele
[TM2020] for strained moiré heterostructures. We consider two honeycomb
lattices to which layer antisymmetric shear strain is applied to periodically
modulate the tunneling between the lattices in one distinguished direction.
This effectively reduces the model to one spatial dimension and makes it
amenable to the theory of matrix-valued quasi-periodic operators. We then
study the transport and spectral properties of this system, explaining the
appearance of a Hofstadter-type butterfly. For sufficiently incommensurable
moiré length and strong coupling between the lattices this leads to the
occurrence of localization phenomena.

Figure 1.  Superposition of two honeycomb lattices with one of the lattices
(with red/blue vertices) exposed to uniaxial strain in $x_{1}$ direction
(left) and shear strain (right).

##  1\. Introduction

We consider a one-dimensional armchair model for bilayer graphene proposed in
[TM2020]. Due to periodic strain-modulation, the bilayer graphene is periodic
in one direction and is, depending on the arithmetic properties of the strain,
periodic or quasi-periodic in the orthogonal direction exhibiting moiré-type
pattern. Using the periodicity in the strained direction, Floquet theory
provides then a family of _one-dimensional_ Hamiltonians depending on some
quasi-momentum $\theta\in{\mathbb{R}}/{\mathbb{Z}}$ that we analyze in this
article.

Many of the applications and motivations of the field of quasi-periodic
Schrödinger operators have been limited to magnetic fields on lattice systems.

The recently emerging field of twistronics provides a variety of examples of
quasi-periodic Hamiltonians due to incommensurable twists of two or several
lattice structures. While such examples exhibit quasi-periodicity in two
spatial dimensions, we shall restrict us here to lattice structures that are
periodic in one dimension and quasi-periodic in only one direction that appear
naturally by superimposing strained lattices. This has the advantage that the
well-developed theory of quasi-periodic one-dimensional discrete operators is
applicable. Similarly to the case of magnetic fields, fractal spectra, the so-
called Hofstadter butterfly [BM11, CNKM20, L21], and metal insulator
transitions [C18] have been observed in moiré structures, too. The latter by
changing the tunneling rate when compressing the lattices. Experimental and
theoretical studies of transport properties for one-dimensional moiré
structures have been considered in [BLB16]. We also discuss model operators
for analogous results in two spatial dimensions. In contrast to some of their
two-dimensional twisted moiré superstructures, one-dimensional models do not
exhibit flat bands, see e.g. our Proposition 7.7.

Unlike in the case of magnetic fields, twisted lattice systems do in general
not allow for an explicit reduction to one-dimensional quasi-periodic
operators. Since the theory of multi-dimensional quasi-periodic operators is
far less developed, this limits the tools available to understand fractal
spectra, see Proposition 6.1, and metal/insulator transitions in depth. Most
results in higher dimensions are limited to establishing Anderson localization
for, in our case, sufficiently strong coupling [BGS02].

In this article, we are concerned with an effectively one-dimensional operator
where the incommensurability is reduced to a single spatial dimension when
physical strain is applied.

After introducing the framework of matrix-valued cocycles, which can be found
for example in [AJS15], we discuss the case of moiré lengths close to rational
numbers in which case point spectrum is absent, see Propositions 3.1 and 3.2.

This extends – with obvious modifications – a classical theorem by Gordon
[G76] (discrete operators) and Simon [S82] (continuous operators) to matrix-
valued operators.

In the opposite regime of diophantine moiré length scales, the Hamiltonian
exhibits Anderson localization, see Theorem 1, if the coupling between the two
lattice structures is strong enough, which can be experimentally seen by the
application of physical pressure. In contrast, if the coupling between the
lattices is sufficiently weak, transport and absolutely continuous spectrum
persist, see Theorem 3. This localization argument relies on a matrix
generalization of the theory that has been obtained by Klein [K17] extending
earlier works [BGS02].

Methods and results showing Cantor spectrum are largely limited to scalar one-
dimensional quasi-periodic operators [GS11] and also in our case, we rely on
the diagonalizability of the matrix-valued operator in one of the two
considered cases to establish fractal spectrum.

###  1.1. Main results and organization of the article

Following [TM2020], the tight-binding kinetic Dirac operator is then defined
in terms of
$\gamma_{15}=\operatorname{diag}(\sigma_{1},\sigma_{1}),\gamma_{25}=\operatorname{diag}(\sigma_{2},\sigma_{2})$
where we have Pauli matrices $\sigma_{i}$ as

$(D_{\operatorname{kin}}(\theta)\psi)_{n}=(t(\theta)\psi_{n+1}+t(\theta)\psi_{n-1}+t_{0}\psi_{n}),$

where
$t(\theta)=\left(\cos(2\pi\theta)\gamma_{15}+\sin(2\pi\theta)\gamma_{25}\right)$,
with $\operatorname{det}(t(\theta))=1,$ $\|t(\theta)\|=1,$ and
$t_{0}=\gamma_{15}$. Here $\theta$ indicates the quasi-momentum perpendicular
to the strained direction.

The two honeycomb lattices interact by a tunneling interaction which is
modeled using tunneling potentials $V_{c},V_{\operatorname{ac}}$ by

$\begin{split}V_{\operatorname{ac}}&=\begin{pmatrix}\mathbf{0}&W_{\operatorname{ac}}\\\
W_{\operatorname{ac}}&\mathbf{0}\end{pmatrix},\quad
W_{\operatorname{ac}}=\operatorname{diag}(U,U)\text{ and }\\\
V_{\operatorname{c}}&=\begin{pmatrix}\mathbf{0}&W_{\operatorname{c}}\\\
W_{\operatorname{c}}^{*}&\mathbf{0}\end{pmatrix}\text{ with
}W_{\operatorname{c}}=\begin{pmatrix}\mathbf{0}&U^{-}_{\operatorname{c}}\\\
U^{+}_{\operatorname{c}}&\mathbf{0}\end{pmatrix},\end{split}$ (1.1)

where for coupling strengths
$w=(w_{0},w_{1})\in{\mathbb{R}}_{0}^{+}\times{\mathbb{R}}_{0}^{+}$

$V_{w}(x)=w_{0}V_{\operatorname{ac}}(x)+w_{1}V_{\operatorname{c}}(x).$

The first summand, given in (1.1) in terms of $U(x):=\frac{1+2\cos(2\pi
x)}{3},$ we refer to as the _anti-chiral_ part, and it describes tunneling
between A-A′/B-B′ atoms. The second summand defined by
$U^{\pm}_{\operatorname{c}}(x):=\tfrac{1-\cos(2\pi x)\pm\sqrt{3}\sin(2\pi
x)}{3}=\tfrac{1\pm 2\sin(2\pi x\mp\pi/6)}{3},$ is the _chiral_ part modeling
the tunneling between A-B′/B-A′ atoms. Here, A and B correspond to the two
different representatives of the fundamental cell of a honeycomb lattice and ′
indicates atoms of the second lattice. This terminology is inspired by the
Bistritzer-MacDonald model [BM11]. The Hamiltonian
$H_{w}(\theta,\phi,\vartheta):\ell^{2}({\mathbb{Z}};{\mathbb{C}}^{4})\rightarrow\ell^{2}({\mathbb{Z}};{\mathbb{C}}^{4})$
is then, for some fixed length $L>0$ of the moiré cell, given by

$H_{w}(\theta,\phi,\vartheta)\psi_{n}=D_{\operatorname{kin}}(\theta)\psi_{n}+V_{w}(\vartheta+\tfrac{n}{L},\phi)\psi_{n}$

where $\vartheta\in[0,1].$

######  Remark 1.

We shall occasionally suppress the parameter dependence in the Hamiltonian and
related quantities to simplify the notation.

The length of the fundamental cell is related to the strength of the strain.
Unlike for the almost Mathieu operator, the only physically relevant frequency
in the tunneling potential is $\vartheta=0.$ However, we introduce the
parameter $\vartheta$ for our mathematical analysis. Physically it corresponds
to an additional offset between the lattices at the origin. We also write
$H_{w}(\theta,\phi)\equiv H_{w}(\theta,\phi,0).$

Introducing the shift operator $\tau\psi_{n}:=\psi_{n-1}$, the Hamiltonian
takes the form of a $4$ by $4$ matrix-valued discrete operator that reads in
terms of $K(\theta):=1+e^{-2\pi i\theta}(\tau+\tau^{*})$

$H_{w}(\theta,\phi,\vartheta)=\begin{pmatrix}0&K(\theta)&w_{0}U(\vartheta+\tfrac{\bullet-\phi}{L})&w_{1}U_{c}^{-}(\vartheta+\tfrac{\bullet}{L})\\\
K(\theta)^{*}&0&w_{1}U_{c}^{+}(\vartheta+\tfrac{\bullet}{L})&w_{0}U(\vartheta+\tfrac{\phi+\bullet}{L})\\\
w_{0}U(\vartheta+\tfrac{\bullet-\phi}{L})&w_{1}U_{c}^{+}(\vartheta+\tfrac{\bullet}{L})&0&K(\theta)\\\
w_{1}U_{c}^{-}(\vartheta+\tfrac{\bullet}{L})&w_{0}U(\vartheta+\tfrac{\bullet+\phi}{L})&K(\theta)^{*}&0\end{pmatrix}.$
(1.2)

The parameter $\phi$ incorporates the different tunneling amplitude for A-A′
and B-B′ sides due to their dislocation in space. Finally, when $w_{0}\equiv
0$ we call this mode the _chiral model_ and when $w_{1}\equiv 0$ the _anti-
chiral model_.

The outline of our article is then as follows:

Outline of article.

  * •

In Section 2 we discuss basic spectral properties including Lyapunov exponents
of the Dirac-Harper model (1.2).

  * •

In Section 3 we study with Propositions 3.1 and 3.2 the spectral and transport
properties for moiré lengths $1/L$ that are rational or only mildly irrational
numbers, i.e. _Liouville numbers_.

  * •

In Section 4 we study the regime of strongly irrational (diophantine) moiré
lengths $1/L$ that satisfy a diophantine condition. For strong tunneling
interaction, this is the regime of Anderson localization and absence of
transport proven in Theorem 1.

  * •

In Section 5 we show the existence of absolutely continuous spectrum for weak
coupling of the lattices, see Theorem 3.

  * •

In Section 6 we show the existence of Cantor spectrum for the Dirac-Harper
operator (1.2) in the anti-chiral limiting case, see Proposition 6.1.

  * •

In Section 7 we study the absence of flat bands and perform a spectral
analysis in an effective low-energy model, see Proposition 7.7.

  * •

In Section 8 we give an outlook on 2D generalizations of the Dirac-Harper
model (1.2) for twisted square lattices.

##  2\. Basic properties of the Dirac-Harper model

In this section we start by exhibiting some basic spectral properties about
the Dirac-Harper Hamiltonian (1.2).

######  Lemma 2.1.

In case of the limiting chiral and anti-chiral model, the Hamiltonian
satisfies the particle-hole symmetry

$\operatorname{Spec}(H_{w}(\theta,\phi,\vartheta))=-\operatorname{Spec}(H_{w}(\theta,\phi,\vartheta)).$

###### Proof.

For both the chiral case and anti-chiral model, this follows by conjugating,
for $\lambda=e^{i\frac{\pi}{4}}$, with unitaries

$P_{c}:=\begin{pmatrix}1&0&0&0\\\ 0&0&1&0\\\ 0&-1&0&0\\\
0&0&0&-1\end{pmatrix}\text{ and
}P_{\operatorname{ac}}:=\begin{pmatrix}i\lambda&0&0&0\\\
0&0&0&-\overline{\lambda}\\\ 0&0&-i\lambda&0\\\
0&\overline{\lambda}&0&0\end{pmatrix}$

respectively from the left which turns the Hamiltonian into a block off-
diagonal operator. Then conjugating with the third Pauli matrix implies the
claim. ∎

###  2.1. Ergodic properties of the system

The arithmetic properties of $L$, foremost depending on whether
$L\in\mathbb{Q}^{+}$ (periodic) or $L\in\mathbb{R}^{+}\setminus\mathbb{Q}$
(quasi-periodic), decide on the spectral and dynamical properties of the
system. Let $\psi$ be a solution to $H_{w}\psi=\mathscr{E}\psi$, with
$\mathscr{E}\in{\mathbb{C}}$, then we can write the solution as

$\begin{pmatrix}\psi_{n+1}\\\
\psi_{n}\end{pmatrix}=\begin{pmatrix}t(\theta)^{-1}(\mathscr{E}-t_{0}-V_{w}(\tfrac{n}{L}))&-\operatorname{id}_{{\mathbb{C}}^{4\times
4}}\\\ \operatorname{id}_{{\mathbb{C}}^{4\times
4}}&0\end{pmatrix}\begin{pmatrix}\psi_{n}\\\ \psi_{n-1}\end{pmatrix}.$

Since $t(\theta)$ is self-involutive, $t(\theta)=t(\theta)^{-1}$, we find that
the associated Schrödinger cocycle $(1/L,A^{\mathscr{E},\theta,w})$ where
$L>0$ and $A^{\mathscr{E},\theta,w}\in
C^{\omega}({\mathbb{R}}/{\mathbb{Z}},\operatorname{SL}(8,{\mathbb{C}}))$ is
given as

$A^{\mathscr{E},\theta,w}(x):=\begin{pmatrix}Q^{\mathscr{E},\theta,w}(x)&-\operatorname{id}_{{\mathbb{C}}^{4\times
4}}\\\ \operatorname{id}_{{\mathbb{C}}^{4\times 4}}&0\end{pmatrix},$ (2.1)

where $Q^{\mathscr{E},\theta,w}(x):=t(\theta)(\mathscr{E}-t_{0}-V_{w}(x)).$
For $w_{1}=0$, we denote the cocycle by
$A^{\mathscr{E},\theta,w_{0}}_{\operatorname{ac}}$ and for $w_{0}=0$ by
$A^{\mathscr{E},\theta,w_{1}}_{\operatorname{c}}.$ The top left block matrix
of the cocycle (2.1) reads

$Q^{\mathscr{E},\theta,w}=\begin{pmatrix}-e^{-2\pi i\theta}&e^{-2\pi
i\theta}\mathscr{E}&-e^{-2\pi i\theta}w_{1}U^{+}_{\operatorname{c}}&e^{-2\pi
i\theta}w_{0}U\\\ e^{2\pi i\theta}\mathscr{E}&-e^{2\pi i\theta}&-e^{2\pi
i\theta}w_{0}U&e^{2\pi i\theta}w_{1}U^{-}_{\operatorname{c}}\\\ -e^{-2\pi
i\theta}w_{1}U^{-}_{\operatorname{c}}&e^{-2\pi i\theta}w_{0}U&-e^{-2\pi
i\theta}&e^{-2\pi i\theta}\mathscr{E}\\\ -e^{2\pi i\theta}w_{0}U&e^{2\pi
i\theta}w_{1}U^{+}_{\operatorname{c}}&e^{2\pi i\theta}\mathscr{E}&-e^{2\pi
i\theta}\end{pmatrix}.$ (2.2)

Let $L>0,$ we introduce the shift $Tx:=x+1/L$ and the $n$-th cocycle iterate

$A_{n}^{\mathscr{E},\theta,w}(x):=\prod_{i=n-1}^{0}A^{\mathscr{E},\theta,w}(T^{i}x)=A^{\mathscr{E},\theta,w}(x+\tfrac{n-1}{L})\cdot...\cdot
A^{\mathscr{E},\theta,w}(x).$

We observe that $A_{n}^{\mathscr{E},\theta,w}(x)$ is, for $n\geq 2$, of the
form

$\begin{split}A_{n}^{\mathscr{E},\theta,w}(x)=\begin{pmatrix}S_{n}^{\mathscr{E},\theta,w}(x)&T_{n-1}^{\mathscr{E},\theta,w}(x)\\\
S_{n-1}^{\mathscr{E},\theta,w}(x)&T_{n-2}^{\mathscr{E},\theta,w}(x)\end{pmatrix},\end{split}$
(2.3)

where

$S_{n}^{\mathscr{E},\theta,w}(x)=(\operatorname{det}(H^{\alpha,\beta}_{w,[0,n-1]}(\theta)-\mathscr{E}))_{\alpha\beta\in{\mathbb{C}}^{4}}\text{
and
}T_{n}^{\mathscr{E},\theta,w}(x)=(\operatorname{det}(H^{\alpha,\beta}_{w,[1,n]}(\theta)-\mathscr{E}))_{\alpha\beta\in{\mathbb{C}}^{4}}.$

Here, $H^{\alpha,\beta}_{w,[p,q]}$ is the Hamiltonian (1.2) on
$\ell^{2}(\\{p,..,q\\};\mathbb{C}^{4})$ with boundary conditions
$u_{p-1}=u_{q+1}=0$ and $u_{p}(j)=u_{q}(i)=0$ for $j\neq\alpha$ and
$i\neq\beta.$

In Figure 2 we illustrate the time-evolution of a discrete Gaussian-type wave-
packet $\psi_{n}=\frac{e^{-n^{2}/(2\sigma^{2})}}{\sqrt{2\pi\sigma^{2}}}e_{1}$,
where $e_{1}$ is the first unit vector and $\sigma=\sqrt{70}.$ In this work,
we are mostly concerned with irrational length scales
$L\in\mathbb{R}^{+}\backslash\mathbb{Q}.$ Under this assumption, the dynamics
becomes ergodic and Oseledet’s theorem implies the existence of eight
(possibly degenerate) Lyapunov exponents (LEs) $\gamma_{j}\in[-\infty,\infty)$
such that for almost every $x\in\mathbb{R}/\mathbb{Z}$ there is a filtration
${\mathbb{C}}^{4}=E_{x}^{1}\supset...\supset E_{x}^{k}$ that satisfies
$A^{\mathscr{E},\theta,w}(x)E_{x}^{j}\subset E_{x+L^{-1}}.$

Figure 2. Time-evolution of Gaussian wavepackets. (Top row) Time-evolved
discretized Gaussian state for chiral (left) and anti-chiral (right) model
with $L=3$ with weak coupling $w_{0}=0.1$ after time $T_{\text{fin}}=5000$
(Space-Amplitude plot). Gaussian state for strong coupling $w_{1}=1.9$,
$w_{0}=0$, and $L=\pi$ on the lower left figure for the chiral model after
$T_{\text{fin}}=2\cdot 10^{4}$ Localization effects are clearly visible. On
the lower right we see the time evolution (Time-Amplitude plot) corresponding
to the amplitude for a Gaussian wavepacket started at the upper lattice. Here,
$(1)$ and $(2)$ refer to the respective components labeling atoms of type A
and B, respectively. The wavepacket oscillates between the different layers

It then follows that for every $u\in E_{x}^{j}\setminus E_{x}^{j+1}$ we have
$\limsup_{n\to\infty}\frac{1}{n}\log\|A_{n}^{\mathscr{E},\theta,w}u\|=\gamma_{j}.$
Writing,
$\gamma_{1}(1/L,A^{\mathscr{E},\theta,w})\geq...\geq\gamma_{8}(1/L,A^{\mathscr{E},\theta,w})$
for the LEs of $(1/L,A^{\mathscr{E},\theta,w})$ repeated according to
multiplicity, they are given by

$\gamma_{i}(1/L,A^{\mathscr{E},\theta,w})=\lim_{n\to\infty}\frac{1}{n}\int_{{\mathbb{R}}/{\mathbb{Z}}}\log(\sigma_{i}[A^{\mathscr{E},\theta,w}_{n}(x)])\,dx,$

where $\sigma_{k}[B]$ is the $k$-th singular value of a matrix $B$, with the
convention that $\sigma_{k}[B]\geq\sigma_{k+1}[B].$ We also define
$\gamma^{k}(1/L,A^{\mathscr{E},\theta,w})=\sum_{j=1}^{k}\gamma_{j}(1/L,A^{\mathscr{E},\theta,w})$.
The LEs $\mathbb{R}\ni
t\mapsto\gamma^{k}(1/L,A^{\mathscr{E},\theta,w}(\bullet+it))$ are then convex
and piecewise affine functions [AJS15]. We emphasize that this property may
however not be true for Lyapunov exponents
$\gamma_{k}(1/L,A^{\mathscr{E},\theta,w}).$ In particular, one has Thouless’
formula [KS88]

$\begin{split}\gamma^{4}(1/L,A^{\mathscr{E},\theta,w})&=\int_{{\mathbb{R}}}\log|\mathscr{E}-\mathscr{E}^{\prime}|\,dn_{H_{w}(\theta,\phi)}(\mathscr{E}^{\prime})\\\
&=\lim_{N\to\infty}\frac{\log|\operatorname{det}(H_{w,{[0,N]}}(\theta,\phi,\vartheta)-\mathscr{E})|}{N},\end{split}$
(2.4)

where the last equality holds for almost all $\vartheta$ and
$dn_{H_{w}(\theta,\phi)}$ is the DOS measure for the Hamiltonian
$H_{w}(\theta,\phi,\vartheta)$ which is independent of $\vartheta$ for $1/L$
irrational. Indeed, if we introduce

$u_{N}(\theta,\phi,\vartheta):=\tfrac{\log\left|\det\left(H_{w}^{[N]}(\theta,\phi,\vartheta)-\mathscr{E}\right)\right|}{N},$

then we have by Thouless’ formula (2.4) that the sum of the four largest
Lyapunov exponents satisfies

$\gamma^{4}(1/L,A^{\mathscr{E},\theta,w})=\lim_{N\to\infty}\int_{\mathbb{T}}u_{N}(\theta,\phi,\vartheta)\,d\vartheta.$

Hence, we conclude that for almost all $\vartheta$ there is $\varepsilon$ such
that for $N$ sufficiently large one can estimate the determinant of the
Hamiltonian in terms of the Lyapunov exponent

$|\operatorname{det}(H_{w,[0,N]}(\theta,\vartheta)-\mathscr{E})|\geq
e^{(\gamma^{4}(1/L,A^{\mathscr{E},\theta,w})-\varepsilon)N}.$ (2.5)

Regarding the Lyapunov exponents of the cocycle in (2.3), we make the
following simple observation using the symplectic structure of our cocycle.

######  Lemma 2.2.

The LEs of $A_{n}^{\mathscr{E},\theta,w}(x)$ given by
$\gamma_{n}(1/L,A^{\mathscr{E},\theta,w})$ for $n\in\\{1,..,4\\}$ appear in
pairs satisfying

$\gamma_{n+4}(1/L,A^{\mathscr{E},\theta,w})=-\gamma_{n}(1/L,A^{\mathscr{E},\theta,w}).$

###### Proof.

We observe that using $\Omega=\begin{pmatrix}0&t(\theta)\\\
-t(\theta)&0\end{pmatrix}$ we have $A_{n}^{\mathscr{E},\theta,w}(x)^{*}\Omega
A_{n}^{\mathscr{E},\theta,w}(x)=\Omega.$ Thus,
$(A_{n}^{\mathscr{E},\theta,w}(x))^{-1}=\Omega^{-1}A_{n}^{\mathscr{E},\theta,w}(x)^{*}\Omega.$
On the other hand, $\Omega$ is anti self-adjoint, so the argument also applies
to the adjoint of $A_{n}^{\mathscr{E},\theta,w}(x)$ and then also to the
product $A_{n}^{\mathscr{E},\theta,w}(x)^{*}A_{n}^{\mathscr{E},\theta,w}(x)$
whose eigenvalues are the squared singular values of
$A_{n}^{\mathscr{E},\theta,w}(x).$ ∎

We also recall the characterization of the a.c. spectrum due to Kotani and
Simon [KS88] showing that

$S_{j}=\\{\mathscr{E}\in\mathbb{R};\text{ There are }2j\leq 8\text{ of LEs
}\gamma_{j}(1/L,A^{\mathscr{E},\theta,w})\text{ that vanish}\\},$ (2.6)

is the essential support of the absolutely continuous spectrum of multiplicity
$2j.$ In particular, if $S_{8}$ contains an open interval $I$, then the
spectrum of the Hamiltonian is purely absolutely continuous on $I.$

Figure 3. Hofstadter butterflies (1/L-Spectrum plots). The left figure shows
the spectrum for the anti-chiral ($w_{0}=1$) and the right figure for the
chiral potential ($w_{1}=1$), both for the case $\theta=0$ for $1/L\in[0,1].$
Figure 4. Hofstadter butterflies (1/L-Spectrum plots). Spectrum of chiral
Hamiltonian with subcritical tunneling $w_{1}=\frac{2}{5}.$ Similarly to the
AMO with coupling constant away from the critical coupling, the spectrum
starts to become more dense.

We also observe that by Hölder’s inequality, we have the following bounds

$\|A^{\mathscr{E},\theta,w_{1}}_{\operatorname{c}}(x)\|\leq
2+|\mathscr{E}|+w_{1}\text{ and
}\|A^{\mathscr{E},\theta,w_{0}}_{\operatorname{ac}}(x)\|\leq
2+|\mathscr{E}|+w_{0}.$ (2.7)

In particular, this clearly implies upper bounds on the LEs

$\gamma_{i}(1/L,A_{\operatorname{c}}^{\mathscr{E},\theta,w_{1}})\leq\log\left(2+|\mathscr{E}|+w_{1}\right)\text{
and
}\gamma_{i}(1/L,A_{\operatorname{ac}}^{\mathscr{E},\theta,w_{0}})\leq\log\left(2+|\mathscr{E}|+w_{0}\right).$
(2.8)

###  2.2. Complexification of LEs

After having stated upper bounds on LEs in (2.8), our first proposition gives
lower bounds on LEs. By Kotani-Simon theory, strict positivity of all Lyapunov
exponents, $\gamma_{i}(1/L,A^{\mathscr{E},\theta,w_{0}})>0$ for all
$i\in\\{1,..,4\\}$, this implies the absence of absolutely continuous
spectrum, cf. (2.6).

######  Proposition 2.3.

For the anti-chiral model the $i\in\\{1,..,4\\}$ LEs satisfy

$\begin{split}\gamma_{i}(1/L,A^{\mathscr{E},\theta,w_{0}}_{\operatorname{ac}})&\geq\operatorname{max}\\{i\log(w_{0}/3)-(i-1)\log(2+|\mathscr{E}|+w_{0}),0\\}\\\
\gamma^{i}(1/L,A^{\mathscr{E},\theta,w_{0}}_{\operatorname{ac}})&\geq
4i\log(w_{0}/3)\end{split}$

and analogously for the chiral model

$\begin{split}\gamma_{i}(1/L,A^{\mathscr{E},\theta,w_{1}}_{\operatorname{c}})&\geq\operatorname{max}\\{i\log(w_{1}/3)-(i-1)\log(2+|\mathscr{E}|+w_{1}),0\\}\\\
\gamma^{i}(1/L,A^{\mathscr{E},\theta,w_{1}}_{\operatorname{c}})&\geq
4i\log(w_{1}/3).\end{split}$

###### Proof.

In case of the anti-chiral model and $\varepsilon$ large, we find for (2.2)

$\begin{split}Q^{\mathscr{E},\theta,w_{0}}(x+i\varepsilon)=\frac{w_{0}e^{2\pi(\varepsilon-
ix)}}{3}\begin{pmatrix}0&0&0&e^{-2\pi i\theta}\\\ 0&0&-e^{2\pi i\theta}&0\\\
0&e^{-2\pi i\theta}&0&0\\\ -e^{2\pi
i\theta}&0&0&0\end{pmatrix}+\mathcal{O}(1),\end{split}$ (2.9)

and for $-\varepsilon$ large

$\begin{split}Q^{\mathscr{E},\theta,w_{0}}(x+i\varepsilon)=\frac{w_{0}e^{-2\pi(\varepsilon-
ix)}}{3}\begin{pmatrix}0&0&0&e^{-2\pi i\theta}\\\ 0&0&-e^{2\pi i\theta}&0\\\
0&e^{-2\pi i\theta}&0&0\\\ -e^{2\pi
i\theta}&0&0&0\end{pmatrix}+\mathcal{O}(1).\end{split}$ (2.10)

We then introduce the matrix

$\mathcal{V}=\begin{pmatrix}ie^{-2\pi i\theta}&0&-ie^{-2\pi i\theta}&0\\\
0&-ie^{2\pi i\theta}&0&ie^{2\pi i\theta}\\\ 0&1&0&1\\\ 1&0&1&0\end{pmatrix}$

such that

$\mathcal{V}^{-1}t(\theta)(\mathscr{E}-t_{0}-V_{w}(x+i\varepsilon))\mathcal{V}=\frac{iw_{0}e^{2\pi(\varepsilon-
ix)}}{3}\operatorname{diag}(1,1-1,-1)+\mathcal{O}(1).$

The chiral limit corresponds to the choice $w_{0}=0,$ such that for
$\varepsilon$ large in terms of $\nu_{\pm}(\theta):=e^{2\pi i\theta}(1\pm
i\sqrt{3}),$ we find

$\begin{split}Q(\theta,x+i\varepsilon,\mathscr{E})=\frac{w_{1}e^{-2\pi
i(x+i\varepsilon)}}{6}\begin{pmatrix}0&0&\nu_{-}(-\theta)&0\\\
0&0&0&-\nu_{-}(\theta)\\\ \nu_{+}(-\theta)&0&0&0\\\
0&-\nu_{+}(\theta)&0&0\end{pmatrix}+\mathcal{O}(1).\end{split}$ (2.11)

We then consider the matrix $\mathcal{U}_{\omega}$

$\mathcal{U}_{\omega}=\begin{pmatrix}\omega&-\omega&0&0\\\
0&0&-\omega&\omega\\\ 1&1&0&0\\\ 0&0&1&1\end{pmatrix},$ (2.12)

with $\omega:=e^{2\pi i/3}$, then for $\beta(\theta):=e^{2\pi i\theta},$ we
find

$\mathcal{U}_{\omega}^{-1}Q(\theta,x+i\varepsilon,\mathscr{E})\mathcal{U}_{\omega}=\frac{w_{1}}{3}e^{-2\pi
i(x+i\varepsilon)}\operatorname{diag}(-\beta(\theta)^{-1},\beta(\theta)^{-1},-\beta(\theta),\beta(\theta))+\mathcal{O}(1).$

Conversely, for $-\varepsilon$ large in terms of $\nu_{\pm}(\theta):=e^{2\pi
i\theta}(1\pm i\sqrt{3}),$ we find

$\begin{split}Q(\theta,x+i\varepsilon,\mathscr{E})=\frac{w_{1}e^{2\pi
i(x+i\varepsilon)}}{6}\begin{pmatrix}0&0&\nu_{+}(-\theta)&0\\\
0&0&0&-\nu_{+}(\theta)\\\ \nu_{-}(-\theta)&0&0&0\\\
0&-\nu_{-}(\theta)&0&0\end{pmatrix}+\mathcal{O}(1).\end{split}$ (2.13)

Thus,

$\mathcal{U}^{-1}_{\bar{\omega}}Q(\theta,x+i\varepsilon,\mathscr{E})\mathcal{U}_{\bar{\omega}}=\frac{w_{1}}{3}e^{-2\pi
i(x+i\varepsilon)}\operatorname{diag}(-\beta(\theta)^{-1},\beta(\theta)^{-1},-\beta(\theta),\beta(\theta))+\mathcal{O}(1).$

The four LEs of the complexified anti-chiral, $w_{0}>0=w_{1},$ and chiral,
$w_{1}>0=w_{0}$ cocycle coincide for each model individually and are for
$j\in\\{1,\ldots,4\\}$ given by

$\gamma_{j}(1/L,A_{\operatorname{ac}}^{\mathscr{E},\theta,w_{0}}(\bullet+i\varepsilon))=|\log(w_{0}/3)|+2\pi|\varepsilon|;\quad|\varepsilon|\geq\varepsilon_{0}$

and

$\gamma_{j}(1/L,A_{\operatorname{c}}^{\mathscr{E},\theta,w_{1}}(\bullet+i\varepsilon))=|\log(w_{1}/3)|+2\pi|\varepsilon|;\quad|\varepsilon|\geq\varepsilon_{0}.$

Convexity and piecewise affinity in $\varepsilon$ of the LEs $\gamma^{i}$
together with (2.8) then implies the claim. ∎

##  3\. Rational and almost rational moire lengths

###  3.1. Rational moiré lengths

When $1/L\in\mathbb{Q}$, the Hamiltonian $H_{w}$ is a periodic operator.
Hence, its spectrum can be studied using Floquet-Bloch theory and we obtain
the following spectral decomposition.

######  Proposition 3.1.

The spectrum of the Hamiltonian for $1/L\in\mathbb{Q}$ is purely absolutely
continuous and the density of states is continuous.

###### Proof.

We can apply [K, Theo. $6.10$] to see that
$\sigma_{\operatorname{sc}}(H_{w})=\emptyset,$ and any possible point spectrum
consists only of flat bands after applying the Floquet transform. The non-
existence of flat bands is then shown in Subsection 7.1. ∎

After having studied rational length scales, we shall turn now to irrational
length scales that approximate rational ones well.

###  3.2. Almost-rational (Liouville) moiré lengths

Recall that a number $\alpha\in{\mathbb{R}}^{+}\backslash\mathbb{Q}$ is called
a _Liouville number_ if for all $k\in\mathbb{N}$ there are
$p_{k},q_{k}\in\mathbb{N}$ such that

$|\alpha-p_{k}/q_{k}|<q_{k}^{-1}k^{-q_{k}}.$

For such numbers, it is well-known that quasi-periodic Schrödinger operators
do not exhibit point spectrum. As the next proposition shows, this holds true
for our matrix-valued discrete operators, when $1/L$ is Liouville.

######  Proposition 3.2.

Let $1/L$ be a Liouville number, then the Hamiltonian does not have any point
spectrum. In particular, if in addition
$\gamma_{4}(1/L,A^{\mathscr{E},\theta,w})>0$111this holds e.g. under the
assumptions of Prop. 2.3., then the spectrum of the Hamiltonian is purely
singular continuous.

###### Proof.

We start by estimating

$\sup_{|n|\leq
8q_{k}}\Big{|}A^{\mathscr{E},\theta,w}(x+n/L)-A^{\mathscr{E},\theta,w}(x+np_{k}/q_{k})\Big{|}\leq
C\sup_{|n|\leq 8q_{k}}|n||p_{k}/q_{k}-1/L|\leq Ck^{-q_{k}}.$ (3.1)

Then,

$\begin{split}&\sup_{|n|\leq
8q_{k}}\left\lVert\prod_{i=n}^{1}A^{\mathscr{E},\theta,w}(x+i/L)-\prod_{i=n}^{1}A^{\mathscr{E},\theta,w}(x+ip_{k}/q_{k})\right\rVert\|(\psi_{1},\psi_{0})\|\leq\sup_{|n|\leq
8q_{k}}|n|e^{C^{\prime}|n|}k^{-q_{k}}.\end{split}$ (3.2)

Recall then that if $T\in\operatorname{GL}(8,{\mathbb{C}})$ then by the
Cayley-Hamilton theorem there are $c_{i}$ not all zero such that
$\sum_{i=0}^{8}c_{i}T^{i}=0.$ Normalizing, we can assume that one of the
$c_{i}$ is equal to one and the other ones are strictly smaller in absolute
value. This shows that for any normalized vector $v$ we have

$\max(\|Tv\|,..,\|T^{8}v\|,..,\|T^{-1}v\|,..,\|T^{-8}v\|)\geq 1/8.$

Applying this result, we find writing $[k]=\\{1,2,..,k\\}$

$\max_{n\in\pm[8]q_{k}}\left\|\prod_{i=n}^{1}A^{\mathscr{E},\theta,w}(x+ip_{k}/q_{k})(\psi_{1},\psi_{0})^{t}\right\rVert\geq\tfrac{1}{8}\|(\psi_{1},\psi_{0})\|.$

Thus,

$\limsup_{n\to\infty}\frac{\|(\psi_{n+1},\psi_{n})\|}{\|(\psi_{1},\psi_{0})\|}\geq\limsup_{k\to\infty}\frac{\max_{n\in\pm[8]q_{k}}\left\|\prod_{i=n}^{1}A^{\mathscr{E},\theta,w}(x+ip_{k}/q_{k})(\psi_{1},\psi_{0})^{t}\right\rVert}{\|(\psi_{1},\psi_{0})\|}\geq\frac{1}{8}.$

This shows that if $1/L$ is a Liouville number, the operator does not have any
point-spectrum. The absence of of absolutely-continuous spectrum in case of
positive LEs follows immediately from Kotani-Simon theory (2.6). ∎

##  4\. Diophantine moiré lengths and Anderson localization

We saw in the previous section that for rational numbers and irrational moiré
length scales $1/L$ that are close to rational ones (so-called Liouville
numbers), the Hamiltonian does not exhibit any point spectrum. We will now
focus on moiré lengths $1/L$ described by real numbers on the opposite end,
satisfying for some $t>0$ a diophantine condition ($\operatorname{DC}_{t}$)

$\min_{n\in{\mathbb{Z}}}|n-\tfrac{k}{L}|>\tfrac{t}{|k|^{2}}\text{ for all
}k\in{\mathbb{Z}}\backslash\\{0\\},$ (4.1)

for which the Hamiltonian exhibits, as we will show, Anderson localization,
i.e. exponentially decaying eigenfunctions. We present one method in
Subsection 4.2 originally due to Bourgain [B05, Ch. $10$], which applies to
the Hamiltonian (1.2) in a very general sense and one more refined approach in
Subsection 4.4 for a special case of the anti-chiral Hamiltonian that goes
back to ideas of Jitomirskaya [J99-1]. The latter approach has been originally
introduced for the almost Mathieu operator and applies to a larger range of
Moiré length scales. The first approach has been extended to the matrix-valued
setting by Klein [K17]. Both approaches also imply dynamical localization

$\sup_{t>0}\left(\sum_{n\in\mathbb{Z}}(1+n^{2})\Big{|}(e^{itH}\psi)(n)\Big{|}^{2}\right)^{1/2}<\infty,$
(4.2)

as explained in [B05, Ch.10]. The above diophantine condition (4.1), which
appears naturally in the localization proofs in [B05, K17], applies to a set
of full measure of real numbers.

To start, we shall now recall the definition of _generalized eigenfunctions_
which characterize the spectrum by the Schnol-Simon theorem [CFKS87, Theo
$2.9$].

######  Definition 4.1.

Elements $\mathscr{E}\in\operatorname{Spec}(H_{w}(\theta,\phi,\vartheta))$ are
characterized by the existence of a _generalized eigenfunction_
$u:{\mathbb{Z}}\to{\mathbb{C}}$ with $|u(n)|\lesssim\langle n\rangle$
satisfying $H_{w}(\theta,\phi,\vartheta)u=\mathscr{E}u.$

To establish Anderson localization it therefore suffices to show that any
generalized eigenfunction decays exponentially and the usual proof proceeds
via Green’s function estimates showing exponential decay for the off-diagonal
entries of the Green’s function.

More precisely, one aims to show that $|u_{n}|\leq e^{-c|n|}\text{ as
}|n|\to\infty.$ Our main result of this section is

######  Theorem 1.

Consider the chiral $w=(0,w_{1})$ or anti-chiral $w=(w_{0},0)$ Hamiltonian
$H_{w}(\theta).$ For large enough coupling $|w|\geq C$, for some constant
$C>0,$ the Hamiltonian satisfies for a set of full measure of reciprocal
length scales $1/L\in\mathbb{T}$ dynamical localization (4.2) and therefore
also Anderson localization.

###  4.1. Preliminaries

In the sequel, we shall write for a block matrix $A\in\mathbb{C}^{4n\times
4n}$ where $n\in\mathbb{N}$

$A=(A_{\gamma,\gamma^{\prime}})_{\gamma,\gamma^{\prime}\in[4n]}=(A(i,j))_{i,j\in[n]}$

with $A(i,j)\in{\mathbb{C}}^{4\times 4}$ being themselves matrices, whereas
$A_{\gamma,\gamma^{\prime}}$ are scalars.

Let $\mathcal{N}=[n_{1},n_{2}]=\\{n\in\mathbb{Z};n_{1}\leq n\leq n_{2}\\},$ we
define two canonical restrictions

$\begin{split}P_{\mathcal{N}}^{-}&=\begin{pmatrix}0_{\ell^{2}((-\infty,n_{1}-1];{\mathbb{C}}^{4})}&\operatorname{id}_{\ell^{2}(\mathcal{N};{\mathbb{C}}^{4})}&0_{\ell^{2}([n_{2}+1,\infty);{\mathbb{C}}^{4})}\end{pmatrix}\text{
and }\\\
P_{\mathcal{N}}^{+}&=\begin{pmatrix}0_{\ell^{2}((-\infty,n_{1}-2];{\mathbb{C}}^{4})}\oplus
0_{{\mathbb{C}}^{2\times 2}}&1_{\mathcal{N}}^{+}&0_{{\mathbb{C}}^{2\times
2}}\oplus
0_{\ell^{2}([n_{2}+1,\infty),{\mathbb{C}}^{4})}\end{pmatrix},\end{split}$
(4.3)

where $1_{\mathcal{N}}^{+}=\operatorname{id}_{{\mathbb{C}}^{2\times
2}}\oplus\ell^{2}_{([n_{1},n_{2}-1],{\mathbb{C}}^{4})}\oplus\operatorname{id}_{{\mathbb{C}}^{2\times
2}}$. Thus, $P_{\mathcal{N}}^{+}$ is shifted by two components compared to
$P_{\mathcal{N}}^{-}.$ The operator $P_{\mathcal{N}}^{-}$ is just the
projection onto ${\mathbb{C}}^{4}$-valued elements on $\mathcal{N}.$ On the
other hand, $P_{\mathcal{N}}^{+}$ projects onto ${\mathbb{C}}^{4}$-valued
elements on $[n_{1},n_{2}-1]$ and in addition the last two components at
$n_{1}-1$ and the first two components of $n_{2}.$

In the case that $\mathcal{N}=[0,N-1]$ we also write $[N]$ instead
$\mathcal{N}$ and introduce

$\begin{split}H_{w}^{\pm,\mathcal{N}}(\theta,\phi,\vartheta)&:=P^{\pm}_{\mathcal{N}}H_{w}(\theta,\phi,\vartheta)P^{\pm}_{\mathcal{N}}.\end{split}$
(4.4)

The Hamiltonian $H_{w}^{-,\mathcal{N}}$ is obviously defined on
$\ell^{2}(\mathcal{N},{\mathbb{C}}^{4}).$ The Hamiltonian
$H_{w}^{+,\mathcal{N}}$ is defined on $[n_{1},n_{2}-1]$, but in addition takes
the two last components of the point $n_{1}-1$ and the first two components at
$n_{2}$ into account. Thus, by shifting two components, we can (and shall)
also consider this one as an operator on
$\ell^{2}(\mathcal{N},{\mathbb{C}}^{4}).$

Now, let
$\mathscr{E}\notin\operatorname{Spec}(H_{w}^{\pm,\mathcal{N}}(\theta,\phi,\vartheta))$
and $n,m\in\mathcal{N}$, we define the _Green’s function_
$G^{\pm,\mathcal{N}}_{w}(\theta,\phi,\vartheta,\mathscr{E})\in{\mathbb{C}}^{4|\mathcal{N}|\times
4|\mathcal{N}|}$ by

$G^{\pm,\mathcal{N}}_{w}(\theta,\phi,\vartheta,\mathscr{E})(n,m):=(H_{w}^{\pm,\mathcal{N}}(\theta,\phi,\vartheta)-\mathscr{E})^{-1}(n,m).$

The Green’s function is a $|\mathcal{N}|\times|\mathcal{N}|$ block matrix,
with blocks that are themselves $4\times 4$ matrices over ${\mathbb{C}}.$ Let
$\varphi$ be a solution to
$(H_{w}(\theta,\phi,\vartheta)-\mathscr{E})\varphi=0$ with
$\mathscr{E}\notin\operatorname{Spec}(H_{w}^{\pm,\mathcal{N}}(\theta,\phi,\vartheta))$,
then it follows as for discrete Schrödinger operators, that for $n$ located
between $n_{1}$ and $n_{2}$, we have

$f_{n}=-G^{\pm,\mathcal{N}}_{w}(\theta,\phi,\vartheta,\mathscr{E})(n,n_{1})t(\theta)f_{n_{1}-1}-G^{\pm,\mathcal{N}}_{w}(\theta,\phi,\vartheta,\mathscr{E})(n,n_{2})t(\theta)f_{n_{2}+1}.$

Hence,

$\|f_{n}\|\leq\|t(\theta)\|\left(\|G^{\pm,\mathcal{N}}_{w}(\theta,\phi,\vartheta,\mathscr{E})(n,n_{1})\|\|f_{n_{1}-1}\|+\|G^{\pm,\mathcal{N}}_{w}(\theta,\phi,\vartheta,\mathscr{E})(n,n_{2})\|\|f_{n_{2}+1}\|\right).$
(4.5)

From this identity it is clear that good decay estimates on the Green’s
function implies the decay of eigenfunctions.

In terms of
$\mathscr{V}_{i}(w,\mathscr{E})=t_{0}+V_{w}(\tfrac{i}{L})-\mathscr{E},$ we can
write

$H_{w}^{[N]}(\theta)-\mathscr{E}=\begin{pmatrix}\mathscr{V}_{1}(w,E)&t(\theta)&0&\cdots&0\\\
t(\theta)&\mathscr{V}_{2}(w,\mathscr{E})&t(\theta)&\ddots&\vdots\\\
0&t(\theta)&\mathscr{V}_{3}(w,\mathscr{E})&\ddots&0\\\
\vdots&\ddots&\ddots&\ddots&t(\theta)\\\
0&\cdots&0&t(\theta)&\mathscr{V}_{n}(w,\mathscr{E})\end{pmatrix}.$

Each entry of this block matrix is a $4\times 4$ matrix and $H_{w}^{[N]}$ is a
matrix of size $4N\times 4N.$

###  4.2. Almost sure Anderson localization

We start by introducing

$\mu^{[N]}(\theta,\phi,\vartheta,w,\mathscr{E})_{\alpha,\alpha^{\prime}}:=\operatorname{det}((H_{w}^{[N]}(\theta,\phi,\vartheta)-\mathscr{E})_{\alpha,\alpha^{\prime}}),$
(4.6)

where $()_{\alpha,\alpha^{\prime}}$ indicates the
$\alpha,\alpha^{\prime}$-minor of the matrix.

The importance of the minors is due to Cramer’s rule

$\begin{split}(H_{w}^{[N]}(\theta,\phi,\vartheta)-\mathscr{E})^{-1}_{\alpha,\alpha^{\prime}}=\frac{\mu^{[N]}(\theta,\phi,\vartheta,w,\mathscr{E})_{\alpha,\alpha^{\prime}}}{\operatorname{det}(H_{w}^{[N]}(\theta,\phi,\vartheta)-\mathscr{E})}.\end{split}$
(4.7)

For $\gamma\in[4n]$ we introduce $n(\gamma)\in\mathbb{N}$ such that
$\gamma=4n(\gamma)+r\text{ with }-4\leq r<0.$ Then, by [K17, Prop. $2$] there
is $C<\infty$ such that for all $N$, all $\alpha,\alpha^{\prime}\in[4N]$, and
all $\vartheta,\phi,\theta,$ we have

$\frac{1}{4N}\log\left(|\mu^{[N]}(\theta,\phi,\vartheta,w,\mathscr{E})_{\alpha,\alpha^{\prime}}|\right)\leq\left(1-\frac{|n(\alpha)-n(\alpha^{\prime})|}{4N}\right)\log|w/3|+C,$
(4.8)

where $w$ is either $w_{0}$ (anti-chiral) or $w_{1}$ (chiral).

Next, we turn our attention to the following set studying the deviations of
the ergodic mean in the Thouless formula (2.4)

$\mathcal{B}_{N}^{M}(1/L,\theta,\phi,\mathscr{E}):=\left\\{\vartheta\in\mathbb{T};\frac{1}{M}\sum_{j=0}^{M-1}u_{N}(\theta,\phi,\vartheta+j/L)\leq(1-\delta)\gamma^{4}(1/L,A^{\mathscr{E},\theta,w})\right\\}$

for fixed $\mathscr{E}$ and $1/L\in\operatorname{DC}_{t},$ with
$\operatorname{DC}_{t}$ defined in (4.1). We then have the following
Proposition, similar to [B05, Prop.7.19] and [K17, Prop.$4$], which shows that
the set of bad frequencies at which the Green’s function has no good a priori
decay properties is small.

######  Proposition 4.2.

Fix $t>0$ and let $1/L\in\operatorname{DC}_{t}.$ For any $N,M$ large enough,
the set $\mathcal{B}_{N}^{M}(1/L,\theta,\mathscr{E})$ is exponentially small
in M, such that there is $a>0$ such that
$|\mathcal{B}_{N}^{M}(1/L,\theta,\mathscr{E})|<e^{-M^{a}}$ and is a semi-
algebraic set222see [B05, Ch. $9$] for a comprehensive definition of this
concept. of degree $\mathcal{O}(N^{2}M).$ However, there is $p\in\mathbb{N}$
such that for all $\vartheta$

$\left\lvert\left\\{0\leq
n<N^{p}:(H_{w}^{[N]}(\theta,\phi,\vartheta+n/L)-\mathscr{E})^{-1}\text{ is not
a good Green's function}.\right\\}\right\rvert\ll N^{p}$

where the Green’s function is called _good_ , if for some $\varepsilon>0$ the
decay is

$\frac{\log((H_{w}^{[N]}(\theta,\phi,\vartheta+n/L)-\mathscr{E})^{-1}_{\alpha,\alpha^{\prime}})}{N}<-\left(\frac{|n(\alpha)-n(\alpha^{\prime})|}{N}-\varepsilon\right)\gamma^{4}(1/L,A^{\mathscr{E},\theta,w}).$

###### Proof.

The diophantine condition enters in the following quantitative version of
Birkhoff’s ergodic theorem [DK16, Theo. $6.5$]: Let
$1/L\in\operatorname{DC}_{t}$ and $M\geq t^{-2}$, then for
$S=\mathcal{O}(\gamma^{4}(1/L,A^{\mathscr{E},\theta,w}))$ there is $a>0$ such
that

$\left\lvert\left\\{\vartheta\in\mathbb{T}:\left\lvert\frac{1}{M}\sum_{j=0}^{M-1}u_{N}(\theta,\phi,\vartheta+j/L)-\int_{\mathbb{T}}u_{N}(\theta,\phi,\vartheta)\,d\vartheta\right\rvert>SM^{-a}\right\\}\right\rvert<e^{-M^{a}}.$
(4.9)

In other words, the set of points where the ergodic average is far away from
the average is exponentially small. If $\vartheta$ is not in the set in (4.9),
then by Thouless’ formula (2.4)

$\begin{split}\frac{1}{M}\sum_{j=0}^{M-1}u_{N}(\theta,\phi,\vartheta+j/L)&\geq\int_{\mathbb{T}}u_{N}(\theta,\phi,\vartheta)\,d\vartheta-
SM^{-a}\\\
&=\gamma^{4}(1/L,A^{\mathscr{E},\theta,w})(1-\mathcal{O}(1)M^{-a})-o(1)\\\
&\geq(1-\delta)\gamma^{4}(1/L,A^{\mathscr{E},\theta,w}),\end{split}$ (4.10)

where $\delta$ can be chosen arbitrarily small for $M$ and $N$ large enough.
This implies the estimate on the measure of
$B_{N}^{M}(1/L,\theta,\mathscr{E}).$

One can then estimate the Green’s function by Cramer’s rule (4.7). Thus,

$\frac{\log((H_{w}^{[N]}(\theta,\phi,\vartheta)-\mathscr{E})^{-1}_{\alpha,\alpha^{\prime}})}{N}=\frac{\log(\mu^{[N]}(\theta,\phi,\vartheta,w,\mathscr{E})_{\alpha,\alpha^{\prime}})}{N}-u_{N}(\theta,\phi,\vartheta).$

Let $\vartheta\notin\mathcal{B}_{N}^{M}(1/L,\theta,\mathscr{E}),$ then we have
for some $j\in\\{0,...,M-1\\}$ that

$u_{N}(\theta,\phi,\vartheta+j/L)>(1-\delta)\gamma^{4}(1/L,A^{\mathscr{E},\theta,w}).$

This implies that for this choice of $j$, we have together with [K17, Prop.
$2$] for some $C>0$ and $|w|$ large enough

$\begin{split}\frac{\log((H_{w}^{[N]}(\theta,\phi,\vartheta+j/L)-\mathscr{E})^{-1}_{\alpha,\alpha^{\prime}})}{N}&\leq\gamma^{4}(1/L,A^{\mathscr{E},\theta,w})\Big{(}1-\tfrac{|n(\alpha)-n(\alpha^{\prime})|}{N}\Big{)}\\\
&\quad+C-(1-\delta)\gamma^{4}(1/L,A^{\mathscr{E},\theta,w})\\\
&<-\left(\tfrac{|n(\alpha)-n(\alpha^{\prime})|}{N}-\varepsilon\right)\gamma^{4}(1/L,A^{\mathscr{E},\theta,w})\end{split}$
(4.11)

for some $\varepsilon$ sufficiently small. This shows that the Green’s
function satisfies an exponential decay estimate. We now observe that for
$\vartheta\in\mathcal{B}_{N}^{M}(1/L,\theta,\mathscr{E})$

$\begin{split}\prod_{j=0}^{M-1}\operatorname{det}\left(H_{w}^{[N]}(\theta,\phi,\vartheta+j/L)-\mathscr{E}\right)\leq
e^{(1-\delta)MN\gamma^{4}(1/L,A^{\mathscr{E},\theta,w})},\end{split}$ (4.12)

where the left hand side is a Fourier polynomial of degree at most $4N^{2}M.$
Setting $M=N^{1/2}$ we see that
$\mathcal{B}_{N}(1/L,\mathscr{E})=\mathcal{B}_{N}^{N^{1/2}}(1/L,\mathscr{E})$
is a semi-algebraic set of degree at most $4N^{3/2}.$ We can then use [B05,
Corr. $9.7$] to see that for $N_{1}:=N^{p}$ with $p>0$ large enough

$\left|\\{k=0,...,N_{1};\vartheta+\tfrac{k}{L}\in\mathcal{B}_{N}(1/L,\theta,\mathscr{E})\\}\right|<N_{1}^{(1-\delta(L))}$

for some small $\delta(L)>0$ and $N$ large enough. Thus,
$\vartheta+\tfrac{k}{L}\notin\mathcal{B}_{N}(1/L,\theta,\mathscr{E})$ is
common. This implies by (4.11) that for every $\vartheta\in\mathbb{T}$ we can
find $n\in[0,N_{1})$ such that
$G_{N}(\theta,\phi,\vartheta+\tfrac{n}{L},\mathscr{E})$ is a good Green’s
function by (4.10). ∎

###  4.3. Paving good Green’s functions

To finish the localization argument one wants to cover any large interval
$\mathcal{N}^{\prime}$ by smaller intervals $\\{\mathcal{N}_{n}\\}$, with
$\mathcal{N}_{n}=[N]+j_{n}$, of length $|\mathcal{N}_{n}|=N$ as discussed in
the previous subsection on which the Green’s function exhibits good decay
properties. More precisely, let $N^{\prime}=N^{c}\gg N\gg 1$ with $N^{\prime}$
a much larger number for $C>1$ than $N$, then, as we will see, any interval
$\mathcal{N}^{\prime}\supset[\tfrac{N^{\prime}}{2},2N^{\prime}]$ of length of
order $N^{\prime}$, i.e. $N^{\prime}\lesssim|\mathcal{N}^{\prime}|\lesssim
N^{\prime}$ can be covered by a collection $\\{\mathcal{N}_{n}\\}$ of length
$|\mathcal{N}_{n}|=N$ such that $G_{w}^{\mathcal{N}_{n}}(\theta,E)$ exhibits
exponential decay as in Proposition 4.2.

The paving of Green’s function follows by studying
$\mathcal{N}=\mathcal{N}_{1}\cup\mathcal{N}_{2}$,
$\mathcal{N}_{1}\cap\mathcal{N}_{2}=\emptyset.$ Then by the resolvent identity
[B05, p.60]

$G_{w}^{\mathcal{N}}(\theta,\mathscr{E})=(G_{w}^{\mathcal{N}_{1}}(\theta,\mathscr{E})+G_{w}^{\mathcal{N}_{2}}(\theta,\mathscr{E}))(\operatorname{id}-(H^{\mathcal{N}}_{w}(\theta)-(H^{\mathcal{N}_{1}}_{w}(\theta)+H^{\mathcal{N}_{2}}_{w}(\theta)))G_{w}^{\mathcal{N}}(\theta,\mathscr{E})).$

Assuming $m\in\mathcal{N}_{1}$ and $n\in\mathcal{N},$ this implies [B05, p.60]

$|G_{w}^{\mathcal{N}}(m,n)|\leq|G_{w}^{\mathcal{N}}(m,n)|\delta_{n\in\mathcal{N}_{1}}+\sum_{\begin{subarray}{c}n^{\prime}\in\mathcal{N}_{1}\\\
n^{\prime\prime}\in\mathcal{N}_{2},|n^{\prime}-n^{\prime\prime}|=1\end{subarray}}|G_{w}^{\mathcal{N}}(m,n^{\prime})||G_{w}^{\mathcal{N}}(n^{\prime\prime},n)|.$

This estimate shows that the concatenation of good Green’s function is again a
good Green’s function.

The final part of the argument for localization then consists of removing the
energy dependence in the exceptional set
$\mathcal{B}_{N}(1/L,\theta,\mathscr{E}).$ We know that
$\mathcal{B}_{N}(1/L,\theta,\mathscr{E})$ is a small set for every fixed
energy, but as the sets could be disjoint for different energies, this could
imply that for example every $\theta$ will eventually be in one of these sets
for some energy. A key observation is now that it suffices to consider a
finite set of energies determined by the union of spectra of finite-rank
approximations of the Hamiltonian. This restriction is sufficient, since we
already know that for any $\vartheta$ the Green’s function will be
(eventually) good by the last point in Proposition 4.2, we just do not know if
good Green’s functions can be paved together for general $\vartheta.$ Indeed,
we shall study sets

$S_{N}(1/L,\theta)=\bigcup_{\mathscr{E}\in\mathcal{E}(\theta,\vartheta)}\mathcal{B}_{N}(1/L,\theta,\mathscr{E}),$
(4.13)

with $\mathcal{B}_{N}$ as defined in the proof of Proposition 4.2. To exhibit
when the Green’s function exhibits good decay, we must therefore study when
$\vartheta+n/L\notin S_{N}(1/L,\theta)$ for $\sqrt{N^{\prime}}\leq n\leq
2N^{\prime},$ where $N^{\prime}$ is the large constant from the paving
argument. This property is linked to the Green’s function decay by Proposition
4.2. The set of energies considered in (4.13) is just the union of all finite
rank approximations

$\mathcal{E}(\theta,\vartheta)=\bigcup_{1\leq j\leq
N^{p}}\operatorname{Spec}(H^{[-j,j]}_{w}(\theta,\vartheta)).$

Then, by a simple union bound
$|\mathcal{E}(\theta,\vartheta)|\leq\sum_{j=1}^{N^{p}}4(2j+1)=4N^{p}(N^{p}+2).$

Notice that since sets $\mathcal{B}_{N}$ are of exponentially small measure,
this also implies that $|S_{N}(1/L,\theta)|=\mathcal{O}(e^{-N^{a^{\prime}}})$
for any $0<a^{\prime}<a.$

We then consider
$\mathscr{S}_{N}(\theta):=\\{(1/L,\vartheta);1/L\in\operatorname{DC}\text{ and
}\vartheta\in S_{N}(1/L,\theta)\\}$ and aim to show that by discarding a
suitable zero set of $1/L$, we may ensure that $\vartheta+n/L\notin
S_{N}(1/L,\theta)$ for some $\sqrt{N^{\prime}}\leq n\leq 2N^{\prime}$.

One then has by [B05, Ch.$10$], invoking the semi-algebraic sets, the estimate

$\Omega_{N}:=\\{1/L\in\mathbb{T};(1/L,n/L)\in\mathscr{S}_{N}(\theta)\text{ for
}n\sim N^{\prime}\\}\Rightarrow|\Omega_{N}|\leq 1/\sqrt{N^{\prime}}.$

Hence, since $N^{\prime}=N^{C}$ with $C$ large, it follows that
$\Omega:=\limsup_{N\to\infty}\Omega_{N}$ (in the measure-theoretic sense) is
of zero measure. Hence, as long as a diophantine $1/L\notin\Omega$, then
$1/L\notin\Omega_{N}$ for $N$ large enough. This yields the localization
result of Theorem 1.

Figure 5. Lowest eigenfunction of chiral Hamiltonian restricted to interval
$\\{-100,-99,...,100\\}$. Figures on the left are for _rational_ length scales
$L=20,$ whereas on the right we study the strongly _irrational_ (diophantine)
$L=1/\text{golden mean}.$ The top figures correspond to $w_{0}=\frac{3}{2}$,
the bottom ones to $w_{0}=6.$

###  4.4. Arithmetic version of Anderson localization

We shall now show how particle-hole symmetry can be used to establish
localization along the lines of Jitomirskaya’s arithmetic argument for the AMO
[J99-1]. This argument heavily relies on the cosine-nature of the potential
and requires non-resonant tunneling phases for $A$ and $B$ atoms of the
potential. In particular, the argument does not seem to carry over easily to
the case of chiral coupling. We consider the Hamiltonian in the anti-chiral
limit, for $\theta=0$, but take $\phi=1/4$ with plain cosine potential
$U(x)=\cos(2\pi x)$. The Hamiltonian then reads in terms of
$K(\theta):=1+e^{2\pi i\theta}(\tau+\tau^{*})$

$H_{w_{0}}(\theta,\tfrac{1}{4},\vartheta)=\begin{pmatrix}0&K(\theta)&w_{0}U(\tfrac{x-\phi}{L})&0\\\
K(\theta)^{*}&0&0&w_{0}U(\tfrac{x+\phi}{L})\\\
w_{0}U(\tfrac{x-\phi}{L})&0&0&K(\theta)\\\
0&w_{0}U(\tfrac{x+\phi}{L})&K(\theta)^{*}&0\end{pmatrix}.$ (4.14)

Let us first comment on the applicability of Jitomirskaya’s method for matrix-
valued operators. In general, this method does not seem to apply well to
matrix-valued operators. However, for the particular Hamiltonian (4.14), the
characteristic polynomial of $H_{w_{0}}$ restricted to $N$ lattice sites will
be, as we will show, a Fourier polynomial of degree $4N.$ Since the polynomial
is also even, it suffices to study this polynomial at $2N$ distinct points.
The definition of the Hamiltonian with $\phi=1/4$, implies that there are
natural $2N$ values of the characteristic polynomial of shifts of the matrix
at which we can interpolate the characteristic polynomial of the matrix.

######  Theorem 2.

Let $w_{0}$ be sufficiently large and $1/L\in\mathbb{T}$ diophantine. Then the
Hamiltonian (4.14) exhibits Anderson localization.

We shall now sketch the proof of Theorem 2 emphasizing the main steps and
differences compared with [J99-1]. In order to flip lattices $2$ and $3$, we
conjugate the Hamiltonian by $P=\operatorname{diag}(1,\sigma_{1},1)$ such that
$\mathscr{H}_{w_{0}}(\vartheta):=PH_{w_{0}}(\theta,\phi=\tfrac{1}{4},\vartheta)P$
becomes

$\mathscr{H}_{w_{0}}(\vartheta)=\begin{pmatrix}0&w_{0}U(\tfrac{x-\phi}{L})&K(\theta)&0\\\
w_{0}U(\tfrac{x-\phi}{L})&0&0&K(\theta)\\\
K(\theta)^{*}&0&0&w_{0}U(\tfrac{x+\phi}{L})\\\
0&K(\theta)^{*}&w_{0}U(\tfrac{x+\phi}{L})&0\end{pmatrix}.$

We recall the definition of $(\gamma,k)$-regularity which we shall apply to
our operator $\mathscr{H}_{w_{0}}(\vartheta).$

######  Definition 4.3 ($(\gamma,k)$-regularity).

Let $\mathscr{E},\gamma\in\mathbb{R}$ and $k\geq 1$. We call a number
$n\in\mathbb{Z}$ then $(\gamma,k)$-regular if there is
$\mathcal{N}=[n_{1},n_{2}]\subset{\mathbb{Z}}$, with $n\in\mathcal{N}$ such
that

  * •

$n_{2}=n_{1}+k-1,$ $I=\\{4n_{1},4n_{2}+3\\},$

  * •

$\alpha=4n\in[4n_{1},4n_{2}+3]$, $d(I,4n)>\frac{4k}{5},$

  * •

$|G^{[n_{1},n_{2}]}_{w}(\vartheta,E)_{\alpha,\alpha^{\prime}}|<e^{-\gamma|\alpha-\alpha^{\prime}|}$
where $\alpha^{\prime}\in I.$

If $(\gamma,k)$ is not _regular_ , we call it _singular_. In particular, for
$k$ sufficiently large and $\gamma$ fixed, it is clear that any point of
$y\in\mathbb{Z}$ such that $u(y)\neq 0$, for $u$ a generalized eigenfunction,
is $(\gamma,k)$-singular.

Observe that for $\mathcal{N}=[0,N-1]$ the characteristic polynomial

$p^{\mathcal{N}^{\pm}}(\vartheta)=\det(\mathscr{H}_{w_{0}}^{\mathcal{N}^{\pm}}(\vartheta)-\mathscr{E})$

has the property that $p^{\mathcal{N}^{\pm}}$ is an even function of
$\vartheta+\tfrac{N-1}{2L}$ and
$p^{\mathcal{N}^{\pm}+1}(\vartheta)=p^{\mathcal{N}^{\pm}}(\vartheta+\tfrac{1}{L}).$
In addition, in the case of $\phi=1/4$ we have
$p^{\mathcal{N}^{+}}(\vartheta)=p^{\mathcal{N}^{-}}(\vartheta-\tfrac{1}{2L})$.
Hence $\vartheta\mapsto p^{\mathcal{N}^{-}}(\vartheta-\tfrac{N-1}{2L})$ is an
even function, satisfies
$p^{\mathcal{N}^{-}}(\vartheta)=p^{\mathcal{N}^{-}}(\vartheta+\frac{1}{2})$,
which is an additional symmetry that does not exist in the case of the AMO,
and therefore by the orthogonality of the Fourier basis

$p^{\mathcal{N}^{-}}(\vartheta-\tfrac{N-1}{2L})=\sum_{j\in[0,2N]}\tilde{b}_{j}(L)\cos\left(4\pi
j\vartheta\right)\text{ for }\tilde{b}_{j}(L)\in{\mathbb{R}},$

such that for some new $b_{j}\in{\mathbb{R}}$

$q\big{(}\cos\left(2\pi(\vartheta+\tfrac{N-1}{2L})\right)^{2}\big{)}:=p^{\mathcal{N}^{-}}(\vartheta)=\sum_{j\in[0,2N]}b_{j}\cos^{2j}\left(2\pi(\vartheta+\tfrac{N-1}{2L})\right).$

We observe that $q$ is a polynomial of degree $2N.$

######  Lemma 4.4.

Suppose $n\in\mathbb{Z}$ is $(\gamma,k)$-singular, then for any $j$ with

$n-\lceil\tfrac{3}{4}k\rceil\leq j\leq
n-\lfloor\tfrac{3}{4}k\rfloor+\tfrac{k+1}{2},$

we have that for $\mathcal{N}=[j,k+j]$

$|\operatorname{det}(\mathscr{H}_{w_{0}}^{\mathcal{N}^{\pm}}(\vartheta)-\mathscr{E})|\leq
e^{4k\big{(}|\log(w_{0}/3)|+\frac{\gamma-4|\log(w_{0}/3)|}{5}+\mathcal{O}(1)\big{)}}\text{
for all }\vartheta\in\mathbb{T}_{1}.$

###### Proof.

Let $n_{1}:=j$ and $n_{2}:=k+j$. By the definition of singularity, Cramer’s
rule (4.8), and (4.7), we have for $n_{i}\in\\{n_{1},n_{2}\\}$ and $n$ as in
the definition of singularity

$\begin{split}|\operatorname{det}(\mathscr{H}_{w_{0}}^{\mathcal{N}^{\pm}}(\vartheta)-\mathscr{E})|&\leq
e^{\gamma|\alpha-\alpha_{i}|}\|\mu^{\mathcal{N}}(w,\vartheta,E)(n,n_{i})\|\\\
&\leq
e^{\gamma|\alpha-\alpha_{i}|}e^{4k|\log(w_{0}/3)|\left(1-\tfrac{|n(\alpha)-n(\alpha_{i})|}{4k}\right)+\mathcal{O}(4k)}\\\
&\leq
e^{4k|\log(w_{0}/3)|}e^{(\gamma-4|\log(w_{0}/3)|)|\alpha-\alpha_{i}|+\mathcal{O}(4k)}\\\
&\leq
e^{4k\big{(}|\log(w_{0}/3)|+\frac{\gamma-4|\log(w_{0}/3)|}{5}+\mathcal{O}(1)\big{)}},\end{split}$
(4.15)

where we used that $|\alpha-\alpha_{i}|\geq\frac{4k}{5}$ and
$\alpha_{i}=4n_{i}$ is the index that corresponds to $n_{i}.$ ∎

Let $n_{1}$ and $n_{2}$ be both $(\gamma,k=N)$-singular,
$d:=n_{2}-n_{1}>\frac{k+1}{2}$, and
$x_{i}=n_{i}-\lfloor{\tfrac{3k}{4}\rfloor}.$ We now set, for $\phi=1/4$

$\begin{split}\vartheta_{j}:=\begin{cases}&\vartheta+\tfrac{x_{1}+\frac{k-1}{2}+\frac{j}{2}}{L},\qquad\qquad
j=0,...,2\lceil\frac{k+1}{2}\rceil-1\\\
&\vartheta+\tfrac{x_{2}+\frac{k-1}{2}+\frac{j}{2}-\lceil\frac{k+1}{2}\rceil}{L},\
\quad j=2\lceil\frac{k+1}{2}\rceil,..,2k.\end{cases}\end{split}$ (4.16)

By the assumption that $d>\frac{k+1}{2}$ all $\vartheta_{j}$ are distinct.
Lagrange interpolation yields then, since $p^{\mathcal{N}^{-}}$ is even with
respect to $\vartheta+\tfrac{N-1}{2L}$,

$q(z^{2})=\sum_{j\in[0,2N]}q(\cos(2\pi\vartheta_{j})^{2})\frac{\prod_{l\neq
j}(z^{2}-\cos(2\pi\vartheta_{l}))}{\prod_{l\neq
j}(\cos(2\pi\vartheta_{j})-\cos(2\pi\vartheta_{l}))}.$ (4.17)

Finally, we have by [J99-1, Lemma $7$], [AJ10, Lemma 5.8] for $d<k^{\alpha}$
with $\alpha<2$ that for any $\varepsilon>0$ there is $K>0$ such that for
$k>K$ and all $z\in[-1,1]$

$\left\lvert\frac{\prod_{l\neq j}(z-\cos(2\pi\vartheta_{l}))}{\prod_{l\neq
j}(\cos(2\pi\vartheta_{j})-\cos(2\pi\vartheta_{l}))}\right\rvert\leq
e^{k\varepsilon}.$

Combining this estimate with Lemma 4.4, which applies to the above choice of
(4.16), and the interpolation formula (4.17), we thus conclude that for all
$\vartheta\in[-1,1]$

$|p^{\mathcal{N}^{-}}(\vartheta)|\leq(2k+1)e^{4k\big{(}|\log(w_{0}/3)|+\frac{\gamma-|\log(w_{0}/3)|}{5}+\mathcal{O}(1)\big{)}}.$
(4.18)

In addition, we have by (2.5) and Proposition 2.3 the existence of some
$\vartheta_{0}\in\mathbb{T}_{1}$ such that for $k$ large enough

$|p^{\mathcal{N}^{-}}(\vartheta_{0})|\geq
e^{4k(|\log(w_{0}/3)|-\varepsilon)}.$ (4.19)

Having both (4.18) and (4.19) leads to a contradiction to our assumption
$d<k^{\alpha}$ for two singular clusters, once $w_{0}$ is large enough with
$\gamma=\gamma^{4}$. Hence, singular points are far apart. Fixing an energy
$E\in\mathbb{R}$, and $u_{E}$ a generalized eigenfunction to $E$ of the
operator with $u_{E}(0)\neq 0.$ The last condition can be assumed without loss
of generality, as $u_{E}$ may not vanish at $u_{E}(-1)$ and $u_{E}(0)$ at the
same time. As mentioned before, since $u_{E}(0)\neq 0$, $0$ has to be
$(\gamma^{4},k)$ singular for $k$ sufficiently large. Hence, repulsion of
singular clusters shows that any $n$ sufficiently large will be
$(\gamma^{4},k)$ regular for some suitable $k.$ Thus, we obtain an interval
$\mathcal{N}=[n_{1},n_{2}]$ of length $|\mathcal{N}|=k$ with $n\in\mathcal{N}$
such that

$\frac{1}{5}(|n|-1)\leq|n_{i}-n|\leq\frac{4}{5}(|n|-1)$

with $n_{i}\in\\{n_{1},n_{2}\\}$ and the decay bound

$\|G^{\pm,\mathcal{N}}_{w}(\theta,\phi,\vartheta,\mathscr{E})_{\alpha,\alpha_{i}}\|\leq
e^{-(\gamma^{4}-\varepsilon)|\alpha-\alpha_{i}|}.$

Combining this with (4.5), we find for any generalized eigenfunction $u_{E}$

$|u_{E}(n)|\leq 2C\langle n\rangle e^{-(\gamma^{4}-\varepsilon)(|n|-1)/5}.$

This implies exponential localization and finishes the sketch of proof of
Theorem 2.

##  5\. Weak coupling and AC spectrum

We now study the regime where the coupling of the honeycomb lattices is weak
and see that the AC spectrum that is present in case of non-interacting sheets
persists. We saw in the previous section that in the strong coupling regime,
the Hamiltonian exhibits Anderson localization (point spectrum) at almost
every diophantine moiré lengths $1/L$. In this section, we show for fixed
diophantine moiré lengths $1/L$, $H_{w}(\theta)$ has some AC spectrum if the
coupling is weak enough. Our main theorem is then as follows.

######  Theorem 3.

Consider the chiral or anti-chiral Hamiltonian $H_{w}(\theta).$ For
$\frac{1}{L}\in\operatorname{DC}_{t}$ and small enough coupling $|w|\leq c$,
for some constant $c(L)>0,$ the AC spectrum of the Hamiltonian $H_{w}(\theta)$
is non-empty.

Recall that the famous Schnol’s theorem says the spectrum of
$H_{w}(\theta,\phi,\vartheta)$ is given by the closure of the set of
generalized eigenvalues of $H_{w}(\theta,\phi,\vartheta)$. Actually, one can
also characterize the AC spectrum based on more concrete descriptions of
growth of the generalized eigenfunctions. The theory was first built for one
dimensional discrete Schrödinger operators. Let $H$ be a discrete Schrödinger
operator on $\ell^{2}(\mathbb{Z})$:

$(Hu)(n)=u(n-1)+u(n+1)+V_{n}u(n),\ \ n\in\mathbb{Z},$ (5.1)

where $\\{V_{n}\\}_{n\in\mathbb{Z}}$ is a sequence of real numbers (the
potential). A non-trivial solution $u$ of $Hu=Eu$ is called subordinate at
$\infty$ if

$\lim\limits_{L\rightarrow\infty}\frac{\|u\|_{L}}{\|v\|_{L}}=0$

for any linearly independent solution $v$ of $Hu=Eu$, where

$\|u\|_{L}=\left[\sum\limits_{n=1}^{[L]}|u(n)|^{2}+(L-[L])|u([L]+1)|^{2}\right]^{\frac{1}{2}},$

here $[L]$ denotes the integer part of $L$. The absolutely continuous spectrum
of $H$, denoted by $\operatorname{Spec}_{ac}(H)$, has the following
characterization,

$\displaystyle\operatorname{Spec}_{ac}(H)=\overline{\left\\{E\in\mathbb{R}:\text{at
$\infty$ or $-\infty$, $Hu=Eu$ has no subordinate solution}\right\\}}^{ess},$

known as the subordinate theory [GP89, JL99, LS99]. For our purpose, we will
use the subordinate theory [OC2021] for the following matrix-valued Jacobi
operators,

$(J{u})(n)=D_{n-1}{u}(n-1)+D_{n}{u}(n+1)+V_{n}{u}(n),\ \ n\in\mathbb{Z},$
(5.2)

where $(D_{n})_{n},(V_{n})_{n}$ are bilateral sequences of $m\times m$ self-
adjoint matrices.

We define the Dirichlet and Neumann solutions as the solutions to $J{u}=E{u}$
that satisfy, respectively, the initial conditions

$\begin{cases}\phi_{0}=0_{m},\\\ \phi_{1}=I_{m},\end{cases}\ \ \ \
\begin{cases}\psi_{0}=I_{m},\\\ \psi_{1}=0_{m}.\end{cases}$

where $I_{m}$ and $0_{m}$ are the m-dimensional identity and zero matrices,
respectively.

######  Theorem 4 ([OC2021]).

Let, for each $r\in\left\\{1,2\cdots,m\right\\}$,

$\mathcal{S}_{r}=\left\\{E\in\mathbb{R}:\liminf\limits_{L\rightarrow\infty}\frac{1}{L}\sum\limits_{n=1}^{L}\sigma^{2}_{m-r+1}[\phi_{n}(E)]+\sigma^{2}_{m-r+1}[\psi_{n}(E)]<\infty\right\\},$

where $\sigma_{k}[T]$ stands for the $k$-th singular value of $T$. Then the
set $\overline{\mathcal{S}_{r+1}\backslash\mathcal{S}_{r}}^{ess}$ corresponds
to the absolutely continuous component of multiplicity $r$ of any self-adjoint
extension of the operator $J^{+}$ which is $J$ restricted to
${\operatorname{Dom}}(J^{+}_{max}):=\\{{u}\in\ell^{2}(\mathbb{N};\mathbb{C})|J^{+}{u}\in\ell^{2}(\mathbb{N};\mathbb{C})\\}$
(satisfying any admissible boundary condition at $n=0$).

Thus, to characterize AC spectrum for matrix-valued Schrödinger operators, one
needs study the singular values of the transfer matrix. In this section, we
are mainly interested in the quasi-periodic case. We consider a rather general
quasiperiodic model and the Hamiltonian in (1.2) is one of the typical
examples. For our purpose, consider the following multi-frequency matrix-
valued Schrödinger operators,

$(J_{\lambda,\theta}{u})_{n}=C{u}_{n+1}+C{u}_{n-1}+\lambda
V(\theta+n\alpha){u}_{n},$ (5.3)

acting on $\ell^{2}(\mathbb{Z},\mathbb{C}^{m})$ where
$\alpha\in(\mathbb{R}\backslash\mathbb{Q})^{d}$, $\lambda\in\mathbb{R}$, $C$
is a $m\times m$ invertible self-adjoint matrix and $V$ is an analytic self-
adjoint matrix which is 1-periodic in each variable. In particular,
$\alpha=1/L$, $\lambda=w$ in the case of the Hamiltonian (1.2).

Our approach is the so-called reducibility method, which was initially
developed by Dinaburg-Sinai [DS75], Eliasson [E92], further developed by Hou-
You [HY12], Avila-Jitomirskaya [AJ10] and Avila [Avila1, Avila2]. Our result
is based on the reducibility results in [E01, C13, GYZ2020, HY2006] for higher
dimensional quasi-periodic cocycles.

###  5.1. Preliminaries

Recall that $\alpha\in\mathbb{R}^{d}$ is called Diophantine if there are $t>0$
such that $\alpha\in{\operatorname{DC}}_{t}^{d}$, where

${\operatorname{DC}}^{d}_{t}:=\left\\{\alpha\in\mathbb{R}^{d}:\inf_{j\in\mathbb{Z}}\left|\langle
n,\alpha\rangle-j\right|>\frac{t}{|n|^{d+1}},\quad\forall\,n\in\mathbb{Z}^{d}\backslash\\{0\\}\right\\}.$
(5.4)

Given $A\in C^{\omega}(\mathbb{T}^{d},\operatorname{GL}(2m,\mathbb{C}))$ and
rational independent $\alpha\in\mathbb{R}^{d}$, we define the quasi-periodic
$\operatorname{GL}(2m,\mathbb{C})$ cocycle $(\alpha,A)$:

$(\alpha,A)\colon\left\\{\begin{array}[]{rcl}\mathbb{T}^{d}\times\mathbb{C}^{2m}&\to&\mathbb{T}^{d}\times\mathbb{C}^{2m}\\\\[2.84526pt]
(x,v)&\mapsto&(x+\alpha,A(x)\cdot v).\end{array}\right..$

We denote by $L_{1}(\alpha,A)\geq L_{2}(\alpha,A)\geq...\geq L_{m}(\alpha,A)$
the Lyapunov exponents of $(\alpha,A)$ repeatedly according to their
multiplicities, i.e.,

$L_{k}(\alpha,A)=\lim\limits_{n\rightarrow\infty}\frac{1}{n}\int_{\mathbb{T}}\ln(\sigma_{k}(A_{n}(x)))dx.$

$(\alpha,A)$ is said to be reducible if there exist $B\in
C^{\omega}(\mathbb{T}^{d},\operatorname{GL}(2m,\mathbb{C}))$,
$\widetilde{A}\in\operatorname{GL}(2m,\mathbb{C})$ such that

$B(x+\alpha)A(x)B^{-1}(x)=\widetilde{A}.$

The following are two general facts on the Lyapunov exponents, which were
proved in [GYZ2020] (see Proposition 2.2 and Proposition 2.3).

######  Proposition 5.1.

Assume $(\alpha,A)\in\mathbb{T}^{d}\times
C^{0}(\mathbb{T}^{d},\operatorname{GL}(2m,\mathbb{C})),\ \ B\in
C^{0}(\mathbb{T}^{d},\operatorname{GL}(2m,\mathbb{C})),$ and
$\widetilde{A}(x)=B(x+\alpha)A(x)B^{-1}(x)$, we have

$L_{i}(\alpha,\widetilde{A})=L_{i}(\alpha,A),\ \ 1\leq i\leq 2m.$

######  Proposition 5.2.

If we denote the eigenvalues of $A\in\operatorname{GL}(2m,\mathbb{C})$ by
$\\{e^{-2\pi i\rho_{j}}\\}_{j=1}^{2m}$, then

$\\{L_{j}(\alpha,A)\\}_{j=1}^{2m}=\\{2\pi\operatorname{Im}\rho_{j}\\}_{j=1}^{2m}.$

Now we consider the eigen-equation $J_{\lambda,\theta}{u}=E{u}$ with
$J_{\lambda,\theta}$ as in (5.3). To obtain a first order system and the
corresponding linear skew product we use the fact that $C$ in (5.3) is
invertible and write

$\begin{pmatrix}{u}_{k+1}\\\
{u}_{k}\end{pmatrix}=\begin{pmatrix}C^{-1}(EI_{m}-\lambda
V(\theta+k\alpha))&-I_{m}\\\ I_{m}&0_{m}\end{pmatrix}\begin{pmatrix}{u}_{k}\\\
{u}_{k-1}\end{pmatrix}.$

Denote

$L^{\lambda V}_{E}(\theta)=\begin{pmatrix}C^{-1}(EI_{m}-\lambda
V(\theta))&-I_{m}\\\ I_{m}&0_{m}\end{pmatrix}.$ (5.5)

Note that $(\alpha,L_{E}^{\lambda V})$ is a symplectic cocycle. As a
corollary, the Lyapunov exponents of $(\alpha,L_{E}^{\lambda V})$ come in
pairs $\pm L_{i}(\alpha,L_{E}^{\lambda V})$ ($1\leq i\leq m$).

Let ${\operatorname{S}p_{2m}}(\mathbb{C})$ denote the set of $2m\times 2m$
complex symplectic matrices. Given any $A\in
C^{0}(\mathbb{T}^{d},{\operatorname{S}p_{2m}}(\mathbb{C}))$, we say the
cocycle $(\alpha,A)$ is uniformly hyperbolic if for every
$x\in\mathbb{T}^{d}$, there exists a continuous splitting
$\mathbb{C}^{2m}=E^{s}(x)\oplus E^{u}(x)$ such that for some constants
$C>0,c>0$, and for every $n\geqslant 0$,

$\displaystyle\lvert A_{n}(x)v\rvert\leqslant Ce^{-cn}\lvert v\rvert,$
$\displaystyle v\in E^{s}(x),$ $\displaystyle\lvert
A_{n}(x)^{-1}v\rvert\leqslant Ce^{-cn}\lvert v\rvert,$ $\displaystyle v\in
E^{u}(x+n\alpha).$

This splitting is invariant by the dynamics, which means that for every
$x\in\mathbb{T}^{d}$, $A(x)E^{\ast}(x)=E^{\ast}(x+\alpha)$, for $\ast=s,u$.
The set of uniformly hyperbolic cocycles is open in the $C^{0}$-topology.

Let $\Sigma_{\lambda}$ be the spectrum of $J_{\lambda,\theta}$.
$\Sigma_{\lambda}$ is closely related to the dynamical behavior of the
symplectic cocycle $(\alpha,L_{E}^{\lambda V})$. $E\notin\Sigma_{\lambda}$ if
and only if $(\alpha,L_{E}^{\lambda V})$ is uniformly hyperbolic.

###  5.2. Existence of AC spectrum

In this subsection, we will prove the following theorem.

######  Theorem 5.

Let $\alpha\in{\operatorname{DC}}^{d}_{t}$ and $V$ be an analytic self-adjoint
$m\times m$ matrix. There is $\lambda_{0}(m,\alpha,C,V)$ such that if
$|\lambda|<\lambda_{0}$, then the ac part of $J_{\lambda,\theta}$ is non-empty
for any $\theta\in\mathbb{T}^{d}$.

The proof is based on a positive measure reducibility theorem for higher
dimensional quasi-periodic cocycles and subordinate theory for matrix-valued
Schrödinger operators.

######  Theorem 6.

Let $\alpha\in{\operatorname{DC}}^{d}_{t}$ and $V$ be an analytic self-adjoint
$m\times m$ matrix. There is $\lambda_{0}(m,\alpha,C,V)$ and
$\mathcal{E}_{\lambda}\subset\Sigma_{\lambda}$ such that if
$|\lambda|<\lambda_{0}$, then for any $E\in\mathcal{E}_{\lambda}$,
$\left(\alpha,L_{E}^{\lambda V}\right)$ is reducible. Moreover,
$\left|\Sigma_{\lambda}\backslash\mathcal{E}_{\lambda}\right|\rightarrow 0$ as
$\lambda\rightarrow 0$.

Proof of Theorem 5. Under the assumption, by Theorem 6, for any
$E\in\mathcal{E}_{\lambda}$, there is $B\in
C^{\omega}(\mathbb{T}^{d},\operatorname{GL}(2m,\mathbb{C}))$,
$A_{E}\in\operatorname{GL}(2m,\mathbb{C})$ such that

$B(x+\alpha)L_{E}^{\lambda V}(x)B^{-1}(x)=A_{E}.$ (5.6)

By Proposition 5.1 and Proposition 5.2,

$\\{L_{j}(\alpha,L_{E}^{\lambda
V})\\}_{j=1}^{m}=\\{\ln|\lambda_{j}(E)|\\}_{j=1}^{m},$

where $\lambda_{1}(E),\cdots,\lambda_{d}(E)$ are the eigenvalues of $A_{E}$
outside the unit circle, counting the multiplicity.

Since $\mathcal{E}_{\lambda}\subset\Sigma_{\lambda}$, we claim
$|\lambda_{d}(E)|=1$ for all $E\in\mathcal{E}_{\lambda}$. Otherwise, by the
definition, $(\alpha,L_{E}^{\lambda V})$ is uniformly hyperbolic which
contradicts that $E\in\Sigma_{\lambda}$. We denote

$2\omega(E)=\\#\\{\text{The number of unit eigenvalues of $A_{E}$}\\}.$

Then by the above argument and the complex symplectic structure we have
$\omega(E)$ is always an integer and $\omega(E)\geq 1$.

To characterize the AC spectrum, we need to apply Theorem 4. We define the
Dirichlet and Neumann solutions as the solutions to
$J_{\lambda,\theta}{u}=E{u}$ that satisfy, respectively, the initial
conditions

$\begin{cases}\phi_{0}(x,E)=0_{m},\\\ \phi_{1}(x,E)=I_{m},\end{cases}\ \ \ \
\begin{cases}\psi_{0}(x,E)=I_{m},\\\ \psi_{1}(x,E)=0_{m}.\end{cases}$

Note that

$\begin{pmatrix}\phi_{n+1}(x,E)\\\ \phi_{n}(x,E)\end{pmatrix}=(L_{E}^{\lambda
V})_{n}(x+\alpha)\begin{pmatrix}I_{m}\\\ 0_{m}\end{pmatrix},$
$\begin{pmatrix}\psi_{n+1}(x,E)\\\ \psi_{n}(x,E)\end{pmatrix}=(L_{E}^{\lambda
V})_{n}(x+\alpha)\begin{pmatrix}0_{m}\\\ I_{m}\end{pmatrix}.$

On the other hand, by (5.6)

$(L_{E}^{\lambda V})_{n}(x)=B^{-1}(x+n\alpha)A_{E}^{n}B(x).$

Thus there is $\bar{C}$ depending $C,V$ such that

$\sigma^{2}_{m-r+1}[\phi_{n}(E)]+\sigma^{2}_{m-r+1}[\psi_{n}(E)]\leq\bar{C},\
\ 1\leq r\leq\omega(E).$

which implies $\mathcal{E}_{\lambda}\subset\mathcal{S}_{1}$. By Theorem 4, we
have

$\overline{\mathcal{E}_{\lambda}}^{ess}\subset\operatorname{Spec}_{ac}(J^{+}_{\lambda,\theta})$

where $J^{+}_{\lambda,\theta}$ is the restriction of $J_{\lambda,\theta}$ on
$\ell^{2}(\mathbb{Z}^{+})$ with Dirichlet boundary condition. Actually, one
can prove in the exact same way that

$\overline{\mathcal{E}_{\lambda}}^{ess}\subset\operatorname{Spec}_{ac}(J^{-}_{\lambda,\theta})$

where $J^{-}_{\lambda,\theta}$ is the restriction of $J_{\lambda,\theta}$ on
$\ell^{2}(\mathbb{Z}^{-})$ with Dirichlet boundary condition. Finally, we need
the following proposition in [OC2021],

######  Proposition 5.3.

We have
$\operatorname{Spec}_{ac}(J_{\lambda,\theta})=\operatorname{Spec}_{ac}(J_{\lambda,\theta}^{+}\oplus
J_{\lambda,\theta}^{-})$.

Hence

$\overline{\mathcal{E}_{\lambda}}^{ess}\subset\operatorname{Spec}_{ac}(J_{\lambda,\theta}).$

Theorem 5 follows from the fact that $|\mathcal{E}_{\lambda}|>0$. $\square$

Proof of Theorem 6. First, we introduce a positive measure reducibility
theorem proved in [HY2006]. We consider $(\alpha,A(E)+F(E,\cdot))\in
C^{\omega}(\mathbb{T},\operatorname{GL}(2m,\mathbb{C}))$, $A(E)+F(E,\cdot)$
also depends analytically on $E\in I$ where $I$ is an interval. Let
$\\{\lambda_{i}(E)\\}_{i=1}^{2m}$ be the eigenvalues of $A(E)$. For any
$u\in\mathbb{R}$, we denote

$h(E,u)=\prod\limits_{i\neq j}\left(\lambda_{i}(E)-\lambda_{j}(E)-iu\right).$

We say $A(E)$ is non-degenerate if there is $p\in\mathbb{N}$ such that for all
$u\in\mathbb{R}$, we have

$\displaystyle\max_{1\leq i\leq p}\left|\frac{\partial^{i}h(E,u)}{\partial
E^{i}}\right|\geq c>0.$ (5.7)

######  Theorem 7 ([HY2006]).

Assume $\alpha\in\operatorname{DC}_{t}^{d}$ and $I$ is a parameter interval,
$A\in C^{\omega}(I,\operatorname{GL}(2m,\mathbb{R}))$ satisfies the non-
degeneracy condition (5.7), $F\in C^{\omega}(T\times
I,\operatorname{GL}(2m,\mathbb{R}))$ and there is $M>0$ such that $|A|_{s}<M$.
Then there exists $\varepsilon_{0}$ such that if
$|F|_{s,\delta}<\varepsilon_{0}$333Here
$|F|_{s,\delta}=\sup_{|\operatorname{Im}\theta|<s,|\operatorname{Im}E|<\delta}|F(E,\theta)|$.,
the measure of the set of parameter $I$ for which $(\alpha,A(E)+F(E,\cdot))$
is not reducible is no larger than $CL(10\varepsilon_{1})^{c}$, where $C,c$
are some positive constants, $L$ is the length of the parameter interval $I$.

Actually, Theorem 6 is a special case of Theorem 7. To see this, we denote the
characteristic polynomial of $L^{0}_{E}$ by
$p(E,z)=\det\left({L^{0}_{E}-zI_{2m}}\right)$. Let $\\{z_{i}(E)\\}_{i=1}^{2m}$
be the zeros of $p(E,z)$. For any $u\in\mathbb{R}$, we denote

$g(E,u)=\prod\limits_{i\neq j}\left(z_{i}(E)-z_{j}(E)-iu\right).$

It is easy to check that

$\displaystyle g(E,u)$
$\displaystyle=\det{\left[iuI_{16m^{2}}-\left(I_{4m}\otimes
L_{E}^{0}-(L_{E}^{0})^{T}\otimes I_{4m}\right)\right]}/(iu)^{4m}$
$\displaystyle=E^{2m}+g^{\prime}(E,u)$

where $g^{\prime}(E,u)$ is a polynomial of $E$ with degree $\leq 2m-1$. Thus
$g(E,u)$ satisfies the following non-degeneracy condition for all
$E\in\mathbb{R}$

$\max_{1\leq i\leq{2m}}\left|\frac{\partial^{i}g(E,u)}{\partial
E^{i}}\right|\geq 1.$

Moreover, $|L_{E}^{0}|\leq M$ for all $E\in\Sigma_{\lambda}$. Thus all
conditions in Theorem 7 are satisfied. This completes the proof of Theorem 6.
$\square$

##  6\. Cantor spectrum for anti-chiral Hamiltonian

The key property of the anti-chiral Hamiltonian that allows us to establish
Cantor spectrum is the operator-valued diagonalizability of the matrix-valued
discrete operator. In this section, we show that we can block-diagonalize the
Hamiltonian into four Schrödinger operators for $\theta\in{\mathbb{Z}}/2.$

To this end, we define the matrix

$\mathcal{U}=\frac{1}{2}\begin{pmatrix}1&-1&-1&1\\\ -1&1&-1&1\\\ -1&-1&1&1\\\
1&1&1&1\end{pmatrix}.$

This implies that

$\mathcal{U}^{*}H_{(w_{0},0)}(\theta,0)\mathcal{U}=1+e^{2\pi
i\theta}(\tau+\tau^{*})\operatorname{diag}(-1_{{\mathbb{C}}^{2\times
2}},1_{{\mathbb{C}}^{2\times
2}})-w_{0}U(\vartheta+\tfrac{\bullet}{L})\operatorname{diag}(\sigma_{3},\sigma_{3}).$

By flipping matrix entries $(1,1),(3,3)$ and $(2,2),(4,4)$ of each individual
block, appropriately, we see that the Hamiltonian is equivalent to the
following operator on $\ell^{2}({\mathbb{Z}};{\mathbb{C}}^{4}):$

$w_{0}\left(\frac{\tau^{*}+\tau-1}{3}\right)\operatorname{diag}(\sigma_{3},\sigma_{3})+\left(2\cos\left(2\pi(\frac{\bullet}{L}+\vartheta)\right)\pm
1\right)\operatorname{diag}(-1_{{\mathbb{C}}^{2\times
2}},1_{{\mathbb{C}}^{2\times 2}}),$

where the choice of the $\pm$ sign depends on whether we are studying
$\theta=0$ or $\theta=1/2.$ We readily conclude:

######  Proposition 6.1 (AMO).

Let $1/L\notin\mathbb{Q},\theta\in\\{0,1/2,1\\}$. Let $w_{0}=3$, then the
spectrum of the anti-chiral Hamiltonian is purely singular continuous and a
Cantor set of zero measure. Let $w_{0}>3$ then the spectral measure is
absolutely continuous and for $w_{0}<3$ it is pure point if $1/L$ satisfies a
diophantine condition. In either case, the spectrum is a Cantor set of
positive measure.

###### Proof.

The spectrum, as a set, of a finite direct sum of operators is the finite
union of the spectra of all operators on the diagonal. These are in all cases
we consider here Cantor sets, i.e. closed, nowhere dense sets, without
isolated points. Thus, the finite union of such sets will still be Cantor sets
and clearly the spectral type is also preserved under finite direct sums of
operators. The same argument applies to the Lebesgue decomposition of the
spectral measure. ∎

##  7\. Spectral analysis of flat bands in effective models

We start by studying the spectral properties of a linearized low-energy model,
proposed in [TM2020], close to zero energy. The proposed continuum Hamiltonian
is given by

$\mathscr{L}=\begin{pmatrix}0&D_{x}-ik_{\perp}&w_{0}U(x/L)&w_{1}U^{-}(x/L)\\\
D_{x}+ik_{\perp}&0&w_{1}U^{+}(x/L)&w_{0}U(x/L)\\\
w_{0}U(x/L)&w_{1}U^{+}(x/L)&0&D_{x}-ik_{\perp}\\\
w_{1}U^{-}(x/L)&w_{0}U(x/L)&D_{x}+ik_{\perp}&0\end{pmatrix}.$

Since this Hamiltonian $\mathscr{L}$ is periodic, we can apply standard Bloch-
Floquet theory to equivalently study the spectrum on
$L^{2}(\mathbb{R}/L{\mathbb{Z}})$ of

$\mathscr{L}(k_{x})=\begin{pmatrix}0&D_{x}+k_{x}-ik_{\perp}&w_{0}U(x/L)&w_{1}U^{-}(x/L)\\\
D_{x}+k_{x}+ik_{\perp}&0&w_{1}U^{+}(x/L)&w_{0}U(x/L)\\\
w_{0}U(x/L)&w_{1}U^{+}(x/L)&0&D_{x}+k_{x}-ik_{\perp}\\\
w_{1}U^{-}(x/L)&w_{0}U(x/L)&D_{x}+k_{x}+ik_{\perp}&0\end{pmatrix}$

such that

$\operatorname{Spec}(\mathscr{L})=\bigcup_{k_{x}\in[0,2\pi/L]}\operatorname{Spec}(\mathscr{L}(k_{x})).$
(7.1)

The study of the nullspace of the Hamiltonian $\mathscr{L}(k_{x})-\lambda$ is
equivalent to the study of the nullspace of the operator
$\widehat{\mathscr{L}}_{\lambda}(k_{x})=\operatorname{diag}(\sigma_{1},\sigma_{1})(\mathscr{L}(k_{x})-\lambda)$
given by

$\widehat{\mathscr{L}}_{\lambda}(k_{x})=\begin{pmatrix}D_{x}+k_{x}+ik_{\perp}&w_{1}U^{+}(x/L)&-\lambda&w_{0}U(x/L)\\\
w_{1}U^{-}(x/L)&D_{x}+k_{x}+ik_{\perp}&w_{0}U(x/L)&-\lambda\\\
-\lambda&w_{0}U(x/L)&D_{x}+k_{x}-ik_{\perp}&w_{1}U^{-}(x/L)\\\
w_{0}U(x/L)&-\lambda&w_{1}U^{+}(x/L)&D_{x}+k_{x}-ik_{\perp}\end{pmatrix}.$
(7.2)

######  Proposition 7.1.

The Hamiltonian $\mathscr{L}$ does not possess any flat bands in $k_{x}$ for
any fixed $k_{\perp}$, i.e. there is no $\lambda\in\mathbb{R}$ such that
$\lambda\in\operatorname{Spec}(\mathscr{L}(k_{x}))$ for all
$k_{x}\in{\mathbb{R}}.$

###### Proof.

This is an easy consequence of (7.2) and Bloch-Floquet theory. In fact, by
(7.2) we have $\lambda\in\operatorname{Spec}(\mathscr{L}(k_{x}))$ if and only
if $k_{x}\in\operatorname{Spec}(\widehat{\mathscr{L}}_{\lambda}(0))$. Since
the Hamiltonian $\widehat{\mathscr{L}}_{\lambda}(0))$ has compact resolvent,
its spectrum is discrete and therefore it is impossible that
$k_{x}\in\operatorname{Spec}(\widehat{\mathscr{L}}_{\lambda}(0))$ for all
$k_{x}\in\mathbb{R}$, which proves the claim. ∎

We shall now restrict us to the case $k_{\perp}=0$ and study the spectrum of
the continuous Hamiltonian. In particular, we shall analyze under what
conditions $0$ is in the spectrum. We start with the anti-chiral Hamiltonian
for which our spectral analysis is rather complete:

######  Proposition 7.2.

The spectrum of the anti-chiral Hamiltonian
$\mathscr{L}_{\operatorname{ac}}(k_{x})=\mathscr{L}(k_{x})$ with $w_{1}=0$ for
$k_{\perp}=0$ satisfies
$\bigcup_{k_{x}\in[0,2\pi/L]}\operatorname{Spec}(\mathscr{L}_{\operatorname{ac}}(k_{x}))={\mathbb{R}}$.
In particular, $0\in\operatorname{Spec}(\mathscr{L}_{\operatorname{ac}})$ for
all $w_{0}\in\mathbb{R}$.

###### Proof.

According to (7.2) it suffices to consider the equation

$D_{x}\varphi+A(\lambda,x)\varphi=0,$ (7.3)

where

$A(\lambda,x)=\begin{pmatrix}k_{x}&0&-\lambda&w_{0}U(x/L)\\\
0&k_{x}&w_{0}U(x/L)&-\lambda\\\ -\lambda&w_{0}U(x/L)&k_{x}&0\\\
w_{0}U(x/L)&-\lambda&0&k_{x}\end{pmatrix}.$

Let $W(x/L)=iL\Big{(}\frac{x}{L}+\frac{\sin(\frac{2\pi x}{L})}{\pi}\Big{)}$,
so that $D_{x}W(x/L)=U(x/L)$. With

$B(\lambda,x)=\begin{pmatrix}ik_{x}x&0&-i\lambda x&w_{0}W(x/L)\\\
0&ik_{x}x&w_{0}W(x/L)&-i\lambda x\\\ -i\lambda x&w_{0}W(x/L)&ik_{x}x&0\\\
w_{0}W(x/L)&-i\lambda x&0&ik_{x}x\end{pmatrix}.$

we thus have $D_{x}B(\lambda,x)=A(\lambda,x)$. Using the unitary matrix

$\mathscr{U}=\frac{1}{2}\begin{pmatrix}-1&1&-1&1\\\ -1&-1&1&1\\\ 1&-1&-1&1\\\
1&1&1&1\end{pmatrix},$

we see that

$\begin{split}\mathscr{U}B(\lambda,x)\mathscr{U}^{*}=\operatorname{diag}&(i\lambda
x-w_{0}{W(x/L)},i\lambda x+w_{0}{W(x/L)},\\\ &-i\lambda
x-w_{0}{W(x/L)},-i\lambda x+w_{0}{W(x/L))}+ik_{x}x\end{split}$

is diagonal. Since
$\mathscr{U}e^{-B}\mathscr{U}^{\ast}=e^{-\mathscr{U}B\mathscr{U}^{\ast}}$ it
then follows that

$D_{x}e^{-B}=\mathscr{U}^{\ast}D_{x}e^{-\mathscr{U}B\mathscr{U}^{\ast}}\mathscr{U}=\mathscr{U}^{\ast}D_{x}(-\mathscr{U}B\mathscr{U}^{\ast})e^{-\mathscr{U}B\mathscr{U}^{\ast}}\mathscr{U}=-Ae^{-B}.$

Hence, the solutions to (7.3) are of the form
$\varphi(x)=e^{-B(\lambda,x)}\varphi_{0}.$ For $\lambda$ to be an eigenvalue,
such a solution is required to be $L$-periodic. Inspecting the expression for
$\mathscr{U}B(\lambda,x)\mathscr{U}^{*}$ we see that it is necessary that, for
$w_{0}$ fixed, we can find $k_{x}\in[0,2\pi/L]$ such that for any of the
combinations $\lambda\pm w_{0}\pm k_{x}\in\frac{2\pi}{L}{\mathbb{Z}}$. Thus
$\operatorname{Spec}(\mathscr{L}_{\operatorname{ac}})={\mathbb{R}}.$ ∎

For the chiral Hamiltonian $\mathscr{L}_{c}=\mathscr{L}$ with $w_{0}=0$, we do
not have an explicit description of the full spectrum, however we can still
locate the zero energy spectrum.

######  Proposition 7.3.

For all $w_{1}\in\mathbb{R}$ it follows that
$0\in\operatorname{Spec}(\mathscr{L}_{c})$ but there is $\varepsilon>0$ such
that for all $w_{1}\in(-\varepsilon,\varepsilon)\backslash\\{0\\}$ we have
$0\notin\operatorname{Spec}(\mathscr{L}_{c}(k_{x}=0)).$

###### Proof.

To prove the first part of the statement, note that it suffices to show that
there is $\mu\in\mathbb{C}$ such that for some function $\psi\in
H^{1}({\mathbb{R}}/{\mathbb{Z}};{\mathbb{C}}^{2})$

$D_{x}\psi+w_{1}\begin{pmatrix}0&U^{+}(x)\\\
U^{-}(x)&0\end{pmatrix}\psi=\mu\psi.$

Let then $x\mapsto X(x,w_{1})$ be the fundamental solution with
$\mathcal{U}(x):=\begin{pmatrix}0&U^{+}(x)\\\ U^{-}(x)&0\end{pmatrix}$,
satisfying

$D_{x}X(x)+w_{1}\mathcal{U}(x)X(x)=0,\quad X(0)=\operatorname{id}.$

The matrix $M(w_{1}):=X(1,w_{1})$, with $\operatorname{det}(M)=1$, is then
called the monodromy matrix and let $\rho\in\operatorname{Spec}(M(w_{1}))$
where $\rho\neq 0.$ Thus, there is $v\neq 0$ such that $\phi(x):=X(x)v$
satisfies the periodicity condition $\phi(1)=M(w_{1})v=\rho v=\rho
X(0,w_{1})v=\rho\phi(0).$ We then define $\mu\in{\mathbb{C}}$, such that
$\psi(x):=e^{i\mu x}\phi(x)$ is the desired solution.

If $\lambda=0$ was protected in the $k_{x}=0$ sector, then this would imply
that there is always a solution

$D_{x}\psi+w_{1}\mathcal{U}(x)\psi=0.$

In this case, our claim amounts to showing that
$1\notin\operatorname{Spec}(M(w_{1}))$ for $w_{1}$ small but non-zero. Since
$D_{x}(\operatorname{det}(X(x,w_{1})))=w_{1}\operatorname{tr}(\mathcal{U}(x))=0$
it follows that $\operatorname{det}(X(x,w_{1}))=1.$ On the other hand, since
$X(\bullet,x)$ is analytic, such that $X(\bullet,x)=\sum_{i\geq
0}w_{1}^{i}X_{i}(x),$ we find the recurrence equation

$D_{x}X_{i+1}(x)=\mathcal{U}(x)X_{i}(x),\quad X_{i+1}(0)=0,$

and $X_{0}=\operatorname{id}_{\mathbb{C}^{2}}.$ Hence, all $X_{i}$ with $i$
odd are trace-free. We compute $\operatorname{tr}(X_{2}(1))=1.$ This implies
that
$\operatorname{tr}(M(w_{1}))=\operatorname{tr}(X_{0})+w_{1}^{2}\operatorname{tr}(X_{2})+\mathcal{O}(w_{1}^{4})\neq
2$ for $w_{1}$ small but non-zero. Hence,
$1\notin\operatorname{Spec}(M(w_{1})).$ ∎

###  7.1. Absence of flat bands for TB Hamiltonians

Similarly as in Proposition 7.1 for the linearized model, the absence of flat
bands also holds for the tight-binding model, but before stating that result,
we need some preliminary discussions:

We associate to $H_{w}$ in (1.2) a semiclassical $\operatorname{\Psi DO}$ on
$L^{2}(\mathbb{S}^{1})$. The semiclassical parameter is the Moiré length
$h=(2\pi L)^{-1}$, i.e., we are concerned with the limit of large moiré
lengths $L\gg 1$.

######  Lemma 7.4.

The Hamiltonian $H_{w}$ is unitarily equivalent to the semiclassical
$\operatorname{\Psi DO}$ $H_{\operatorname{\Psi
DO}}(w):L^{2}(\mathbb{S}^{1})\to L^{2}(\mathbb{S}^{1})$ defined as

$H_{\operatorname{\Psi DO}}(w)u(x):=(2\mathbf{t}(k_{\perp})\cos(2\pi
hD_{x})+\mathbf{t}_{0}+V_{w}(x))u(x).$ (7.4)

We define the regularized trace for $f\in C_{c}^{\infty}({\mathbb{R}})$ of the
tight-binding model $H$ by using the equivalence between the tight-binding
Hamiltonian and the pseudodifferential operator from Lemma 7.4

$\widehat{\operatorname{tr}}(f(H)):=\lim_{n\to\infty}\frac{\operatorname{tr}_{\ell^{2}({\mathbb{Z}})}(\operatorname{1\hskip-2.75ptl}_{\\{-n,..,n\\}}f(H))}{2n+1}.$

######  Proposition 7.5.

The regularized trace for the tight-binding model satisfies for
$1/L=\frac{p}{q}\in\mathbb{Q}$

$\widehat{\operatorname{tr}}(f(H))=\frac{\sum_{\gamma\in\\{0,..,q-1\\}}\operatorname{tr}_{{\mathbb{C}}^{4}}(\int_{\mathbb{T}}\sigma(f(H_{\operatorname{\Psi
DO}}))(x,\gamma L^{-1})\,dx)}{q}.$ (7.5)

For all other $1/L$

$\widehat{\operatorname{tr}}(f(H))=\int_{{\mathbb{R}}^{2}/{\mathbb{Z}}^{2}}\operatorname{tr}_{{\mathbb{C}}^{4}}\sigma(f(H_{\operatorname{\Psi
DO}}))(x,\xi)\,dx\,d\xi.$ (7.6)

###### Proof.

The formula for the Weyl symbol [Z12, Theorem $4.19$] implies that with
$\mathcal{U}$ being the unitary map used in the Proof of Lemma 7.4

$\begin{split}\widehat{\operatorname{tr}}(f(H))&=\lim_{n\to\infty}\frac{\operatorname{tr}_{\ell^{2}({\mathbb{Z}})}(\operatorname{1\hskip-2.75ptl}_{\\{-n,...,n\\}}f(H))}{2n+1}\\\
&=\lim_{n\to\infty}\frac{\operatorname{tr}_{\ell^{2}({\mathbb{Z}})}(\operatorname{1\hskip-2.75ptl}_{\\{-n,...,n\\}}\mathcal{U}^{-1}f(H_{\operatorname{\Psi
DO}})\mathcal{U})}{2n+1}\\\
&=\lim_{n\to\infty}\frac{\sum_{\gamma\in\\{-n,..,n\\}^{2}}\operatorname{tr}_{{\mathbb{C}}^{4}}(\int_{\mathbb{T}_{1}}e^{-i\gamma
x}f(H_{\operatorname{\Psi DO}}(x,hD_{x},\theta))e^{i\gamma x}\,dx)}{2n+1}\\\
&=\lim_{n\to\infty}\frac{\sum_{\gamma\in\\{-n,..,n\\}^{2}}\operatorname{tr}_{{\mathbb{C}}^{4}}(\int_{\mathbb{T}_{1}}\sigma(f(H_{\operatorname{\Psi
DO}}))(x,\gamma L^{-1})\,dx)}{2n+1}.\end{split}$ (7.7)

When $1/L$ is rational

$\gamma\mapsto\operatorname{tr}_{{\mathbb{C}}^{4}}\left(\int_{\mathbb{T}_{1}}\sigma(f(H_{\operatorname{\Psi
DO}}))(x,\gamma L^{-1})\,dx\right)$

is periodic and thus we obtain

$\widehat{\operatorname{tr}}(f(H))=\sum_{\gamma\in\\{0,..,q-1\\}}\operatorname{tr}_{{\mathbb{C}}^{4}}\int_{\mathbb{T}_{1}}\sigma(f(H_{\operatorname{\Psi
DO}}))(x,\gamma L^{-1})\,dx.$

If $1/L$ does not satisfy this rationality condition, then the translation
$(T^{\gamma}u)(x)=u(x+1/L)$ is a _uniquely_ ergodic endomorphism on the
probability space ${\mathbb{R}}/{\mathbb{Z}}$ and therefore using the
continuity of the Weyl symbol, it follows that [W82, Theo. $6.19$]

$\widehat{\operatorname{tr}}(f(H))=\int_{{\mathbb{R}}^{2}/{\mathbb{Z}}^{2}}\operatorname{tr}_{{\mathbb{C}}^{4}}\sigma(f(H_{\operatorname{\Psi
DO}}))(x,\xi)\,dx\,d\xi.$ (7.8)

∎

In the sequel, we shall write

$\widehat{\operatorname{tr}}_{\Omega}(\operatorname{Op}(a)):=\int_{{\mathbb{R}}^{2}/{\mathbb{Z}}^{2}}\operatorname{tr}a(x,\xi)\,dx\,d\xi.$

To see that the density of states for commensurable angles coincides with
formula (7.6), we use the following Lemma which we actually state for one-
dimension Schrödinger operators, but whose proof carries immediately over to
arbitrary dimensions and matrix-valued operators, whose kinetic and potential
operators are sums and products of exponential functions, including our
operator of interest.

######  Lemma 7.6.

Let $S:\ell^{2}({\mathbb{Z}})\rightarrow\ell^{2}({\mathbb{Z}})$ be a discrete
Schrödinger operator with a potential that has a finite Fourier representation

$Su_{n}=u_{n+1}+u_{n-1}+\sum_{j=-m}^{m}a_{j}e^{2\pi ijn/L},$

then $S$ is unitarily equivalent to a pseudodifferential operator

$\begin{split}S_{\operatorname{\Psi DO}}f(x)&=2\cos(2\pi
x)f(x)+\sum_{j=-m}^{m}a_{j}e^{2\pi ij/LD}f(x),\end{split}$ (7.9)

with $D=-i\partial_{x}$, and its density of states, defined by the regularized
trace
$\widehat{\operatorname{tr}}(f(S))=\lim_{n\to\infty}\frac{\operatorname{tr}_{\ell^{2}({\mathbb{Z}})}(\operatorname{1\hskip-2.75ptl}_{\\{-n,..,n\\}}f(S))}{2n+1}$
satisfies

$\widehat{\operatorname{tr}}(f(S))=\int_{{\mathbb{R}}^{2}/{\mathbb{Z}}^{2}}\sigma(f(S_{\operatorname{\Psi
DO}}))(x,\xi)\,dx\,d\xi.$

###### Proof.

We consider the one-dimensional operator
$S:\ell^{2}({\mathbb{Z}})\rightarrow\ell^{2}({\mathbb{Z}})$

$Su_{n}=u_{n+1}+u_{n-1}+\sum_{j=-m}^{m}a_{j}e^{2\pi ijn/L}$

where we assume that $a_{j}=\bar{a}_{-j}.$ Then this operator is equivalent to
a pseudodifferential operator on $\mathbb{S}$ given as

$\begin{split}S_{\operatorname{\Psi DO}}f(x)&=2\cos(2\pi
x)f(x)+\sum_{j=-m}^{m}a_{j}e^{ij/LD}f(x)\\\ &=2\cos(2\pi
x)f(x)+\sum_{j=-m}^{m}a_{j}f(x+2\pi j/L).\end{split}$ (7.10)

On the level of the symbol of the operator, the commensurable and
incommensurable expressions for the integrated density of states always
coincide due to $\sum_{k=0}^{n-1}e^{\frac{2\pi ik}{n}}=0.$ Similar reasoning
and the composition formula for symbols of operators implies that the two
formulas coincide for $f$ in the functional calculus being any polynomial.
Thus, Weierstrass’s theorem implies that the two formulations coincide for any
continuous function and since the map from operators defined in the symbol
class $\mathcal{S}(1)$ to their Weyl symbols is continuous under uniform
convergence, the result follows. ∎

######  Proposition 7.7.

The density of states of the tight-binding Hamiltonian $H$ is a continuous
function. In particular, the Hamiltonian does not possess any flat bands at
commensurable length scales $L\in\mathbb{Q}^{+}$.

###### Proof.

Fix $\lambda\in{\mathbb{R}}$, it then suffices to show that we have
$\widehat{\operatorname{tr}}(\operatorname{1\hskip-2.75ptl}_{\\{\lambda\\}}(H))=0.$
Since translations only appear at leading order in $H$, one then observes that
a solution $H\psi=\lambda\psi$ is uniquely determined inside
$\\{-n,-n+1,...,n\\}$ by specifying it on $\\{\pm n\\}.$ Since these are only
two points, we find that

$\operatorname{tr}_{\ell^{2}({\mathbb{Z}})}(\operatorname{1\hskip-2.75ptl}_{\\{-n,-n+1,...,n\\}}\operatorname{1\hskip-2.75ptl}_{\\{\lambda\\}}(H))=\mathcal{O}(1)$

and hence
$\widehat{\operatorname{tr}}(\operatorname{1\hskip-2.75ptl}_{\\{\lambda\\}}(H))=0.$
Thus, there cannot be any flat band at $\lambda$, as this would imply that
$\widehat{\operatorname{tr}}(\operatorname{1\hskip-2.75ptl}_{\\{\lambda\\}}(H))>0.$
∎

##  8\. A two-dimensional example

We now consider the case of $2D$ twisted lattice structures. For simplicity,
we shall consider two square lattices with moiré lengths $L_{1},L_{2}>0$, as
discussed for example in [KV19]. The kinetic energy is described by a discrete
Laplacian on each of the lattices in terms of

$(-\Delta_{{\mathbb{Z}}^{2}}u)_{n}=(u_{n+e_{1}}+u_{n-e_{1}}+u_{n+e_{2}}+u_{n-e_{2}})\text{
with }n\in{\mathbb{Z}}^{2}$

such that
$D_{\operatorname{kin}}\psi_{n}=-\operatorname{diag}(\Delta_{{\mathbb{Z}}^{2}},\Delta_{{\mathbb{Z}}^{2}})\psi_{n}$
is the discrete Laplacian of the individual lattices without any additional
interaction. The interaction is then modeled by a tunneling potential

$V_{w}(n)=w\begin{pmatrix}0&U(\tfrac{n_{1}}{L_{1}},\tfrac{n_{2}}{L_{2}})\\\
U(\tfrac{n_{1}}{L_{1}},\tfrac{n_{2}}{L_{2}})&0\end{pmatrix}$

with coupling strength $w>0,$ where we assume that $U$ is a real-valued smooth
$1$-periodic function in both components. This defines a Hamiltonian
$H:\ell^{2}({\mathbb{Z}}^{2};{\mathbb{C}}^{4})\to\ell^{2}({\mathbb{Z}}^{2};{\mathbb{C}}^{4})$

$H\psi_{n}=D_{\operatorname{kin}}\psi_{n}+V_{w}\psi_{n}.$ (8.1)

Figure 6. Twisted square lattices exhibit yet another macroscopic (moiré)
square lattice.

We then introduce

$P=\left(P_{\tfrac{n_{1}}{L_{1}},\tfrac{n_{2}}{L_{2}}}\right)_{n\in{\mathbb{Z}}^{2}}\text{
with }P_{X}=\frac{1}{\sqrt{2}}\begin{pmatrix}-\operatorname{sgn}(U(X))&1\\\
1&\operatorname{sgn}(U(X))\end{pmatrix},$

then conjugating by $P$ yields, after swapping entries according to the sign
of $U$, an equivalent block-diagonal Hamiltonian
$\tilde{H}:\ell^{2}({\mathbb{Z}}^{2};{\mathbb{C}}^{4})\to\ell^{2}({\mathbb{Z}}^{2};{\mathbb{C}}^{4})$

$\tilde{H}\psi_{n}=\operatorname{diag}\left(-\Delta_{{\mathbb{Z}}^{2}}+U(\tfrac{n_{1}}{L_{1}},\tfrac{n_{2}}{L_{2}}),-\Delta_{{\mathbb{Z}}^{2}}-U(\tfrac{n_{1}}{L_{1}},\tfrac{n_{2}}{L_{2}})\right)\psi_{n}.$
(8.2)

This leads us to the following result which shows that for a set of moiré
length scales of large measure, the model actually exhibits Anderson
localization.

######  Theorem 8.

[BGS02, Theo $6.2$] Let $U:\mathbb{T}^{2}\to\mathbb{R}$ be real analytic such
that the marginals

$\theta_{1}\mapsto U(\theta_{1},\theta_{2})\text{ and }\theta_{2}\mapsto
U(\theta_{1},\theta_{2})$

are non-degenerate. Moreover, let $\varepsilon>0,$ then for any $w\geq
w_{0}(\varepsilon)$ there exists a set of moiré length scales
$(1/L_{1},1/L_{2})\in\mathbb{T}^{2}$ of measure $1-\varepsilon$ such that the
Hamiltonian exhibits full Anderson localization, i.e. the spectrum consists
only of exponentially decaying eigenfunctions.

Related questions on the existence of absolutely continuous spectrum for small
coupling $w$ and Cantor spectrum are widely open. The situation simplifies,
once one imposes a separability condition, $U(x)=U_{1}(x_{1})U_{2}(x_{2})$
with $U_{1},U_{2}$ real-analytic and non-degenerate. In this case, the
operator (8.2) decomposes into the direct sum of two Hamiltonians

$H_{1}=\operatorname{diag}(-\Delta_{{\mathbb{Z}}}+U_{1},-\Delta_{{\mathbb{Z}}}-U_{1})\text{
and
}H_{2}=\operatorname{diag}(-\Delta_{{\mathbb{Z}}}+U_{2},-\Delta_{{\mathbb{Z}}}-U_{2}).$

## Acknowledgments

The research of Jens Wittsten was partially supported by The Swedish Research
Council grant 2019-04878.

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