# From incommensurate bilayer heterostructures to Allen-Cahn: An exact
thermodynamic limit

Michael Hott E-mail<EMAIL_ADDRESS>School of Mathematics,
University of Minneapolis, Twin Cities,
Minneapolis, MN 55455, USA  0000-0003-4243-6585 Alexander B. Watson E-mail:
<EMAIL_ADDRESS>School of Mathematics,
University of Minneapolis, Twin Cities,
Minneapolis, MN 55455, USA  0000-0002-7566-4851 Mitchell Luskin E-mail:
<EMAIL_ADDRESS>School of Mathematics,
University of Minneapolis, Twin Cities,
Minneapolis, MN 55455, USA  0000-0003-1981-199X

###### Abstract

We give a complete and rigorous derivation of the mechanical energy for
twisted 2D bilayer heterostructures without any approximation beyond the
existence of an empirical many-body site energy. Our results apply to both the
continuous and discontinuous continuum limit. Approximating the intralayer
Cauchy-Born energy by linear elasticity theory and assuming an interlayer
coupling via pair potentials, our model reduces to a modified Allen-Cahn
functional. We rigorously control the error, and, in the case of sufficiently
smooth lattice displacements, provide a rate of convergence for twist angles
satisfying a Diophantine condition.

##### Statements and Declarations

The authors do not declare financial or non-financial interests that are
directly or indirectly related to the work submitted for publication.

##### Acknowledgements

MH’s and ML’s research was partially supported by Targeted Grant Award No.
896630 through the Simons Foundation. AW’s, and ML’s research was supported in
part by DMREF Award No. 1922165 through the National Science Foundation. The
authors are grateful to Eric Cancès and Xiaowen Zhu for helpful comments on
the first version of the manuscript. The authors would like to thank Dmitriy
Bilyk, Max Chaudkhari, Max Engelstein, Fabian Faustlich, and Ziyan (Zoe) Zhu
for helpful comments, stimulating discussions, and for pointing out helpful
references.

###### Contents

  1. 1 Introduction
  2. 2 Description of the model
    1. 2.1 Relaxed lattice
      1. 2.1.1 Monolayer potential
      2. 2.1.2 Interlayer coupling
    2. 2.2 Thermodynamic limit
      1. 2.2.1 Monolayer potential
      2. 2.2.2 Interlayer coupling
      3. 2.2.3 Convergence rate
      4. 2.2.4 Regularity assumptions
    3. 2.3 Cauchy-Born Approximation
    4. 2.4 Relationship with Allen-Cahn energy functional
    5. 2.5 Outlook
    6. 2.6 Sketch of the proof
    7. 2.7 Comparison with previous result
  3. 3 Preliminaries
    1. 3.1 Equivalence of disregistries
    2. 3.2 Notation
    3. 3.3 Lattice reduction formulae
    4. 3.4 Ergodic Theorems
      1. 3.4.1 General setting
      2. 3.4.2 Incommensurate bilayer system
      3. 3.4.3 Dirichlet kernel as a discrepancy function
      4. 3.4.4 Properties of the Dirichlet kernel
  4. 4 Statement of results
    1. 4.1 Convergence without a rate
    2. 4.2 Pointwise convergence with rate
  5. 5 Proofs of lattice reduction formulae
  6. A Lattice calculus
  7. B Existence of Diophantine 2D rotations
  8. C Proofs of ergodic theorems

##  1 Introduction

Setting a two-dimensional lattice on another with a small rotation creates
quasi-periodic moiré patterns on a superlattice scale given by the period of
oscillation of the interlayer disregistry. In recent years, moiré materials
consisting of a few relatively twisted layers of 2D-crystal structures have
attracted great attention since the moiré length scale offers the possibility
of discovering previously inaccessible electronic phases and other phenomena
[35].

The seminal paper in 2011 by Bistritzer and MacDonald [15] predicted that
twisted bilayer graphene (TBG), at an angle of $\theta\approx 1.1^{\circ}$ –
the so-called magic angle – exhibits almost flat electronic bands near the
charge neutrality point. As a consequence of the flat bands, there is a high
density of states, which has been known to provide a platform for emergent
electronic phases involving strong electron interactions. This has been
experimentally verified, e.g., in [23, 22], where both a superconducting and a
correlated insulating phase have been established. In particular, moiré
materials have become quantum simulators, i.e., experimental realizations for
effective theories predicted by theorists. For overviews on moiré materials,
we refer to, e.g., [25, 52, 67].

Structural relaxation, the process by which atoms deviate from their monolayer
equilibrium positions in order to minimize the total mechanical energy of the
twisted bilayer, can significantly modify the electronic properties of moiré
materials. For example, relaxation enhances the band gap between the flat
electronic bands and other bands in twisted bilayer graphene [81]. In a
simplified model of twisted bilayer graphene with strong relaxation, the
chiral model, the almost flat bands become exactly flat [6, 5, 80, 78]. It is
thus critical to develop systematic methods for modeling structural relaxation
and its effects on electronic properties of moiré materials.

Modeling structural relaxation is complicated by the fact that the atomic
structure of twisted bilayer 2D materials is generally aperiodic
(incommensurate) except at special commensurate angles [79]. Phenomenological
continuum models for structural relaxation at commensurate twist angles have
been proposed in [83, 68, 72, 82].

A systematic approach not restricted to commensurate twist angles has been
proposed in [27]. In this approach, the displacements $u_{j}$ of atoms in
layer $j$ of an incommensurate twisted 2D bilayer structure are assumed to be
continuous functions of the local configuration, or disregistry, which can be
identified with a two-dimensional torus. This is based on the more general
concept of a hull, a compact parametrization of all possible local
environments, which has been introduced in the context of electronic
properties of general aperiodic structures, see, e.g., [7, 9, 8] and
references therein.

Such a parametrization is known to be valid for the Frenkel-Kontorova model in
one dimension, using Aubry-Mather theory [2, 3, 4]. In addition, as a function
of the disregistry, a minimizer, if it exists, is smooth as long as the
intralayer potential dominates the interlayer coupling; see [2, 3, 4] and also
[28] for a simplified model. When this condition does not hold, the minimizer
can be discontinuous at a dense set of points [2, 3, 4, 28]. In two
dimensions, we are not aware of any proof that minimizers of the twisted
bilayer energy necessarily admit such a continuous parameterization, although
the assumption is well-motivated given that intralayer lattice bonds are known
to be much stronger than interlayer couplings in TBG and other van der Waals
2D heterostructures.

The total mechanical energy of the bilayer is, then, assumed to be
decomposable into a monolayer and an interlayer coupling contribution. For the
monolayer part, [27] starts with many-body site potentials and then passes
directly to the Cauchy-Born approximation, inspired by, e.g., [70, 16]. For
the interlayer coupling, they utilize a stacking penalization, the
_generalized stacking fault energy_ (GSFE), see also [33], dependent on a
modulated disregistry, i.e., on the relaxed local environment. Their modulated
disregistry is obtained by projecting an interpolated relaxed position onto
the relaxed structure of the other layer, see figure 1.

Figure 1: Modulated disregistry ${\bf\hat{D}}_{j\to 3-j}(R_{j})$ chosen in
[27]. In comparison, the sequence $\vec{\Delta}_{j\to 3-j}(R_{j})$ of all
modulated disregistries chosen in the present work.

To define the modulated disregistry more precisely, let $A_{j}$ denote the
matrix whose columns are a basis of lattice ${\mathcal{R}}_{j}$ describing
layer $j$, and

$\displaystyle D_{j\to 3-j}\,:=\,I-A_{3-j}A_{j}^{-1}$ (1.1)

denote the _disregistry matrices_. Then the modulated disregistry chosen in
[27] is given by

$\displaystyle{\bf\hat{D}}_{j\to 3-j}(R_{j})=D_{j\to
3-j}R_{j}+u_{j}(R_{j})-u_{3-j}(R_{j})\,,\quad R_{j}\in{\mathcal{R}}_{j}\,.$
(1.2)

In particular, since $u_{3-j}$ is not evaluated at a lattice vector of layer
$3-j$, the chosen modulated disregistry in [27] only approximates the
difference vectors of nearby atoms from different layers for small angles
$\theta$, as then the positions of the two layers align.

In order to pass to a thermodynamic limit, [27] employs an ergodic property
for continuous functions of the interlayer disregistry in incommensurate
lattices, derived in [21, 65] in the context of 2D incommensurate systems –
systems with a relative irrational angle, see Assumption 1 below. After they
approximate the Cauchy-Born energy by linear elasticity, their resulting model
is given by

$\displaystyle\begin{split}&\frac{1}{2}\sum_{j=1}^{2}\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
\Big{(}Du_{j}(x):{\mathcal{E}}_{j}:Du_{j}(x)\\\
&\qquad+\Phi^{\mathrm{(GSFE)}}_{j}\big{(}D_{j\to
3-j}x+u_{j}(x)-u_{3-j}(x)\big{)}\Big{)}\,,\end{split}$ (1.3)

where $\Gamma_{\mathcal{M}}$ denotes a moiré unit cell, see (2.11) below, and
the GSFE $\Phi^{\mathrm{(GSFE)}}_{j}:{\mathbb{R}}^{2}\to{\mathbb{R}}$ is
periodic w.r.t. to lattice ${\mathcal{R}}_{3-j}$ of the other layer.
Furthermore, we abbreviate by

$\displaystyle{\mathcal{E}}_{j,abcd}\,:=\,\lambda_{j}\delta_{ab}\delta_{cd}\,+\,\mu_{j}(\delta_{ad}\delta_{bc}+\delta_{ac}\delta_{bd})\,,$
(1.4)

the elasticity tensor where $\lambda_{j},\mu_{j}>0$ are referred to as _Lamé
parameters_. We adjusted the sign-convention in (1.3) to that chosen in the
present work.

In applications, the GSFE is obtained via a density functional theory (DFT)
approach by minimizing the energy of two untwisted parallel layers at a given
$xy$-shift corresponding to the disregistry, while the system is allowed to
relax in the $z$-direction, see [33, 24]. Periodic boundary conditions can
then be chosen to account for the lattice periodicity.

The relaxation model obtained in [27] has been very successfully applied to
determine the relaxed structure of twisted bilayer systems at small twist
angles, see, e.g., [24]. The goal of the present work is to develop a new
approach to modeling relaxation of twisted bilayer 2D materials which builds
on and refines several aspects of the approach of [27]. In particular, we
expect our model to retain accuracy even at relatively large twist angles, and
lend itself better to the systematic study of lattice vibrations (phonons),
which are known to play a significant role in the electronic properties of
many materials.

The starting point of our approach is a weaker, namely111To be more precise,
we pass to the thermodynamic limit for any $(u_{1},u_{2})$ such that the
limiting site energy is in $L^{1}(\Gamma_{\mathcal{M}})$, where
$\Gamma_{\mathcal{M}}$ is the moiré cell; see Proposition 2.7, and Theorems
4.2 and 4.3. We show that $u_{j}\in L^{\infty}(\Gamma_{\mathcal{M}})\subset
L^{1}(\Gamma_{\mathcal{M}})$ is sufficient for this in Lemma 3.5. $L^{1}$,
parametrization of the atomic displacement functions $u_{j}$ as functions of
their relative position w.r.t. the moiré unit cell $\Gamma_{\mathcal{M}}$,
modulo a moiré lattice shift. This parametrization is given meaning by
embedding the twisted bilayer model within the family of models defined by all
possible layer shifts (modulo shifts which leave the bilayer invariant) and
applying Birkhoff-type ergodic theorems which hold up to a measure zero set of
shifts. In particular, this perspective allows us to pass to an almost sure
thermodynamic limit without assuming a continuous parametrization of the
atomic displacements as in [27].

The second key point of our approach is that we start from an exact (not
interpolated) interlayer penalization based on the site potential dependent on
the entire sequence of differences

$\displaystyle\vec{\Delta}_{j\to
3-j}(R_{j})\,=\,\big{(}R_{j}+u_{j}(R_{j})-R_{3-j}-u_{3-j}(R_{3-j})\big{)}_{R_{3-j}\in{\mathcal{R}}_{3-j}}$
(1.5)

of displaced lattice positions. Passing to the thermodynamic limit with such a
penalization requires a more general form of the Birkhoff theorem used in
[27]. Since we do not invoke any interpolation or small-angle approximation
here, we expect this interlayer energy to retain accuracy for arbitrary twist
angles. We additionally generalize [27] in allowing for finitely-many
independent sublattice degrees of freedom, and relaxation in the $z$
direction. Let $\alpha_{j}\in{\mathcal{A}}_{j}$ be a sublattice degree of
layer $j$ with corresponding shift vector
$\tau^{(\alpha_{j})}_{j}\in{\mathbb{R}}^{2}\times\\{0\\}$, $\gamma_{j}$ be the
position of the origin of layer $j$, $A_{j}\in\mathrm{GL}_{2}({\mathbb{R}})$
the matrix consisting of the basis vectors of layer $j$,
$u_{j}(R_{j}+\gamma_{j}\mod\Gamma_{\mathcal{M}},\alpha_{j})$ be the
displacement of the lattice point in layer $j$ labeled by its position $R_{j}$
and its sublattice degree of freedom $\alpha_{j}$. Moreover, let
$u_{j}(\cdot,\alpha_{j})$ denote the moiré-periodic extension of $u_{j}$, and
denote $Y_{j}(x,\alpha_{j}):=x+\tau^{(\alpha_{j})}_{j}+u_{j}(x,\alpha_{j})$.
Let $V^{\mathrm{(mono)}}_{j,\alpha_{j}}$ be an empirical many-body monolayer
energy for layer $j$, and $V^{\mathrm{(inter)}}_{j,\alpha_{j}}$ an empirical
many-body interlayer coupling potential for layer $j$. Then the resulting
limiting energy is given by

$\displaystyle\sum_{j=1}^{2}\frac{1}{|{\mathcal{A}}_{j}||\Gamma_{j}|}\sum_{\alpha_{j}\in{\mathcal{A}}_{j}}\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{A_{j}^{-1}\Gamma_{\mathcal{M}}}\mathrm{d}x\
\Big{[}V^{\mathrm{(mono)}}_{j,\alpha_{j}}\Big{(}\big{(}u_{j}(A_{j}x+R_{j}^{\prime},\alpha_{j}^{\prime})-u_{j}(A_{j}x,\alpha_{j})\big{)}_{\begin{subarray}{c}R_{j}^{\prime}\in{\mathcal{R}}_{j},\\\
\alpha_{j}^{\prime}\in{\mathcal{A}}_{j}\end{subarray}}\Big{)}$ (1.6)
$\displaystyle\,+\,V^{\mathrm{(inter)}}_{j,\alpha_{j}}\Big{(}\big{(}Y_{j}(A_{j}x+\gamma_{j},\alpha_{j})-Y_{3-j}(A_{3-j}x+\gamma_{3-j}+R_{3-j},\alpha_{3-j})\big{)}_{\begin{subarray}{c}R_{3-j}\in{\mathcal{R}}_{3-j},\\\
\alpha_{3-j}\in{\mathcal{A}}_{3-j}\end{subarray}}\Big{)}\Big{]}\,.$ (1.7)

Although we emphasize the generality of our approach, it is illuminating to
present our configuration-space model under simplifying assumptions. In
particular, this allows for comparison between our model and (1.3). Let us
assume the interlayer many-body potential is given by an even pair potential
$v$. We ignore sublattice degrees of freedom and lattice shifts, and restrict
$u_{j}$ to take values in ${\mathbb{R}}^{2}$, i.e., we ignore out-of-plane
relaxation, and assume smooth $u_{j}$ so that the Cauchy-Born approximation
applies. After further approximation by linear elasticity, our model is given
by

$\displaystyle\begin{split}&\frac{1}{2}\sum_{j=1}^{2}\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
Du_{j}(x):{\mathcal{E}}_{j}:Du_{j}(x)\\\
&\quad+\frac{1}{|\Gamma_{\mathcal{M}}|}\int_{{\mathbb{R}}^{2}}\mathrm{d}x\
v\big{(}A_{1}x+u_{1}(A_{1}x)-A_{2}x-u_{2}(A_{2}x)\big{)}\,.\end{split}$ (1.8)

For the most general case of an interlayer coupling energy, we refer to (2.54)
and Section 2.2.2. We show the difference between the models for a specific
choice of GSFE depending on $v$ in Section 2.7. In Section 2.3, we provide
some more details on the Cauchy-Born and linear elasticity approximation.

The other contribution of the present work is, under additional regularity
assumptions on the diplacements $u_{j}$, an estimate on the convergence rate
for passage to the thermodynamic limit. For this, we introduce the notion of
Diophantine 2D rotations, a two-dimensional analogue of Diophantine numbers,
to quantify the incommensurability of twisted 2D bilayer lattices. We show
that an ergodic average can be rewritten in Fourier representation as a
weighted sum involving an appropriately rescaled Dirichlet kernel. The leading
order term in orders of the truncation corresponds to the limit of the ergodic
average. The lower order terms can be bounded by the tail-behavior of the
Dirichlet kernel, if we assume an appropriate Diophantine angle condition,
defining Diophantine 2D rotations.

Our concept of Diophantine 2D rotations extends the ideas developed in [29] in
the context of coupled one-dimensional chains. Metric Diophantine
approximation theory is a vibrant research area in analytic number theory. We
refer to [12, 56, 10] and references therein for historic reviews, and, e.g.,
to [13, 14, 31, 47, 69] and references therein for more recent results.

One of the promising properties of the models presented here and in [27] are
that they lend themselves to methods from calculus of variations for
continuous systems, even though the underlying system is discrete. As we will
discuss below in Section 2.4, the resulting model is connected to the Allen-
Cahn or Ginzburg-Landau energy functional [41, 42]. For references on the
study of the Allen-Cahn equation, we refer to, e.g., [20, 32, 43, 62, 77] and
references therein, for the Ginzburg-Landau equations, e.g., to [46, 49, 73,
75] and references therein, and for results on more general semilinear
elliptic equations, e.g., to [1, 30, 36, 38, 39, 50], and references therein.

An interesting question is whether an existence and regularity theory can be
developed for the thermodynamic limit energy involving many-body monolayer
potentials, before passing to the Cauchy-Born approximation. We do not address
this question in the present work, instead developing the existence and
regularity theory only for displacements for which the intralayer energy can
be simplified using the Cauchy-Born approximation, such as
$C^{2}$-displacements. After this simplification, the existence and regularity
theory are standard, since the Euler-Lagrange equation is, then, elliptic;
see, e.g., [27].

We comment on the practical application of our results, especially to the
study of lattice vibrations (phonons) in twisted bilayer materials. First, we
note that passing to the thermodynamic limit without restricting to smooth or
even continuous displacements $u_{j}$ should allow us to model more subtle
relaxation patterns and vibrations than those which can be described using the
Cauchy-Born or linear elasticity approximation. The use of the GSFE in [55] is
limited to nearest neighbor interactions and does not include out-of-plane
displacement. The phonon dispersion method given in [61] includes longer-
ranged interactions in the GSFE and out-of-plane displacements. Our interlayer
energy is exact, but it needs approximation since it includes infinite range
interactions. A systematic approximation of our interlayer energy can give a
method to compute the phonon dispersion with controlled accuracy. Finally, we
note that our new interlayer misfit energy has the potential to be highly
accurate if implemented with the new generation of machine learned empirical
potentials [66, 76]. These issues will be the subjects of forthcoming works.

##  2 Description of the model

Let $\theta\in{\mathbb{R}}$, $q>0$, and $A\in\text{GL}_{2}({\mathbb{R}})$.
Define

$\displaystyle A_{1}\,:=\,q^{-1/2}R_{-\theta/2}A\,,\quad
A_{2}\,:=\,q^{1/2}R_{\theta/2}A\,,\quad
R_{\phi}\,:=\,\begin{pmatrix}\cos\phi&-\sin\phi\\\
\sin\phi&\cos\phi\end{pmatrix}\,.$ (2.1)

Let

$\displaystyle{\mathcal{R}}_{j}\,:=\,A_{j}{\mathbb{Z}}^{2}\,,\,j\in\\{1,2\\},$
(2.2)

denote two crystal layers at a relative twist angle $\theta$ and with a
relative mismatch of $q$, and abbreviate the respective unit cells

$\displaystyle\Gamma_{j}\,:=\,A_{j}[0,1)^{2}\,,\,j\in\\{1,2\\}\,.$ (2.3)

In addition, denoting $\llbracket
n_{1},n_{2}\rrbracket:=[n_{1},n_{2}]\cap{\mathbb{Z}}$ whenever
$n_{1},n_{2}\in{\mathbb{Z}}$, $n_{1}\leq n_{2}$, we introduce the truncated
lattices

$R_{j}(N)\,:=\,A_{j}\llbracket-N,N\rrbracket^{2}\,,\,j\in\\{1,2\\}\,,\,N\in{\mathbb{N}}\,.$
(2.4)

Furthermore, let

$\displaystyle B_{j}\,:=\,2\pi A_{j}^{-T}\,,\,j\in\\{1,2\\},$ (2.5)

denote the matrix of reciprocal vectors, and define the reciprocal lattices

$\displaystyle{\mathcal{R}}_{j}^{*}\,:=\,B_{j}{\mathbb{Z}}^{2}\,,\,j\in\\{1,2\\}\,.$
(2.6)

Throughout this work, we let $j\in\\{1,2\\}$.

Figure 2: Twisted bilayer with its moiré lattice highlighted in yellow.

$\xrightarrow[\cong]{\Pi_{\Gamma_{{\mathcal{M}}}}}$

The _moiré frequency vectors_ are given by

$\displaystyle B_{\mathcal{M}}\,:=\,B_{1}-B_{2}\,,$ (2.7)

generating the _moiré frequency lattice_

$\displaystyle{\mathcal{R}}^{*}_{\mathcal{M}}\,:=\,B_{\mathcal{M}}{\mathbb{Z}}^{2}\,.$
(2.8)

The corresponding moiré direct lattice vectors are given by

$\displaystyle A_{\mathcal{M}}\,$ $\displaystyle:=\,2\pi
B_{\mathcal{M}}^{-T}\,=\,(A_{1}^{-1}-A_{2}^{-1})^{-1}\,.$ (2.9)

Then we define _moiré lattice_

$\displaystyle{\mathcal{R}}_{\mathcal{M}}\,:=\,A_{\mathcal{M}}{\mathbb{Z}}^{2}$
(2.10)

together with its unit cell

$\displaystyle\Gamma_{\mathcal{M}}\,:=\,A_{\mathcal{M}}[0,1)^{2}\,.$ (2.11)

As can be seen in figure 2, the moiré lattice is the lattice w.r.t. which the
union of the lattices ${\mathcal{R}}_{1}\cup{\mathcal{R}}_{2}$ appears almost
periodic.

######  Lemma 2.1 (Moiré length scale).

For any $x\in{\mathbb{R}}^{2}$, we have that

$\displaystyle|A_{\mathcal{M}}x|\,=\,[(q^{1/2}-q^{-1/2})^{2}+4\sin^{2}(\theta/2)]^{-\frac{1}{2}}|Ax|\,.$
(2.12)

Similarly, we have for any $k\in{\mathbb{R}}^{2}$ that

$\displaystyle|B_{\mathcal{M}}k|\,=\,[(q^{1/2}-q^{-1/2})^{2}+4\sin^{2}(\theta/2)]^{\frac{1}{2}}|2\pi
A^{-T}k|\,.$ (2.13)

For a proof, we refer to Appendix A. Motivated by this lemma, we define the
characteristic _moiré length scale ratio_ given by

$\rho_{\mathcal{M}}\,:=\,[(q-q^{-1})^{2}+4\sin^{2}(\theta/2)]^{-\frac{1}{2}}\,.$
(2.14)

In order to specify almost periodicity, we impose the following restrictions.

######  Assumption 1.

$({\mathcal{R}}_{1},{\mathcal{R}}_{2})$ is _incommensurate_ , i.e.,

$\displaystyle G_{1}\,+\,G_{2}\,=\,0\mbox{\ \emph{for}\
}G_{j}\in{\mathcal{R}}_{j}^{*}\quad\mbox{\emph{iff}}\quad G_{1}=G_{2}=0\,.$
(2.15)

Assumption 1 has been introduced in [21, 65]. In addition, we impose the
following condition.

######  Assumption 2.

$({\mathcal{R}}_{1}^{*},{\mathcal{R}}_{2}^{*})$ is incommensurate.

######  Remark 2.2.

Under assumption 1, it can be readily verified that also ${\mathcal{R}}_{j}$
and ${\mathcal{R}}_{\mathcal{M}}$ are incommensurate. In fact, let
$G_{\mathcal{M}}=B_{\mathcal{M}}m\in{\mathcal{R}}^{*}_{\mathcal{M}}$,
$m\in{\mathbb{Z}}^{2}$, and $G_{1}=B_{1}n\in{\mathcal{R}}_{1}^{*}$,
$n\in{\mathbb{Z}}^{2}$. Then we have that

$G_{\mathcal{M}}\,+\,G_{1}\,=\,B_{1}(m+n)\,-\,B_{2}n\,=\,0$ (2.16)

iff $m=n=0$, see (2.7). Analogously,
$({\mathcal{R}}_{1}^{*},{\mathcal{R}}_{2}^{*})$ being incommensurate implies
that $({\mathcal{R}}_{j}^{*},{\mathcal{R}}_{\mathcal{M}}^{*})$, $j=1,2$ is
incommensurate.

A straight-forward calculation yields the following result.

######  Lemma 2.3.

We have that $({\mathcal{R}}_{1},{\mathcal{R}}_{2})$ is incommensurate iff
$(q,\theta)\in{\mathbb{R}}^{+}\times[0,2\pi)$ satisfy

$qA^{T}R_{\theta}A^{-T}{\mathbb{Z}}^{2}\cap{\mathbb{Z}}^{2}\,=\,\\{0\\}\,.$
(2.17)

As a consequence of Assumption 2, the relative positions of
${\mathcal{R}}_{j}$ w.r.t. the moiré lattice never repeat. We want to exploit
this fact in order to relabel lattice positions by their relative position
within a translate of a moiré unit cell.

Assumption 1 yields that positions in the lattice
$\gamma_{j}+{\mathcal{R}}_{j}$ can be relabelled by positions in
$\Gamma_{\mathcal{M}}$. Consequently, for $x\in{\mathbb{R}}^{2}$, define

$\left\\{x\right\\}_{\mathcal{M}}\,:=\,\sum_{R_{\mathcal{M}}\in{\mathcal{R}}_{\mathcal{M}}}(x-R_{\mathcal{M}})\mathds{1}_{\Gamma_{\mathcal{M}}}(x-R_{\mathcal{M}})$
(2.18)

and

$\left\lfloor
x\right\rfloor_{\mathcal{M}}\,:=\,x\,-\,\left\\{x\right\\}_{\mathcal{M}}\,.$
(2.19)

Observe that

$\mathds{1}_{\Gamma_{\mathcal{M}}}(x-R_{\mathcal{M}})=0\quad\Leftrightarrow\quad
x\in\Gamma_{\mathcal{M}}+R_{\mathcal{M}}\,.$ (2.20)

Analogously, we define the decomposition w.r.t. the lattices
${\mathcal{R}}_{j}$

$x\,=\,\left\lfloor x\right\rfloor_{j}\,+\,\left\\{x\right\\}_{j}$ (2.21)

with $\left\lfloor x\right\rfloor_{j}\in{\mathcal{R}}_{j}$,
$\left\\{x\right\\}_{j}\in\Gamma_{j}$. An analogous notation has already been
used in [21]. We call $\left\\{\cdot\right\\}_{\mathcal{M}}$ the _disregistry_
w.r.t. ${\mathcal{R}}_{\mathcal{M}}$ and $\left\\{\cdot\right\\}_{j}$ the
_disregistry_ w.r.t. ${\mathcal{R}}_{j}$.

As a consequence of incommensurability of
$({\mathcal{R}}_{1}^{*},{\mathcal{R}}_{2}^{*})$ and
$({\mathcal{R}}_{j}^{*},{\mathcal{R}}_{\mathcal{M}}^{*})$, respectively, both

$\displaystyle\Pi_{\Gamma_{3-j}}\,$
$\displaystyle:\,{\mathcal{R}}_{j}+\gamma_{j}\to\Gamma_{3-j}\,,\quad
R_{j}+\gamma_{j}\mapsto\left\\{R_{j}+\gamma_{j}\right\\}_{3-j}\,,$ (2.22)
$\displaystyle\Pi_{\Gamma_{{\mathcal{M}}}}\,$
$\displaystyle:\,{\mathcal{R}}_{j}+\gamma_{j}\to\Gamma_{\mathcal{M}}\,,\quad
R_{j}+\gamma_{j}\mapsto\left\\{R_{j}+\gamma_{j}\right\\}_{\mathcal{M}}\,,$
(2.23)

are one-to-one, $j=1,2$. For a discussion of this bijection, we refer, e.g.,
to [64]. We call both, $\Pi_{\Gamma_{3-j}}$ and
$\Pi_{\Gamma_{{\mathcal{M}}}}$, _configuration space projections_ , see also
figure 2.

Before continuing, we briefly want to comment why we can simultaneously choose
$(\mathcal{R}_{1},\mathcal{R}_{2})$ _and_
$(\mathcal{R}_{1}^{*},\mathcal{R}_{2}^{*})$ to be incommensurate. For that, we
introduce the notion of _Diophantine $2D$-rotations_, that corresponds to a
quantified version of incommensurability.

######  Definition 2.4.

Given $q>0$, we call $\theta\in{\mathbb{R}}$ a $(q,K,\sigma)$-_Diophantine 2D
rotation_ iff $K>0,\sigma>0$, and for all
$n\in{\mathbb{Z}}^{2}\setminus\\{0\\}$

$\operatorname{dist}\big{(}q^{\pm
1}A^{T}R_{\pm\theta}A^{-T}n,{\mathbb{Z}}^{2}\big{)}\geq\frac{K}{|n|^{2\sigma}}\,.$
(2.24)

A physically important property is that almost every angle $\theta$ is a 2D
Diophantine rotation.

######  Proposition 2.5.

For all $q>0$, $\sigma>\frac{1433}{1248}\approx 1.1482$ and for Lebesgue-
almost all $\theta\in{\mathbb{R}}$ there exists $K>0$ s.t. $\theta$ is a
$(q,K,\sigma)$-Diophantine 2D rotation.

Since almost every angle $\theta$ is a Diophantine 2D rotation, the
corresponding lattice $(\mathcal{R}_{1},\mathcal{R}_{2})$ is incommensurate.
By reciprocity, we also have that for almost every $\theta$,
$(\mathcal{R}_{1}^{*},\mathcal{R}_{2}^{*})$ is incommensurate. In particular,
for (Lebesgue-)almost every $\theta$, both,
$(\mathcal{R}_{1},\mathcal{R}_{2})$ and
$(\mathcal{R}_{1}^{*},\mathcal{R}_{2}^{*})$, are incommensurate.

As the classical results for the one-dimensional case, our proof of
Proposition 2.5, see Appendix B, relies on the Borel-Cantelli Lemma.

######  Remark 2.6.

Bourgain-Watt [18] proposed the bound

$\displaystyle|B_{r}\cap{\mathbb{Z}}^{2}|\,=\,\pi
r^{2}\,+\,O_{r\to\infty}(r^{\frac{517}{824}+{\varepsilon}})$ (2.25)

for any ${\varepsilon}>0$, where $\frac{517}{824}\approx 0.6274$, compared to
$\frac{131}{208}\approx 0.6298$ in B.11 in the proof of Proposition 2.5. In
particular, this would allow us to lower the lower bound of $\sigma$ to
$\frac{5671}{4944}\approx 1.1471$ compared to $\frac{1433}{1248}\approx
1.1482$ above.

The general lattice problem is a very active research area. For related works,
we refer to [17, 40, 19, 45] and references therein.

###  2.1 Relaxed lattice

Instead of the rigid lattices ${\mathcal{R}}_{j}$, we want to allow the
lattice positions to move, in order to attain an energetically favorable
configuration.

In addition to the Bravais lattices ${\mathcal{R}}_{j}$, we also introduce the
sublattice degrees of freedom ${\mathcal{A}}_{j}$, $j\in\\{1,2\\}$.
${\mathcal{A}}_{j}$ is assumed to be a finite set, and to each
$\alpha_{j}\in{\mathcal{A}}_{j}$, we assign a unique shift vector
$\tau^{(\alpha_{j})}_{j}\in{\mathbb{R}}^{2}$. Moreover, as an additional
degree of freedom, we allow layer $j$ to be shifted by
$\gamma_{j}\in\Gamma_{j}$.

We start by stating an almost sure and mean ergodic theorem that will allow us
to rigorously analyze thermodynamic limits of elastic energies that are
functions of nonsmooth displacements $u_{j}\in
L^{1}\big{(}\Gamma_{\mathcal{M}};({\mathbb{R}}^{3})^{{\mathcal{A}}_{j}}\big{)}.$

######  Proposition 2.7 (Ergodic Theorem).

For all $f\in L^{1}(\Gamma_{\mathcal{M}})$

$\lim_{N\to\infty}\frac{1}{(2N+1)^{2}}\sum_{R_{j}\in{\mathcal{R}}_{j}(N)}f(\left\\{R_{j}+\gamma_{j}\right\\}_{\mathcal{M}})\,=\,\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
f(x)\,.$ (2.26)

converges almost everywhere in $\Gamma_{j}$ and also in the
$L^{1}(\Gamma_{j})$-sense.

Proposition 2.7 follows from the general ergodic theorem stated here as
Proposition 3.8. Our first use of Proposition 2.7 will be to prove in
Proposition 2.9 that the Fourier transform of $u_{j}\in
L^{1}\big{(}\Gamma_{\mathcal{M}};({\mathbb{R}}^{3})^{{\mathcal{A}}_{j}}\big{)}$
can be reconstructed as a thermodynamic limit from
$u_{j}(\left\\{\gamma_{j}+{\mathcal{R}}_{j}\right\\}_{\mathcal{M}},{\mathcal{A}}_{j})$
for almost every $\gamma_{j}\in\Gamma_{j}$.

We now define lattice deformations based on $u_{j}\in
L^{1}\big{(}\Gamma_{\mathcal{M}};({\mathbb{R}}^{3})^{{\mathcal{A}}_{j}}\big{)}.$

######  Assumption 3.

There exist displacements $u_{j}\in
L^{1}\big{(}\Gamma_{\mathcal{M}};({\mathbb{R}}^{3})^{{\mathcal{A}}_{j}}\big{)}$
such that the total positions on the reference collection of lattices

$\displaystyle\big{(}(\gamma_{j}+{\mathcal{R}}_{j})\times{\mathcal{A}}_{j}\big{)}_{\gamma_{j}\in\Gamma_{j}}\,,\quad
j\in\\{1,2\\}\,,$ (2.27)

are given by the deformation

$\displaystyle
Y_{j}(\gamma_{j}+R_{j},\alpha_{j})\,:=\,\gamma_{j}\,+\,R_{j}\,+\,\tau_{j}^{(\alpha_{j})}\,+\,u_{j}(\left\\{\gamma_{j}+R_{j}\right\\}_{\mathcal{M}},\alpha_{j})\,.$
(2.28)

Here, we identified the ${\mathbb{R}}^{2}$-valued points $R_{j}$,
$\gamma_{j}$, $\tau_{j}^{(\alpha_{j})}$ with their respective
${\mathbb{R}}^{3}$-embeddings $(R_{j},0)$ resp. $(\gamma_{j},0)$ resp.
$(\tau_{j}^{(\alpha_{j})},0)$. We allow the lattice displacement $u_{j}$ to
take values in ${\mathbb{R}}^{3}$, in order to account for relaxation in
$z$-direction.

Assumption 3 implies that whenever
$\left\\{\gamma_{j}+R_{j}\right\\}_{\mathcal{M}}=\left\\{\gamma_{j}^{\prime}+R_{j}^{\prime}\right\\}_{\mathcal{M}}$,
we have that the associated displacements

$u_{j}(\left\\{\gamma_{j}+R_{j}\right\\}_{\mathcal{M}},\alpha_{j})=u_{j}(\left\\{\gamma_{j}^{\prime}+R_{j}^{\prime}\right\\}_{\mathcal{M}},\alpha_{j})$
(2.29)

coincide. Since $({\mathcal{R}}_{1}^{*},{\mathcal{R}}_{2}^{*})$ is
incommensurate, this can only happen for $\gamma_{j}\neq\gamma_{j}^{\prime}$,
ensuring that we do not impose a moiré-periodicity constraint.

Observe that the total lattice positions are well-defined only up to a nullset
of lattice shifts $\gamma_{j}\in\Gamma_{j}$. Next we will discuss in which
sense the data of $u_{j}$ on a single lattice suffices to reconstruct $u_{j}$.

For $f\in L^{1}(\Gamma_{\mathcal{M}})$ and
$G_{\mathcal{M}}\in{\mathcal{R}}^{*}_{\mathcal{M}}$, we define the Fourier
transform

$\displaystyle\hat{f}(G_{\mathcal{M}})\,:=\,{\mathcal{F}}_{\Gamma_{\mathcal{M}}}(f)(G_{\mathcal{M}})\,:=\,\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
e^{-iG_{\mathcal{M}}\cdot x}f(x)\,.$ (2.30)

######  Definition 2.8.

We say that $u_{j}\in
L^{1}\big{(}\Gamma_{\mathcal{M}};({\mathbb{R}}^{3})^{{\mathcal{A}}_{j}}\big{)}$
is _reconstructable_ along $\gamma_{j}+{\mathcal{R}}_{j}$ iff for all
$G_{\mathcal{M}}\in{\mathcal{R}}^{*}_{\mathcal{M}}$,
$\alpha_{j}\in{\mathcal{A}}_{j}$, we have that

$\displaystyle\begin{split}\lim_{N\to\infty}\frac{1}{(2N+1)^{2}}\sum_{R_{j}\in{\mathcal{R}}_{j}(N)}e^{-iG_{\mathcal{M}}\cdot(\gamma_{j}+R_{j})}u_{j}(\left\\{\gamma_{j}+R_{j}\right\\}_{\mathcal{M}},\alpha_{j})\,=\,\hat{u}_{j}(G_{\mathcal{M}},\alpha_{j})\,.\end{split}$
(2.31)

In particular, if $u_{j}\in
L^{p}\big{(}\Gamma_{\mathcal{M}};({\mathbb{R}}^{3})^{{\mathcal{A}}_{j}}\big{)}$,
$p>1$, is reconstructable along $\gamma_{j}+{\mathcal{R}}_{j}$, the Carleson-
Hunt Theorem, see, e.g., [37, Chapter 1], implies that for almost all
$x\in\Gamma_{\mathcal{M}}$ and all $\alpha_{j}\in{\mathcal{A}}_{j}$

$\displaystyle u_{j}(x,\alpha_{j})\,=\,$ (2.32)
$\displaystyle\sum_{G_{\mathcal{M}}\in{\mathcal{R}}^{*}_{\mathcal{M}}}e^{iG_{\mathcal{M}}\cdot
x}\lim_{N\to\infty}\frac{1}{(2N+1)^{2}}\sum_{R_{j}\in{\mathcal{R}}_{j}(N)}e^{-iG_{\mathcal{M}}\cdot(\gamma_{j}+R_{j})}u_{j}(\left\\{\gamma_{j}+R_{j}\right\\}_{\mathcal{M}},\alpha_{j})\,.$
(2.33)

If $u_{j},v_{j}\in
L^{1}\big{(}\Gamma_{\mathcal{M}};({\mathbb{R}}^{3})^{{\mathcal{A}}_{j}}\big{)}$,
we have the uniqueness property that

$\hat{u}_{j}(G_{\mathcal{M}},\alpha_{j})\,=\,\hat{v}_{j}(G_{\mathcal{M}},\alpha_{j})$
(2.34)

for all
$(G_{\mathcal{M}},\alpha_{j})\in{\mathcal{R}}^{*}_{\mathcal{M}}\times{\mathcal{A}}_{j}$
implies that $u_{j}=v_{j}$, see [37, Corollary 1.11].

As a consequence of Proposition 2.7, we find the following result.

######  Proposition 2.9 (Almost sure recontructability).

Every $u_{j}\in
L^{1}\big{(}\Gamma_{\mathcal{M}};({\mathbb{R}}^{3})^{{\mathcal{A}}_{j}}\big{)}$
is reconstructable along $\gamma_{j}+{\mathcal{R}}_{j}$ for almost every
$\gamma_{j}\in\Gamma_{j}$.

For a proof, we refer to Appendix A.

By abuse of notation, we identify $u_{j}$ with its periodic extension on
${\mathbb{R}}^{2}$. In particular, the deformations

$Y_{j}(x,\alpha_{j})\,:=\,x\,+\,\tau_{j}^{(\alpha_{j})}\,+\,u_{j}(x,\alpha_{j})\,,\quad(x,\alpha_{j})\in{\mathbb{R}}^{2}\times{\mathcal{A}}_{j}\,,$
(2.35)

are well-defined and extend (2.28).

####  2.1.1 Monolayer potential

Let

$V^{\mathrm{(mono)}}_{j,\alpha_{j}}\,:\,({\mathbb{R}}^{3})^{{\mathcal{R}}_{j}\times{\mathcal{A}}_{j}}\to{\mathbb{R}}$
(2.36)

with $\alpha_{j}\in{\mathcal{A}}_{j}$ be given. For $\gamma_{j}\in\Gamma_{j}$,
we introduce the monolayer energy density

$\displaystyle\begin{split}&e^{\mathrm{(mono)}}_{j,N,\gamma_{j}}(u_{j})\,:=\frac{1}{|{\mathcal{A}}_{j}||\Gamma_{j}|(2N+1)^{2}}\sum_{\alpha_{j}\in{\mathcal{A}}_{j}}\sum_{R_{j}\in{\mathcal{R}}_{j}(N)}\\\
&\quad\quad
V^{\mathrm{(mono)}}_{j,\alpha_{j}}\Big{(}\big{(}u_{j}(R_{j}+\gamma_{j}+R_{j}^{\prime},\alpha_{j}^{\prime})\,-\,u_{j}(R_{j}+\gamma_{j},\alpha_{j})\big{)}_{(R_{j}^{\prime},\alpha_{j}^{\prime})\in{\mathcal{R}}_{j}\times{\mathcal{A}}_{j}}\Big{)}\,,\end{split}$
(2.37)

where

$|A_{j}[-N-1/2,N+1/2]^{2}|\,=\,|\Gamma_{j}|(2N+1)^{2}$ (2.38)

is the area of the corresponding truncated region covered by the nuclei in
layer $j$.

####  2.1.2 Interlayer coupling

######  Definition 2.10.

A function

$V\,:\,({\mathbb{R}}^{3})^{{\mathcal{R}}_{j}\times{\mathcal{A}}_{j}}\to{\mathbb{R}}$
(2.39)

is _translation-invariant_ iff for any
$(a_{R_{j},\alpha_{j}})_{(R_{j},\alpha_{j})\in{\mathcal{R}}_{j}\times{\mathcal{A}}_{j}}\in({\mathbb{R}}^{3})^{{\mathcal{R}}_{j}\times{\mathcal{A}}_{j}}$
and any $R_{j}^{\prime}\in{\mathcal{R}}_{j}$, we have that

$V\big{(}(a_{R_{j}+R_{j}^{\prime},\alpha_{j}})_{(R_{j},\alpha_{j})\in{\mathcal{R}}_{j}\times{\mathcal{A}}_{j}}\big{)}\,=\,V\big{(}(a_{R_{j},\alpha_{j}})_{(R_{j},\alpha_{j})\in{\mathcal{R}}_{j}\times{\mathcal{A}}_{j}}\big{)}\,.$
(2.40)

In addition, $V$ is _even_ iff $V(x)=V(-x)$.

Let

$V^{\mathrm{(inter)}}_{j,\alpha_{j}}\,:\,({\mathbb{R}}^{3})^{{\mathcal{R}}_{3-j}\times{\mathcal{A}}_{3-j}}\to{\mathbb{R}}\,,$
(2.41)

with $\alpha_{j}\in{\mathcal{A}}_{j}$ be even and translation-invariant. As a
special instance, let $v_{{\bm{\alpha}}}:{\mathbb{R}}^{3}\to{\mathbb{R}}$,
with $\alpha_{k}\in{\mathcal{A}}_{k}$, $k=1,2$,
${\bm{\alpha}}:=(\alpha_{1},\alpha_{2})$, denote an even pair potential, and
consider

$\displaystyle\begin{split}&V^{\mathrm{(inter)}}_{j,\alpha_{j}}\big{(}(a_{R_{3-j},\alpha_{3-j}})_{\begin{subarray}{c}R_{3-j}\in{\mathcal{R}}_{3-j},\\\
\alpha_{3-j}\in{\mathcal{A}}_{3-j}\end{subarray}}\big{)}\\\
&=\,\frac{1}{2}\sum_{\begin{subarray}{c}R_{3-j}\in{\mathcal{R}}_{3-j}\\\
\alpha_{3-j}\in{\mathcal{A}}_{3-j}\end{subarray}}v_{{\bm{\alpha}}}(a_{R_{3-j},\alpha_{3-j}})\,.\end{split}$
(2.42)

Such interatomic pair potentials $v_{\bm{\alpha}}$ have been constructed using
DFT, e.g., in [59, 63, 74].

The fact that we are using the ordered index
${\bm{\alpha}}=(\alpha_{1},\alpha_{2})$ to label pair potentials, as opposed
to, e.g., $(\alpha_{j},\alpha_{3-j})$, is to indicate that pair interactions
only depend on the pair of involved atoms, as opposed to the order of the
pair. Whenever there is a $\alpha_{j}$ or $\alpha_{3-j}$ on the LHS of an
equation, and $\alpha_{1}$ or $\alpha_{2}$ on the RHS, the corresponding
equation is to be read individually for $j=1$ or $j=2$, to avoid confusion.

Throughout this work, we abbreviate

$\displaystyle{\bf
u}\,=\,(u_{1},u_{2})\,,\quad{\bm{\gamma}}=(\gamma_{1},\gamma_{2})\in\Gamma_{1}\times\Gamma_{2}\,.$
(2.43)

Recalling (2.35), define the interlayer coupling energy density of the $j$-th
layer

$\displaystyle\begin{split}&e^{\mathrm{(inter)}}_{j,N,{\bm{\gamma}}}({\bf
u})\,:=\\\
&\frac{1}{|{\mathcal{A}}_{j}||\Gamma_{j}|(2N+1)^{2}}\sum_{R_{j}\in{\mathcal{R}}_{j}(N)}\sum_{\alpha_{j}\in{\mathcal{A}}_{j}}V^{\mathrm{(inter)}}_{j,\alpha_{j}}\Big{(}\big{(}Y_{j}(R_{j}+\gamma_{j},\alpha_{j})\\\
&\qquad-\,Y_{3-j}(R_{3-j}+\gamma_{3-j},\alpha_{3-j})\big{)}_{\begin{subarray}{c}R_{3-j}\in{\mathcal{R}}_{3-j},\\\
\alpha_{3-j}\in{\mathcal{A}}_{3-j}\end{subarray}}\Big{)}\,.\end{split}$ (2.44)

###  2.2 Thermodynamic limit

We would like to study the thermodynamic limit of the energy densities (2.44)
by employing an appropriate ergodic theorem. Such an ergodic theorem lets us
map a functional defined on the discrete systems ${\mathcal{R}}_{j}$ to a
functional defined on the continuous system $\Gamma_{\mathcal{M}}$, via the
configuration space isomorphism $\Pi_{\Gamma_{{\mathcal{M}}}}$. As a useful
tool to map in between unit cells of the involved lattices, we recall from
(1.1) the disregistry matrices:

$\displaystyle D_{1\to
2}\,=\,I\,-\,A_{2}A_{1}^{-1}\,=\,-A_{2}A_{\mathcal{M}}^{-1}\,,$ (2.45)
$\displaystyle D_{2\to
1}\,=\,I\,-\,A_{1}A_{2}^{-1}\,=\,A_{1}A_{\mathcal{M}}^{-1}\,.$ (2.46)

In particular, we have that

$D_{j\to 3-j}\Gamma_{\mathcal{M}}\,=\,(-1)^{j}\Gamma_{3-j}\,.$ (2.47)

The disregistry matrices $D_{j\to 3-j}$ have the crucial property that they
map between the different notions of configurations if restricted to the
individual non-shifted lattices, see Lemma 3.2.

####  2.2.1 Monolayer potential

Observe that, since we defined $u_{j}$ as
${\mathcal{R}}_{\mathcal{M}}$-periodic extensions,

$\displaystyle\begin{split}&\Phi^{\mathrm{(mono)}}_{j}[u_{j}](x)\,:=\\\
&\frac{1}{|{\mathcal{A}}_{j}||\Gamma_{j}|}\sum_{\alpha_{j}\in{\mathcal{A}}_{j}}V^{\mathrm{(mono)}}_{j,\alpha_{j}}\Big{(}\big{(}u_{j}(x+R_{j}^{\prime},\alpha_{j}^{\prime})\,-\,u_{j}(x,\alpha_{j})\big{)}_{\begin{subarray}{c}R_{j}^{\prime}\in{\mathcal{R}}_{j},\\\
\alpha_{j}^{\prime}\in{\mathcal{A}}_{j}\end{subarray}}\Big{)}\end{split}$
(2.48)

is ${\mathcal{R}}_{\mathcal{M}}$-periodic. If $R_{j}\in{\mathcal{R}}_{j}$,
$\gamma_{j}\in\Gamma_{j}$,
$\Phi^{\mathrm{(mono)}}_{j}[u_{j}](R_{j}+\gamma_{j})$ denotes the sublattice-
averaged site-potential experienced at $R_{j}+\gamma_{j}$. In particular,
(2.37) becomes

$\displaystyle\begin{split}e^{\mathrm{(mono)}}_{j,N,\gamma_{j}}(u_{j})\,=\,\frac{1}{(2N+1)^{2}}\sum_{R_{j}\in{\mathcal{R}}_{j}(N)}\Phi^{\mathrm{(mono)}}_{j}[u_{j}](\left\\{R_{j}+\gamma_{j}\right\\}_{\mathcal{M}})\,.\end{split}$
(2.49)

Then, the ergodic theorem Proposition 2.7 implies, after substituting
$x\mapsto A_{j}x$, that the thermodynamic limit of the monolayer energy is
given by

$\displaystyle\begin{split}&e^{\mathrm{(mono)}}_{j}(u_{j})\,:=\\\
&\frac{1}{|{\mathcal{A}}_{j}||\Gamma_{j}|}\sum_{\alpha_{j}\in{\mathcal{A}}_{j}}\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{A_{j}^{-1}\Gamma_{\mathcal{M}}}\mathrm{d}x\
V^{\mathrm{(mono)}}_{j,\alpha_{j}}\Big{(}\big{(}u_{j}(A_{j}x+R_{j}^{\prime},\alpha_{j}^{\prime})\,-\,u_{j}(A_{j}x,\alpha_{j})\big{)}_{\begin{subarray}{c}R_{j}^{\prime}\in{\mathcal{R}}_{j},\\\
\alpha_{j}^{\prime}\in{\mathcal{A}}_{j}\end{subarray}}\Big{)}\,.\end{split}$
(2.50)

This result is given in Theorem 4.1.

####  2.2.2 Interlayer coupling

Similarly, we define the sublattice-averaged generalized interlayer site
potential

$\displaystyle\begin{split}&\Psi^{\mathrm{(inter)}}_{j}[{\bf u}](x,y)\,:=\\\
&\frac{1}{|{\mathcal{A}}_{j}||\Gamma_{j}|}\sum_{\alpha_{j}\in{\mathcal{A}}_{j}}V^{\mathrm{(inter)}}_{j,\alpha_{j}}\Big{(}\big{(}y-R_{3-j}+\tau_{j}^{(\alpha_{j})}-\tau_{3-j}^{(\alpha_{3-j})}+u_{j}(x,\alpha_{j})\\\
&-\,u_{3-j}(x-y+R_{3-j},\alpha_{3-j})\big{)}_{\begin{subarray}{c}R_{3-j}\in{\mathcal{R}}_{3-j},\\\
\alpha_{3-j}\in{\mathcal{A}}_{3-j}\end{subarray}}\Big{)}\end{split}$ (2.51)

for all $x,y\in{\mathbb{R}}^{2}$, $\gamma_{3-j}\in\Gamma_{3-j}$. Observe that,
due to the ${\mathcal{R}}_{\mathcal{M}}$-periodic extension of $u_{j}$ and due
to the dependence on the collection of all lattice shifts w.r.t.
${\mathcal{R}}_{3-j}$, $\Psi^{\mathrm{(inter)}}_{j}[u]$ is
${\mathcal{R}}_{\mathcal{M}}\times{\mathcal{R}}_{3-j}$-periodic. Using Lemma
3.4 we show that

$\displaystyle\begin{split}&e^{\mathrm{(inter)}}_{j,N,{\bm{\gamma}}}({\bf
u})\,=\\\
&\frac{1}{(2N+1)^{2}}\sum_{R_{j}\in{\mathcal{R}}_{j}(N)}\Psi^{\mathrm{(inter)}}_{j}[{\bf
u}](\left\\{R_{j}+\gamma_{j}\right\\}_{\mathcal{M}},\left\\{R_{j}+\gamma_{j}-\gamma_{3-j}\right\\}_{3-j})\,.\end{split}$
(2.52)

Here, we prove a mean ergodic theorem, Proposition 3.10: Let $h\in
L^{1}_{\emph{loc}}({\mathbb{R}}^{4})$ be
${\mathcal{R}}_{\mathcal{M}}\times{\mathcal{R}}_{3-j}$-periodic. Then we show
that

$\displaystyle\begin{split}&\frac{1}{(2N+1)^{2}}\sum_{R_{j}\in{\mathcal{R}}_{j}(N)}h(\left\\{R_{j}+\omega_{\mathcal{M}}\right\\}_{\mathcal{M}},\left\\{R_{j}+\omega_{3-j}\right\\}_{3-j})\\\
&\xrightarrow[]{N\to\infty}\,\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{A_{j}^{-1}\Gamma_{\mathcal{M}}}\mathrm{d}x\
h\big{(}A_{j}x+\omega_{\mathcal{M}},(A_{j}-A_{3-j})x+\omega_{3-j}\big{)}\end{split}$
(2.53)

converges for almost all
$(\omega_{\mathcal{M}},\omega_{3-j})\in\Gamma_{\mathcal{M}}\times\Gamma_{3-j}$
and also in $L^{1}(\Gamma_{\mathcal{M}}\times\Gamma_{3-j})$. Both, this result
and also the previously mentioned ergodic theorem Prop. 2.7, are special
instances of a more general ergodic theorem for group actions, as can be
found, e.g., in [34, 51, 53], see also Proposition 3.8.

We then obtain in Theorems 4.2 and 4.3 that, in the thermodynamic limit
$N\to\infty$, the interlayer coupling potential converges to

$\displaystyle\begin{split}e^{\mathrm{(inter)}}_{j,{\bm{\gamma}}}({\bf
u})\,:=&\,\frac{1}{|{\mathcal{A}}_{j}||\Gamma_{j}|}\sum_{\alpha_{j}\in{\mathcal{A}}_{j}}\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{A_{j}^{-1}\Gamma_{\mathcal{M}}}\mathrm{d}x\
V^{\mathrm{(inter)}}_{j,\alpha_{j}}\Big{(}\big{(}Y_{j}(A_{j}x+\gamma_{j},\alpha_{j})\\\
&\quad-
Y_{3-j}(A_{3-j}x+\gamma_{3-j}+R_{3-j},\alpha_{3-j})\big{)}_{\begin{subarray}{c}R_{3-j}\in{\mathcal{R}}_{3-j},\\\
\alpha_{3-j}\in{\mathcal{A}}_{3-j}\end{subarray}}\Big{)}\,.\end{split}$ (2.54)

Lemma 3.6 yields in the special case of even pair potentials
$v_{{\bm{\alpha}}}$ that

$\displaystyle\begin{split}e^{\mathrm{(inter)}}_{j,{\bm{\gamma}}}({\bf
u})\,&=\,\frac{1}{2|{\mathcal{A}}_{j}||\Gamma_{\mathcal{M}}|}\sum_{\begin{subarray}{c}\alpha_{k}\in{\mathcal{A}}_{k}\\\
k=1,2\end{subarray}}\int_{{\mathbb{R}}^{2}}\mathrm{d}x\
v_{{\bm{\alpha}}}\big{(}A_{1}x+\gamma_{1}+\tau_{1}^{(\alpha_{1})}+u_{1}(A_{1}x+\gamma_{1},\alpha_{1})\\\
&\quad-
A_{2}x-\gamma_{2}-\tau_{2}^{(\alpha_{2})}-u_{2}(A_{2}x+\gamma_{2},\alpha_{2})\big{)}\,.\end{split}$
(2.55)

####  2.2.3 Convergence rate

In the special case that i.e. $\gamma_{1}=\gamma_{2}=0$, we show in Lemma 3.2
that one can simplify (2.52) to

$\displaystyle\begin{split}e^{\mathrm{(inter)}}_{j,N,{\bm{\gamma}}}({\bf
u})\,=\,\frac{1}{(2N+1)^{2}}\sum_{R_{j}\in{\mathcal{R}}_{j}(N)}\Psi^{\mathrm{(inter)}}_{j}[{\bf
u}](\left\\{R_{j}\right\\}_{\mathcal{M}},D_{j\to
3-j}\left\\{R_{j}\right\\}_{\mathcal{M}})\,,\end{split}$ (2.56)

see (2.45), (2.46), and also (2.47). This allows us to employ a quantitative
version of ergodic theorem of the form Proposition 2.7, see Proposition 3.12.
The convergence results, for the monolayer potential and the interlayer
coupling, are given in Theorems 4.4 and 4.5, 4.6, respectively.

Our methods would allow us to also derive a convergence rate of $1/N$ in the
case ${\bm{\gamma}}\neq(0,0)$, see Sections 3.4.3 and 3.4.4 for more details.
However, to avoid redundancy, we chose not to present such a result, and leave
the details to the interested reader.

####  2.2.4 Regularity assumptions

We now want to comment on the imposed regularity on $u_{j}$ and
$v_{{\bm{\alpha}}}$. Our results require smoothness of $u_{j}$, i.e.,
$u_{j}\in W^{6,\infty}({\mathbb{R}}^{2})$, in order to show convergence with a
rate. For an almost sure ergodic theorem to hold, see Proposition 3.10, we
only require $u_{j}\in L^{\infty}({\mathbb{R}}^{2})$.

We do not expect general displacements $u_{j}$ to be smooth. As explained
above, if we consider a model consisting of a monolayer contribution and an
interlayer coupling as constructed above, this has been analyzed for the
Frenkel-Kontorova model [4] and a simplified coupled linear chain [28] in the
one dimensional case. Their result is that the energy minimizer is a
continuous function of the disregistry whenever the monolayer energy dominates
the interlayer coupling. However, we do not address the question of smoothness
of minimizers in our work. In the Cauchy-Born approximation, see Section 2.3
below, we believe that standard ideas in the theory of semi-linear elliptic
equations, as applied in [27], can be utilized to show that, in case of
$C^{\infty}$-smooth and polynomially decaying pair potentials
$v_{{\bm{\alpha}}}$, critical points ${\bf u}=(u_{1},u_{2})$ inherit
$C^{\infty}$ smoothness.

In order to obtain a rate of convergence, we require that the weighted Sobolev
norm of $v_{{\bm{\alpha}}}$

$\displaystyle\sum_{j=0}^{6}\|\langle\cdot\rangle^{2r}D^{j}v_{{\bm{\alpha}}}\|_{L^{\infty}({\mathbb{R}}^{2}\times[-L_{{\bf
u}}^{z},L_{{\bf u}}^{z}])}\,<\,\infty$ (2.57)

for some $r>1$, is finite, where
$\langle\cdot\rangle=(1+|x|^{2})^{\frac{1}{2}}$ and

$L_{{\bf u}}^{z}\,\sim\,1+\|u_{1,z}\|_{\infty}+\|u_{2,z}\|_{\infty}\,,$ (2.58)

where the involved constant depends on the lattice, see (3.24) below. In
particular, $v_{{\bm{\alpha}}}$ obeys the tail-behavior

$|v_{{\bm{\alpha}}}({\bf x},z)|\,\lesssim\,\frac{1}{|{\bf
x}|^{2+}}\quad\mbox{as\ }|{\bf x}|\to\infty$ (2.59)

for ${\bf x}\in{\mathbb{R}}^{2}$, $z\in[-L_{{\bf u}}^{z},L_{{\bf u}}^{z}]$. As
a special case, this includes the Yukawa potential $\frac{e^{-\kappa|{\bf
x}|}}{|{\bf x}|}$, which accounts for screening effects. Observe that we need
to regularize the potential near the origin.

Observe that in the perpendicular $z$-direction, we have no tail constraints,
but again we need to regularize the potential near the origin. In particular,
we are free to choose the potential to behave like a Lennard-Jones type
potential in $z$-direction, away from the origin, which accounts for the van
der Waals potential between the layers.

In the case that no rate of convergence is obtained, we impose

$\|\langle\cdot\rangle^{2r}v_{{\bm{\alpha}}}\|_{L^{\infty}({\mathbb{R}}^{2}\times[-L_{{\bf
u}}^{z},L_{{\bf u}}^{z}])}\,<\,\infty$ (2.60)

for some $r>1$, amounting to the same decay rate (2.59).

###  2.3 Cauchy-Born Approximation

We want to further approximate the monolayer contribution
$e^{\mathrm{(mono)}}$ by taking a continuum limit. This leads to the _Cauchy-
Born approximation_ , which is a popular approximation in elasticity theory.
For that, we introduce a length-scale $a_{0}\ll 1$ and rescale

$A\,=:\,a_{0}A_{0}\,,$ (2.61)

where $A_{0}$ is independent of $a_{0}$. Consequently,
${\mathcal{R}}_{j}^{(0)}:={\mathcal{R}}_{j}/a_{0}$ is $a_{0}$-independent.

For simplicity, we assume only a single sublattice degree of freedom,
$|{\mathcal{A}}_{j}|=1$. In addition, we restrict $u_{j}$ to take values in
the $xy$-plane.

Recalling (2.48), and assuming that ${\bf u}$ is sufficiently regular, say
${\bf u}\in C^{2}(\Gamma_{\mathcal{M}};{\mathbb{R}}^{4})$, we then obtain for
$a_{0}\ll 1$ that

$\displaystyle\begin{split}\Phi^{\mathrm{(mono)}}_{j}[u_{j}](x)\,\approx&\,\frac{1}{|\Gamma_{j}|}V^{\mathrm{(mono)}}_{j}\Big{(}a_{0}Du_{j}(x)\cdot{\mathcal{R}}_{j}^{(0)}\Big{)}\\\
=&\,W_{j}\big{(}Du_{j}(x)\big{)}\,,\end{split}$ (2.62)

where $W_{j}$ denotes the _Cauchy-Born energy density function_. In the linear
elasticity approximation for $|Du_{j}|$ small enough, and assuming isotropy,
one has

$\displaystyle W_{j}(M)\,\approx$
$\displaystyle\,\frac{1}{2}M:{\mathcal{E}}_{j}:M\,,\quad\mbox{for\ all\
}M\in{\mathbb{R}}^{2\times 2}\,,$ (2.63)

with the elasticity tensor ${\mathcal{E}}_{j}$ given in (1.4). In particular,
the monolayer energy density (2.50) satisfies for $a_{0}\ll 1$

$\displaystyle
e^{\mathrm{(mono)}}_{j}(u_{j})\,\approx\,\frac{1}{2}\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
Du_{j}(x):{\mathcal{E}}_{j}:Du_{j}(x)\,.$ (2.64)

######  Example 2.11.

Let us, for instance, consider twisted bilayer graphene.

Let $\theta\in{\mathbb{R}}$ be and let

$a_{0}\,:=\,\sqrt{3}\cdot 1.42\mbox{\ nm}\,\approx\,2.46\mbox{\ nm}\,.$ (2.65)

Define

$\displaystyle
A\,:=\,a_{0}\begin{pmatrix}\frac{\sqrt{3}}{2}&\frac{\sqrt{3}}{2}\\\
-\frac{1}{2}&\frac{1}{2}\end{pmatrix}$ (2.66)

and recall from (2.1) that

$\displaystyle A_{1}\,=\,R_{-\theta/2}A\,,\quad A_{2}\,=\,R_{\theta/2}A\,.$
(2.67)

Graphene has two sublattice degrees of freedom, denoted by $A$ and $B$, with
associated shift vectors

$\displaystyle\tau^{(A)}_{j}\,=\,-\frac{a_{0}}{2}R_{(-1)^{j}\theta/2}\hat{e}_{1}\,,\quad\tau^{(B)}_{j}\,=\,\frac{a_{0}}{2}R_{(-1)^{j}\theta/2}\hat{e}_{1}\,.$
(2.68)

For simplicity, we can ignore the sublattice degree of freedom in order to
compute the lattice displacements at the center of mass. Then we can set the
relaxed lattice positions to

$\displaystyle\tilde{Y}_{j}(R_{j}+\gamma_{j},\alpha)\,:=\,R_{j}+\gamma_{j}+\tau_{j}^{(\alpha)}+u_{j}(R_{j}+\gamma_{j})$
(2.69)

for $j=1,2$, $\alpha=A,B$.

The Lamé parameters for a single sheet of graphene are given by

$\lambda\,=\,37,950\mbox{\ meV/unit area}\,,\quad\mu\,=\,47,352\mbox{\
meV/unit area}\,,$ (2.70)

see, e.g., [24]. In order to allow for out-of-plane relaxation, we extend the
monolayer energy model to

$\displaystyle\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
\frac{1}{2}\Big{(}\lambda\sum_{k=1}^{2}{\varepsilon}_{k,k;j}(x)^{2}+2\mu\sum_{k,\ell=1}^{2}{\varepsilon}_{k,\ell;j}(x)^{2}+\kappa|D^{2}u_{j,3}(x)|^{2}\Big{)}\,,$
(2.71)

where
${\varepsilon}_{k,\ell;j}:=\frac{1}{2}(\partial_{k}u_{j,\ell}+\partial_{\ell}u_{j,k}+\partial_{k}u_{j,3}\partial_{\ell}u_{j,3})$
is the strain tensor, $\lambda$, $\mu$ are the Lamé parameters as before, and
$\kappa$ is the bending rigidity, see, e.g., [71]. As an interatomic pair
potential, we choose a Morse potential for the monolayer pair potentials, and
a Lennard-Jones potential accounting for the van der Waals coupling between
the layers, i.e.,

$v_{{\bm{\alpha}}}({\bf x},z)\,\equiv\,v({\bf x},z)\,:=\,v_{xy}(|{\bf
x}|)v_{z}(z)$ (2.72)

with

$\displaystyle v_{xy}(r)\,:=$
$\displaystyle\,E_{0}\Big{(}e^{-2\kappa_{0}(r-r_{0})}\,-\,e^{-\kappa_{0}(r-r_{0})}\Big{)}\,,$
(2.73) $\displaystyle v_{z}(z)\,:=$
$\displaystyle\,A_{0}|z|^{-12}-B_{0}|z|^{-6}\,.$ (2.74)

Here $E_{0}$ denotes the dissociation energy, $\kappa_{0}$ relates to the
stiffness, $r_{0}$ refers to the equilibrium position, see [74],
$A_{0},B_{0}>0$ are van der Waals parameters. Strictly speaking, we need to
regularize $v_{z}$ to remove the singularity at the origin for our methods to
apply.

More specialized interlayers potentials have been developed for graphene [54]
and hBN [63], for example.

###  2.4 Relationship with Allen-Cahn energy functional

As in the previous section, let us restrict to a single sublattice degree of
freedom, $|{\mathcal{A}}_{j}|=1$, and assume that $\tau_{j}^{(\alpha_{j})}=0$.
Assume that $v$ an interlayer pair potential that is radially decaying in
$xy$-direction, and set $\gamma_{j}=0$. In addition, we restrict $u_{j}$ to
take values in the $xy$-plane.

We claim that, in the Cauchy-Born approximation, $e_{\mathrm{tot}}(u_{1},0),$
resp., $e_{\mathrm{tot}}(0,u_{2}),$ are modified Allen-Cahn functionals. For
that, define a _stacking fault potentials_

$\displaystyle\Phi^{\mathrm{(misfit)}}_{j}(x_{3-j})\,:=\,$
$\displaystyle\frac{1}{|\Gamma_{j}|}\sum_{R_{3-j}\in{\mathcal{R}}_{3-j}}v\big{(}x_{3-j}-R_{3-j}\big{)}$
(2.75)

for all $x_{3-j}\in\Gamma_{3-j}$, $j=1,2$. We note that the generalized
stacking fault energy (GSFE) more commonly used in numerical computations of
relaxation [27, 24, 26] approximates the misfit energy by a DFT computation on
a periodic cell at zero twist angle. By construction,
$\Phi^{\mathrm{(misfit)}}_{j}$ is periodic w.r.t. ${\mathcal{R}}_{3-j}$,
$j=1,2$. $\Phi^{\mathrm{(misfit)}}_{j}(x_{3-j})$ is maximal on the lattice
points $R_{3-j}\in{\mathcal{R}}_{3-j}$, which we will refer to as _AA
stacking_ , and minimal at

$R_{3-j}+\frac{1}{3}A_{3-j}\begin{pmatrix}1\\\ 1\end{pmatrix}\,,$ (2.76)

which we will refer to as _AB stacking_ , and at

$R_{3-j}+\frac{2}{3}A_{3-j}\begin{pmatrix}1\\\ 1\end{pmatrix}\,,$ (2.77)

which we will refer to as _BA stacking_. In the AA stacking, nuclei from
different layers are stacked on top of each other, while in the AB/BA
stacking, they sit in between three nuclei.

Recalling (2.51) and (1.1), observe that we have the identities

$\displaystyle\Psi^{\mathrm{(inter)}}_{1,0}[u_{1},0](x,D_{1\to
2}x)\,=\,\Phi^{\mathrm{(misfit)}}_{1}(D_{1\to 2}x+u_{1}(x))\,,$ (2.78)
$\displaystyle\Psi^{\mathrm{(inter)}}_{2,0}[0,u_{2}](x,D_{2\to
1}x)\,=\,\Phi^{\mathrm{(misfit)}}_{2}(D_{2\to 1}x+u_{2}(x))\,.$ (2.79)

Moreover, Lemma 3.6 implies that

$\displaystyle\begin{split}\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
\Phi^{\mathrm{(inter)}}_{1,0}[{\bf
u}](x)\,=\,\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
\Phi^{\mathrm{(inter)}}_{2,0}[{\bf u}](x)\,.\end{split}$ (2.80)

In the Cauchy-Born approximation $a_{0}\ll 1$, see (2.64), the total energy
functional becomes

$\displaystyle\begin{split}e_{\mathrm{tot}}({\bf
u})\,\approx\,\frac{1}{2}\sum_{j=1}^{2}\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
\big{(}Du_{j}(x):{\mathcal{E}}_{j}:Du_{j}(x)\,+\,\Phi^{\mathrm{(inter)}}_{j,0}[{\bf
u}](x)\big{)}\,.\end{split}$ (2.81)

In particular, employing (2.78), we have for $a_{0}\ll 1$ that

$\displaystyle\begin{split}e_{\mathrm{tot}}(u_{1},0)\approx\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
\bigg{(}\frac{1}{2}Du_{1}(x):{\mathcal{E}}_{1}:Du_{1}(x)\,+\,\Phi^{\mathrm{(misfit)}}_{1}\big{(}D_{1\to
2}x+u_{1}(x)\big{)}\bigg{)}\,,\end{split}$ (2.82)

resembling the Allen-Cahn energy.

###  2.5 Outlook

In the Cauchy-Born approximation, we can employ ideas from elliptic theory to
establish existence, uniqueness, and regularity of energy minimizers. This
remains true as long as the interlayer potential is a compact perturbation to
the intralayer Cauchy-Born energy. In particular, one can use a contraction
map argument as in [27]. We do not elaborate on further details here, and
defer to future works instead.

While we develop a model for bilayer systems only, we can extend the model to
systems with an arbitrary finite number of layers, by summing over the pair-
wise coupling of layers. For the case of many-body site potentials, we believe
that the ideas presented in [27] lend themselves to extending our results to
the multi-layer case.

###  2.6 Sketch of the proof

We will next introduce in Section 3 the notation necessary to formulate our
results. The exact results are given in Section 4, by Theorems 4.1 and 4.4 for
the monolayer energy, Theorems 4.2 and 4.5 for interlayer many-body potential,
and Theorems 4.3 and 4.6 for two-body interlayer potentials.

The technical tools used for the thermodynamic limit stem from ergodic theory.
We prove an almost sure and mean ergodic theorem, see Proposition 3.10, and a
quantitative ergodic theorem, see Proposition 3.12. The latter is inspired by
[29] which shows uniform convergence w.r.t. shifts of the lattice origin, for
the one-dimensional case. In contrast to [29], our approach is based on
studying the convergence properties of the approximate Dirichlet kernel
associated with the ergodic average over the truncated lattice. We directly
compute an $L^{1}$-limit for smooth functions, which, by density, yields a
general $L^{1}$-limit. In order to obtain convergence almost everywhere, we
invoke a general ergodic theorem 3.12 as stated, e.g., in [53]. For a more
detailed description of our approach, see section 3.4.

As mentioned above, in order to derive a convergence rate, our approach is
based on analyzing the pointwise decay behavior of the approximate Dirichlet
kernel. In order to derive a pointwise limit, as opposed to an Lebesgue-almost
sure limit, we study Diophantine 2D rotation, see Definition 2.4. Using
standard ideas in the metric theory of Diophantine approximations, we show in
Appendix B that Lebesgue-almost every $\theta\in{\mathbb{R}}$ defines a
Diophantine 2D rotation, see Proposition 2.5. With this notion, we are able to
formulate and prove a quantitative ergodic theorem, Proposition 3.12, with
explicit error bounds, with a proof given in Appendix C.

In order to employ the ergodic theorems, we state in Section 3.3 that the site
potential is a (regular) function of the disregistry, see Lemma 3.4, and in
Lemmata 3.6 and 3.7, that, in case of pair potentials, it inherits
integrability from $v_{{\bm{\alpha}}}$ and $u_{j}$. We prove these results in
Section 5.

The ergodic theorems, Propositions 3.10, 3.12, together with Lemmata 3.4, 3.6
and 3.7, enable us to prove in Section 4 the main theorems.

###  2.7 Comparison with previous result

We now give a more detailed comparison of the models (1.3) and (1.8) in the
simplest possible setting. We restrict to a single sublattice degree of
freedom in each layer, assume that the layers are centered at the origin,
$\gamma_{j}=0$, and not mismatched, $q=1$. The stacking fault energy
$\Phi^{\mathrm{(misfit)}}_{j}$, see (2.75), obtained in this work provides a
microscopic model for the GSFE $\Phi^{\mathrm{(GSFE)}}_{j}$ studied in [27,
24, 26]. Setting $\Phi^{\mathrm{(GSFE)}}_{j}=\Phi^{\mathrm{(misfit)}}_{j}$, we
obtain

$\displaystyle\frac{1}{|\Gamma_{\mathcal{M}}|}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
\Phi^{\mathrm{(GSFE)}}_{j}\big{(}D_{j\to 3-j}x+u_{j}(x)-u_{3-j}(x)\big{)}$
(2.83)
$\displaystyle=\,\frac{1}{|\Gamma_{\mathcal{M}}||\Gamma_{j}|}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
\sum_{R_{3-j}\in{\mathcal{R}}_{3-j}}v\big{(}D_{j\to
3-j}x-R_{3-j}+u_{j}(x)-u_{3-j}(x)\big{)}\,.$ (2.84)

Employing (2.45), (2.46), we may define $R_{\mathcal{M}}:=D_{j\to
3-j}^{-1}R_{3-j}\in{\mathcal{R}}_{\mathcal{M}}$. In particular, we obtain

$\displaystyle\frac{1}{|\Gamma_{\mathcal{M}}|}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
\Phi^{\mathrm{(GSFE)}}_{j}\big{(}D_{j\to 3-j}x+u_{j}(x)-u_{3-j}(x)\big{)}$
(2.85)
$\displaystyle=\,\frac{1}{|\Gamma_{\mathcal{M}}||\Gamma_{j}|}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
\sum_{R_{\mathcal{M}}\in{\mathcal{R}}_{\mathcal{M}}}v\big{(}D_{j\to
3-j}(x-R_{\mathcal{M}})+u_{j}(x)-u_{3-j}(x)\big{)}\,.$ (2.86)

Putting technicalities aside, we interchange orders of integration and
summation and substitute $x-R_{\mathcal{M}}\to x$ and use moiré-periodicity of
$u_{j}$ to obtain

$\displaystyle\begin{split}&\frac{1}{|\Gamma_{\mathcal{M}}|}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
\Phi^{\mathrm{(GSFE)}}_{j}\big{(}D_{j\to 3-j}x+u_{j}(x)-u_{3-j}(x)\big{)}\\\
&=\,\frac{1}{|\Gamma_{\mathcal{M}}||\Gamma_{j}|}\sum_{R_{\mathcal{M}}\in{\mathcal{R}}_{\mathcal{M}}}\int_{\Gamma_{\mathcal{M}}+R_{\mathcal{M}}}\mathrm{d}x\
v\big{(}D_{j\to 3-j}x+u_{j}(x)-u_{3-j}(x)\big{)}\\\
&=\,\frac{1}{|\Gamma_{\mathcal{M}}||\Gamma_{j}|}\int_{{\mathbb{R}}^{2}}\mathrm{d}x\
v\big{(}D_{j\to 3-j}x+u_{j}(x)-u_{3-j}(x)\big{)}\,.\end{split}$ (2.87)

Substituting $A_{j}^{-1}x\to x$ and using evenness of $v$, we arrive at

$\displaystyle\begin{split}&\frac{1}{|\Gamma_{\mathcal{M}}|}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
\Phi^{\mathrm{(GSFE)}}_{j}\big{(}D_{j\to 3-j}x+u_{j}(x)-u_{3-j}(x)\big{)}\\\
&=\,\frac{1}{|\Gamma_{\mathcal{M}}|}\int_{{\mathbb{R}}^{2}}\mathrm{d}x\
v\big{(}A_{j}x+u_{j}(A_{j}x)-A_{3-j}x-u_{3-j}(A_{j}x)\big{)}\,.\end{split}$
(2.88)

In contrast, Lemma 3.6 yields that the interlayer energy in the present work
is given by

$\displaystyle\frac{1}{|\Gamma_{\mathcal{M}}|}\int_{{\mathbb{R}}^{2}}\mathrm{d}x\
v\big{(}A_{j}x+u_{j}(A_{j}x)-A_{3-j}x-u_{3-j}(A_{3-j}x)\big{)}\,.$ (2.89)

Due to the presence of $(A_{j}-A_{3-j})x\approx\theta Ax$ as $\theta\to 0$ in
the argument of $v$, we obtain that the models differ in the different
evaluations of $u_{3-j}$.

##  3 Preliminaries

###  3.1 Equivalence of disregistries

Recall the disregistry matrices

$\displaystyle D_{j\to 3-j}\,$ $\displaystyle=\,I\,-\,A_{3-j}A_{j}^{-1}\,,$
(3.1)

see (2.45), (2.46), and (2.1) above. In particular, we have that

$\displaystyle\begin{split}-D_{3-j\to j}D_{j\to 3-j}^{-1}\,&=\,-D_{j\to
3-j}^{-1}D_{3-j\to j}\,=\,R_{(-1)^{j}\theta}\\\ &=\,I-D_{3-j\to
j}\,=\,A_{3-j}A_{j}^{-1}\,.\end{split}$ (3.2)

######  Lemma 3.1.

The disregistry matrices have the following properties.

  1. (1)

$D_{j\to 3-j}\,=\,(-1)^{j}A_{3-j}A_{\mathcal{M}}^{-1}$,

  2. (2)

$-D_{3-j\to j}D_{j\to 3-j}^{-1}=-D_{j\to 3-j}^{-1}D_{3-j\to
j}=A_{j}A_{3-j}^{-1}=I-D_{3-j\to j}$,

  3. (3)

$|D_{j\to 3-j}^{-1}x|=q^{(-1)^{j+1}1/2}\rho_{\mathcal{M}}^{-1}|x|$,

  4. (4)

$|(D_{j\to 3-j}^{-1}-I)x|=q^{(-1)^{j}1/2}\rho_{\mathcal{M}}^{-1}|x|$.

###### Proof.

We show the computations for $j=1$; $j=2$ can be computed analogously. We have
that

$D_{1\to
2}\,=\,I-A_{2}A_{1}^{-1}\,=\,-A_{2}(A_{1}^{-1}-A_{2}^{-1})\,=\,-A_{2}A_{\mathcal{M}}^{-1}\,,$
(3.3)

proving (1). For (2), we employ (1) and obtain

$\displaystyle-D_{2\to 1}D_{1\to
2}^{-1}\,=\,A_{1}A_{\mathcal{M}}^{-1}A_{\mathcal{M}}A_{2}^{-1}\,=\,A_{1}A_{2}^{-1}\,=\,I-D_{2\to
1}\,,$ (3.4)

where, in the last step, we employ (2.46). Similarly, using (1) again, we have
that

$\displaystyle\begin{split}-D_{1\to 2}^{-1}D_{2\to
1}\,&=\,A_{\mathcal{M}}A_{2}^{-1}A_{1}A_{\mathcal{M}}^{-1}\,=\,A_{\mathcal{M}}(A_{2}^{-1}A_{1}-I)A_{\mathcal{M}}^{-1}\,+\,I\\\
&=\,-A_{\mathcal{M}}A_{\mathcal{M}}^{-1}A_{1}A_{\mathcal{M}}^{-1}\,+\,I\,=\,A_{1}A_{2}^{-1}\,.\end{split}$
(3.5)

Observe that, due to (2.1), $|A_{2}x|=q^{\frac{1}{2}}|Ax|$. By (1), we thus
have that

$\displaystyle|D_{1\to
2}^{-1}x|\,=\,|A_{2}A_{\mathcal{M}}^{-1}x|\,=\,q^{1/2}|AA_{\mathcal{M}}^{-1}x|\,=\,q^{\frac{1}{2}}\rho_{\mathcal{M}}^{-1}|x|\,,$
(3.6)

where, in the last step, we employed Lemma 2.1. By (2), we have that

$D_{1\to 2}^{-1}-I\,=\,D_{1\to 2}^{-1}(I-D_{1\to 2})\,=\,-D_{2\to 1}^{-1}\,,$
(3.7)

which is why (4) follows from (3). This concludes the proof. ∎

Lemma 3.1 establishes a relation between the different disregistry notions.
Next, we will specify this relation. Define

$v_{\Gamma_{j}}\,:=\,A_{j}\begin{pmatrix}1\\\ 1\end{pmatrix}\,.$ (3.8)

######  Lemma 3.2.

Let $R_{j}\in{\mathcal{R}}_{j}$, $j\in\\{1,2\\}$. Then we have that

$\displaystyle\left\\{R_{1}\right\\}_{2}\,$ $\displaystyle=\,D_{1\to
2}\left\\{R_{1}\right\\}_{\mathcal{M}}\,+\,v_{\Gamma_{2}}\,,$ (3.9)
$\displaystyle\left\\{R_{2}\right\\}_{1}\,$ $\displaystyle=\,D_{2\to
1}\left\\{R_{2}\right\\}_{\mathcal{M}}\,.$ (3.10)

###### Proof.

Observe that $A_{2}A_{k}^{-1}R_{k}\in{\mathcal{R}}_{2}$ for
$R_{k}\in{\mathcal{R}}_{k}$, $k\in\\{1,{\mathcal{M}}\\}$. Hence we find that

$\displaystyle\begin{split}\left\\{R_{1}\right\\}_{2}\,&=\,\left\\{D_{1\to
2}R_{1}\right\\}_{2}\,=\,\left\\{-A_{2}A_{\mathcal{M}}^{-1}R_{1}\right\\}_{2}\\\
&=\,\left\\{-A_{2}A_{\mathcal{M}}^{-1}\left\\{R_{1}\right\\}_{\mathcal{M}}\right\\}_{2}\,=\,D_{1\to
2}\left\\{R_{1}\right\\}_{\mathcal{M}}+v_{\Gamma_{2}}\in\Gamma_{2}\,,\end{split}$
(3.11)

where, in the last step, we used the fact that
$v_{\Gamma_{2}}\in{\mathcal{R}}_{2}$. The second identity follows analogously.
This concludes the proof. ∎

######  Remark 3.3.

Lemma 3.2 is sensitive to the lattice origin. More precisely, there is no
affine map between $\left\\{x\right\\}_{3-j}$ and
$\left\\{x\right\\}_{\mathcal{M}}$ for general $x\in{\mathbb{R}}^{2}$. In
fact, assume that there is a matrix $M$ such that for all
$x\in{\mathbb{R}}^{2}$,

$x\,=\,M(x+{\mathcal{R}}_{\mathcal{M}})\,+\,{\mathcal{R}}_{3-j}\,.$ (3.12)

In particular, this requires
$M{\mathcal{R}}_{\mathcal{M}}\subseteq{\mathcal{R}}_{3-j}$, which implies

$M\in A_{\mathcal{M}}^{-1}A_{3-j}{\mathbb{Z}}^{2\times 2}\,.$ (3.13)

However, e.g., $(I-M)\sqrt{2}v_{\Gamma_{3-j}}\not\in{\mathcal{R}}_{3-j}$.
Thus, no such $M$ exists.

###  3.2 Notation

In addition to the Fourier transform defined in (2.30), for $h\in
L^{1}(\Gamma_{\mathcal{M}}\times\Gamma_{j})$ and
$(G_{\mathcal{M}},G_{j})\in({\mathcal{R}}^{*}_{\mathcal{M}},{\mathcal{R}}_{j}^{*})$,
define

$\displaystyle\begin{split}&\hat{h}(G_{\mathcal{M}},G_{j})\,:=\,{\mathcal{F}}_{\Gamma_{\mathcal{M}}\times\Gamma_{j}}(h)(G_{\mathcal{M}})\,:=\\\
&\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\Gamma_{j}}\mathrm{d}y\
\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
e^{-i(G_{\mathcal{M}}\cdot x+G_{j}\cdot y)}h(x,y)\,.\end{split}$ (3.14)

Here, we abuse notation and interpret $\hat{h}$ in the local context.

Let $u_{j}:\Gamma_{\mathcal{M}}\times{\mathcal{A}}_{j}\to{\mathbb{R}}^{3}$,
$j=1,2$, denote lattice displacement vectors, and let ${\bf u}=(u_{1},u_{2})$.
In addition, let
${\bm{\gamma}}=(\gamma_{,}\gamma_{2})\in\Gamma_{1}\times\Gamma_{2}$.

As usual, the absolute value function $|\cdot|$ will have a context-dependent
meaning. For finite sets $M$, $|M|$ denotes the cardinality of $M$, for
bounded measurable sets $\Omega\subseteq{\mathbb{R}}^{n}$, $|\Omega|$ denotes
the respective Lebesgue measure, and for $x\in{\mathbb{R}}^{n}$, $|x|$ denotes
the Euclidean norm.

Moreover, let

$\langle x\rangle\,:=\,(1+|x|^{2})^{\frac{1}{2}}$ (3.15)

for any $x\in{\mathbb{R}}^{n}$, $n\in{\mathbb{N}}$. For $k,n\in{\mathbb{N}}$
and $s>0$ and a Lipschitz domain $\Omega\subseteq{\mathbb{R}}^{n}$, let

$\|f\|_{W^{k,\infty}_{s}(\Omega)}\,:=\,\sum_{j=0}^{k}\|\langle\cdot\rangle^{s}D^{j}f\|_{L^{\infty}(\Omega)}$
(3.16)

denote the weighted Sobolev norm, and let

$W^{k,\infty}_{s}(\Omega)\,:=\,\\{f\in
W^{k,\infty}(\Omega)\,\mid\,\|f\|_{W^{k,\infty}_{s}(\Omega)}\,<\,\infty\\}$
(3.17)

denote the associated weighted Sobolev space. In addition, let

$L^{\infty}_{s}(\Omega)\,:=\,\\{f\in
L^{\infty}(\Omega)\,\mid\,\|\langle\cdot\rangle^{s}f\|_{L^{\infty}(\Omega)}\,<\,\infty\\}$
(3.18)

denote the weighted Lebesgue space.

###  3.3 Lattice reduction formulae

For proofs of the statements in this section, we refer to Section 5.

Recall definitions (2.51) of $\Psi^{\mathrm{(inter)}}_{j}[{\bf u}]$ and (4.10)
of $\Phi^{\mathrm{(inter)}}_{j,{\bm{\gamma}}}[{\bf u}]$.

######  Lemma 3.4.

For all $x\in{\mathbb{R}}^{2}$ and all $\gamma_{3-j}\in\Gamma_{3-j}$, we have
that

$\displaystyle\begin{split}&\frac{1}{|{\mathcal{A}}_{j}||\Gamma_{j}|}\sum_{\alpha_{j}\in{\mathcal{A}}_{j}}V^{\mathrm{(inter)}}_{j,\alpha_{j}}\bigg{(}\Big{(}Y_{j}(x,\alpha_{j})\,-\,Y_{3-j}(R_{3-j}+\gamma_{3-j},\alpha_{3-j})\Big{)}_{\begin{subarray}{c}R_{3-j}\in{\mathcal{R}}_{3-j},\\\
\alpha_{3-j}\in{\mathcal{A}}_{3-j}\end{subarray}}\bigg{)}\\\
&=\,\Psi^{\mathrm{(inter)}}_{j}[{\bf
u}](\left\\{x\right\\}_{\mathcal{M}},\left\\{x-\gamma_{3-j}\right\\}_{3-j})\,.\end{split}$
(3.19)

In the case $x=R_{j}\in{\mathcal{R}}_{j}$ and $\gamma_{3-j}=0$, we even have

$\displaystyle\begin{split}&\frac{1}{|{\mathcal{A}}_{j}||\Gamma_{j}|}\sum_{\alpha_{j}\in{\mathcal{A}}_{j}}V^{\mathrm{(inter)}}_{j,\alpha_{j}}\bigg{(}\Big{(}Y_{j}(R_{j},\alpha_{j})\,-\,Y_{3-j}(R_{3-j},\alpha_{3-j})\Big{)}_{\begin{subarray}{c}R_{3-j}\in{\mathcal{R}}_{3-j},\\\
\alpha_{3-j}\in{\mathcal{A}}_{3-j}\end{subarray}}\bigg{)}\\\
&=\,\Phi^{\mathrm{(inter)}}_{j,0}[{\bf
u}](\left\\{R_{j}\right\\}_{\mathcal{M}})\,.\end{split}$ (3.20)

###### Proof.

By translation-invariance of $V^{\mathrm{(inter)}}_{j,\alpha_{j}}$, we may
shift the index

$R_{3-j}\to R_{3-j}+\left\lfloor x-\gamma_{3-j}\right\rfloor_{3-j}.$ (3.21)

Employing ${\mathcal{R}}_{\mathcal{M}}$-periodicity of $u_{j}$ and definition
(2.21) of $\left\lfloor x\right\rfloor_{3-j}$, the first claim then follows
from definition (2.51).

In the case $x=R_{j}\in{\mathcal{R}}_{j}$ and $\gamma_{3-j}=0$, recall from
Lemma 3.2 that

$\left\\{R_{j}\right\\}_{3-j}\,=\,D_{j\to
3-j}\left\\{R_{j}\right\\}_{\mathcal{M}}\,+\,v_{\Gamma_{2}}\delta_{j,1}\,.$
(3.22)

Then the statement follows from (4.10) after shifting $R_{3-j}\to
R_{3-j}+v_{\Gamma_{2}}\delta_{j,1}$ and using translation-invariance. This
concludes the proof. ∎

Let

$d_{{\mathcal{A}}_{1},{\mathcal{A}}_{2}}\,:=\,\max_{\begin{subarray}{c}\alpha_{\ell}\in{\mathcal{A}}_{\ell}\\\
\ell=1,2\end{subarray}}|\tau_{1}^{(\alpha_{1})}-\tau_{2}^{(\alpha_{2})}|\,,$
(3.23)

and for ${\bf u}=(u_{1},u_{2})\in
L^{\infty}({\mathbb{R}}^{2};{\mathbb{R}}^{6})$, with components $u_{j,k}$,
$j=1,2$, $k\in\\{x,y,z\\}$,

$L_{{\bf
u}}^{z}\,:=\,d_{{\mathcal{A}}_{1},{\mathcal{A}}_{2}}\,+\,\|u_{1,z}\|_{\infty}+\|u_{2,z}\|_{\infty}\,.$
(3.24)

Define for all $y\in{\mathbb{R}}^{2}$

$\displaystyle\begin{split}g_{j,{\bm{\alpha}},\gamma_{3-j}}(y)\,:=&\,\tau_{j}^{(\alpha_{j})}-\tau_{3-j}^{(\alpha_{3-j})}\,+\,u_{j}(D_{j\to
3-j}^{-1}(y+\gamma_{3-j}),\alpha_{j})\\\ &-u_{3-j}(D_{j\to
3-j}^{-1}(y+\gamma_{3-j})-y,\alpha_{3-j})\,.\end{split}$ (3.25)

We now establish sufficient conditions on the pair potentials
$v_{{\bm{\alpha}}}$ for the ergodic theorems above to be applicable.

######  Lemma 3.5.

Let $r>1$. Assume that, for $\alpha_{j}\in{\mathcal{A}}_{j}$, $j=1,2$,
$u_{j}(\cdot,\alpha_{j})\in L^{\infty}(\Gamma_{\mathcal{M}})$, and that $f\in
L^{\infty}_{2r}({\mathbb{R}}^{2}\times[-L_{{\bf u}}^{z},L_{{\bf u}}^{z}])$.
Then we have that

$\displaystyle\begin{split}\int_{{\mathbb{R}}^{2}}\mathrm{d}x\
|f(x+g_{j,{\bm{\alpha}},\gamma_{3-j}}(x))|\,\leq&\,\frac{5^{r-1}\pi}{r-1}\,\big{(}1+d_{{\mathcal{A}}_{1},{\mathcal{A}}_{2}}\,+\,\|u_{1}\|_{\infty}+\|u_{2}\|_{\infty}\big{)}^{2r}\\\
&\,\|\langle\cdot\rangle^{2r}f\|_{L^{\infty}({\mathbb{R}}^{2}\times[-L_{{\bf
u}}^{z},L_{{\bf u}}^{z}])}\,.\end{split}$ (3.26)

######  Lemma 3.6.

Let $r>1$ and $\gamma_{3-j}\in\Gamma_{3-j}$. Assume that, for
$\alpha_{j}\in{\mathcal{A}}_{j}$, $j=1,2$, $u_{j}(\cdot,\alpha_{j})\in
L^{\infty}(\Gamma_{\mathcal{M}})$, and that

$\displaystyle v_{{\bm{\alpha}}}\in
L^{\infty}_{2r}({\mathbb{R}}^{2}\times[-L_{{\bf u}}^{z},L_{{\bf u}}^{z}])\,.$
(3.27)

Then we have that $\Phi^{\mathrm{(inter)}}_{j,{\bm{\gamma}}}[{\bf u}]\in
L^{1}(\Gamma_{\mathcal{M}})$ and that

$\displaystyle\begin{split}&{\mathcal{F}}_{\Gamma_{\mathcal{M}}}(\Phi^{\mathrm{(inter)}}_{j,{\bm{\gamma}}}[{\bf
u}])(G_{\mathcal{M}})\\\
=&\,\frac{1}{2|{\mathcal{A}}_{j}||\Gamma_{\mathcal{M}}|}\sum_{\begin{subarray}{c}\alpha_{k}\in{\mathcal{A}}_{k}\\\
k=1,2\end{subarray}}\int_{{\mathbb{R}}^{2}}\mathrm{d}x\
e^{-iG_{\mathcal{M}}\cdot(A_{j}x+\gamma_{j})}v_{{\bm{\alpha}}}\big{(}A_{1}x+\gamma_{1}+\tau_{1}^{(\alpha_{1})}\\\
&+u_{1}(A_{1}x+\gamma_{1},\alpha_{1})-A_{2}x-\gamma_{2}-\tau_{2}^{(\alpha_{2})}-u_{2}(A_{2}x+\gamma_{2},\alpha_{2})\big{)}\,.\end{split}$
(3.28)

In particular, we have that

$\displaystyle\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
\Phi^{\mathrm{(inter)}}_{j,{\bm{\gamma}}}[{\bf
u}](x)\,=\,\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
\Phi^{\mathrm{(inter)}}_{3-j,{\bm{\gamma}}}[{\bf u}](x)\,.$ (3.29)

Define for almost all $x,y\in{\mathbb{R}}^{2}$

$\displaystyle\begin{split}h_{j,{\bm{\alpha}}}(x,y)\,:=\,\tau_{j}^{(\alpha_{j})}-\tau_{3-j}^{(\alpha_{3-j})}+u_{j}(x,\alpha_{j})\,-\,u_{3-j}(x-y,\alpha_{3-j})\,.\end{split}$
(3.30)

For a pair-potential (2.42), (2.51) becomes

$\displaystyle\begin{split}&\Psi^{\mathrm{(inter)}}_{j}[{\bf u}](x,y)\,=\\\
&\frac{1}{2|{\mathcal{A}}_{j}||\Gamma_{j}|}\sum_{\begin{subarray}{c}\alpha_{k}\in{\mathcal{A}}_{k}\\\
k=1,2\end{subarray}}\sum_{\begin{subarray}{c}R_{3-j}\\\
\in{\mathcal{R}}_{3-j}\end{subarray}}v_{{\bm{\alpha}}}\big{(}y-R_{3-j}+h_{j,{\bm{\alpha}}}(x,y-R_{3-j})\big{)}\,.\end{split}$
(3.31)

######  Lemma 3.7.

Let $r>1$. Assume that, for $\alpha_{j}\in{\mathcal{A}}_{j}$, $j=1,2$,
$u_{j}(\cdot,\alpha_{j})\in L^{\infty}(\Gamma_{\mathcal{M}})$, and that

$\displaystyle v_{{\bm{\alpha}}}\in
L^{\infty}_{2r}({\mathbb{R}}^{2}\times[-L_{{\bf u}}^{z},L_{{\bf u}}^{z}])\,.$
(3.32)

Then we have that $\Psi^{\mathrm{(inter)}}_{j}[{\bf u}]\in
L^{1}(\Gamma_{\mathcal{M}}\times\Gamma_{3-j})$.

###  3.4 Ergodic Theorems

####  3.4.1 General setting

Given a probability space $(\Omega,\Sigma,{\mathbb{P}})$ and a group $G$,
recall that a group action $G\curvearrowright\Omega$ is _measure-preserving_
iff

$g.{\mathbb{P}}(B)\,:=\,{\mathbb{P}}(g^{-1}.B)\,=\,{\mathbb{P}}(B)$ (3.33)

for all $B\in\Sigma$ and $g\in G$. Moreover, we call a measurable map
$f:\Omega\to{\mathbb{R}}$ $G$-_invariant_ iff $g.f:=f\circ g^{-1}=f$ for any
$g\in G$ ${\mathbb{P}}-$almost surely. Similarly, we call measurable set $B$
$G$-invariant iff $\mathds{1}_{B}$ is $G$-invariant. Let $\Sigma_{G}$ denote
the sub $\sigma$-algebra of $\Sigma$ of all $G$-invariant sets in $\Sigma$.

Given a measurable function $f:\Omega\to{\mathbb{R}}$ and a finite subset
$H\subseteq G$, we define the _ergodic average_

${\mathcal{A}}_{H}(f)\,:=\,\frac{1}{|H|}\sum_{g\in H}g^{-1}.f\,.$ (3.34)

A sequence $(H_{N})_{N\in{\mathbb{N}}}$ of non-empty finite subsets of $G$ is
called a _Følner sequence_ iff

$\lim_{N\to\infty}\frac{|gH_{N}\triangle H_{N}|}{|H_{N}|}\,=\,0$ (3.35)

for all $g\in G$, where $A\triangle B$ denotes the symmetric difference of
sets $A,B$. $(H_{N})_{N\in{\mathbb{N}}}$ is _tempered_ iff there is $C>0$ s.t.

$\Big{|}\bigcup_{k=1}^{N-1}H_{k}^{-1}H_{N}\Big{|}\,\leq\,C|H_{N}|$ (3.36)

for all $N\in{\mathbb{N}}$, $N\geq 2$.

We finish this general introduction with the following pointwise ergodic
theorem as stated, e.g., in [53, Theorem 4.28].

######  Proposition 3.8 (Pointwise Ergodic Theorem).

Let $(\Omega,\Sigma,{\mathbb{P}})$ be a probability space and
$G\curvearrowright\Omega$ be a measure-preserving group action. Let
$(H_{N})_{N\in{\mathbb{N}}}$ be a tempered Følner sequence. Then we have that
for all $f\in L^{1}(\Omega)$ and

$\lim_{N\to\infty}{\mathcal{A}}_{H_{N}}(f)\,=\,{\mathbb{E}}[f|\Sigma_{G}]$
(3.37)

${\mathbb{P}}-$almost surely and in the $L^{1}$-sense, where
${\mathbb{E}}[\cdot|\Sigma_{G}]$ denotes the conditional expectation given the
$\sigma$-algebra $\Sigma_{G}$ of $G$-invariant events.

######  Remark 3.9.

Recall that the conditional expectation ${\mathbb{E}}[f|\Sigma_{G}]$ is the
unique $(\Omega,\Sigma_{G})$-measurable function such that

$\int_{C}\mathrm{d}{\mathbb{P}}(\omega)\
{\mathbb{E}}[f|\Sigma_{G}](\omega)\,=\,\int_{C}\mathrm{d}{\mathbb{P}}(\omega)\
f(\omega)$ (3.38)

for all ($G$-invariant) sets $C\in\Sigma_{G}$.

####  3.4.2 Incommensurate bilayer system

In our case, we set $G={\mathcal{R}}_{j}$ and $H_{N}:={\mathcal{R}}_{j}(N)$,
see (2.4). It is straight-forward to verify that
$({\mathcal{R}}_{j}(N))_{N\in{\mathbb{N}}}$ is both Følner and tempered. We
leave the details to the interested reader.

Let ${\mathbb{T}}^{2}={\mathbb{R}}^{2}/{\mathbb{Z}}^{2}$ denote the standard
2-torus, and

$\displaystyle\Gamma_{\mathcal{M}}^{\mathrm{(per)}}\,:=\,A_{\mathcal{M}}{\mathbb{T}}^{2}\,,\quad\Gamma_{3-j}^{\mathrm{(per)}}\,:=\,A_{3-j}{\mathbb{T}}^{2}$
(3.39)

denote the tori associated with the moiré unit cell $\Gamma_{\mathcal{M}}$ and
with the unit cell $\Gamma_{3-j}$ of layer $3-j$, respectively. Let
$\mathcal{B}$ denote the respective Borel $\sigma$-algebra over $\Omega$.

Then we choose
$\Omega=\Gamma_{\mathcal{M}}^{\mathrm{(per)}}\times\Gamma_{3-j}^{\mathrm{(per)}}$
and study the Lebesgue-measure preserving group action of ${\mathcal{R}}_{j}$
on $\Omega$ given by translations, i.e., $R_{j}.\omega:=\omega+(R_{j},R_{j})$.
As a consequence of Proposition 3.8, we prove the following statement.

######  Proposition 3.10 (Pointwise and Mean Ergodic Theorem).

Let $h\in L^{1}_{\emph{loc}}({\mathbb{R}}^{4})$ be
${\mathcal{R}}_{\mathcal{M}}\times{\mathcal{R}}_{3-j}$-periodic. Then

$\displaystyle\begin{split}&\lim_{N\to\infty}{\mathcal{A}}_{{\mathcal{R}}_{j}(N)}(h)(\omega_{\mathcal{M}},\omega_{3-j})\,=\,\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{A_{j}^{-1}\Gamma_{\mathcal{M}}}\mathrm{d}x\
h\big{(}A_{j}x+\omega_{\mathcal{M}},(A_{j}-A_{3-j})x+\omega_{3-j}\big{)}\end{split}$
(3.40)

converges for almost all
$(\omega_{\mathcal{M}},\omega_{3-j})\in\Gamma_{\mathcal{M}}\times\Gamma_{3-j}$
and also in $L^{1}(\Gamma_{\mathcal{M}}\times\Gamma_{3-j})$.

######  Remark 3.11.

Let $h\in
L^{1}_{\emph{loc}}(\Gamma_{\mathcal{M}}^{\mathrm{(per)}}\times\Gamma_{3-j}^{\mathrm{(per)}})$.
In addition, let $\mathcal{B}_{{\mathcal{R}}_{j}}$ denote the $\sigma$-algebra
of ${\mathcal{R}}_{j}$-invariant Borel subsets of
$\Gamma_{\mathcal{M}}^{\mathrm{(per)}}\times\Gamma_{3-j}^{\mathrm{(per)}}$,
where the ${\mathcal{R}}_{j}$-action is given as above by
$R_{j}.\omega=\omega+(R_{j},R_{j})$. Then one can prove directly that

${\mathbb{E}}[h|\Sigma_{{\mathcal{R}}_{j}}](\omega_{\mathcal{M}},\omega_{3-j})\,=\,\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{A_{j}^{-1}\Gamma_{\mathcal{M}}}\mathrm{d}x\
h\big{(}A_{j}x+\omega_{\mathcal{M}},(A_{j}-A_{3-j})x+\omega_{3-j}\big{)}$
(3.41)

Lebesgue-almost everywhere, where the conditional expectation on the l.h.s. is
taken w.r.t. the uniform distribution on
$\Gamma_{\mathcal{M}}^{\mathrm{(per)}}\times\Gamma_{3-j}^{\mathrm{(per)}}$.

We find the argument presented in the proof of Proposition 3.10 appealing
because it provides a means to calculate the conditional expectation.

Using the notion of 2D Diophantine rotations, see Definition 2.4, we obtain
the following quantitative ergodic theorem.

######  Proposition 3.12 (Quantitative Ergodic Theorem).

Let $q>0$, $\sigma>\frac{1433}{1248}$, $s>1$, and let $\theta$ be a
$(q,K,\sigma)$-Diophantine 2D rotation for some $K>0$. Let $f\in
L^{1}(\Gamma_{\mathcal{M}})$ be such that
${\mathcal{F}}_{\Gamma_{\mathcal{M}}}\big{(}|\nabla|^{2(\sigma+s)}f\big{)}\in\ell^{\infty}({\mathcal{R}}^{*}_{\mathcal{M}})$.
Then we have for all $N\in{\mathbb{N}}$ large enough that

$\displaystyle\begin{split}&\bigg{|}\frac{1}{\big{(}2N+1\big{)}^{2}}\sum_{R_{j}\in{\mathcal{R}}_{j}(N)}f(\left\\{R_{j}\right\\}_{\mathcal{M}})\,-\,\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
f(x)\bigg{|}\\\
\leq&\,\frac{2\sqrt{2}}{K}\Big{(}\frac{\rho_{\mathcal{M}}\|A\|_{2}}{2\pi}\Big{)}^{2(\sigma+s)}\Big{(}\frac{2s}{2s-1}\,+\,2^{-s}\big{(}\frac{s}{s-1}\big{)}^{2}\Big{)}\\\
&\,\sup_{G_{\mathcal{M}}\in{\mathcal{R}}^{*}_{\mathcal{M}}}|G_{\mathcal{M}}|^{2(\sigma+s)}|\hat{f}(G_{\mathcal{M}})|\frac{1}{2N+1}\,.\end{split}$
(3.42)

For proofs of these ergodic theorems, we refer to Appendix C.

######  Remark 3.13.

Our methods would allow us to adapt Proposition 3.12 to the more general
setting in Proposition 3.10. However, for presentational purposes, we chose to
omit such a result and leave the details to the interested reader.

We now want to motivate how to prove Propositions 3.10 and 3.12.

####  3.4.3 Dirichlet kernel as a discrepancy function

Let $h\in L^{p}(\Gamma_{\mathcal{M}}\times\Gamma_{3-j})$, $p>1$. Using the
Fourier transform defined in (3.14), we obtain that for almost all
$(\omega_{\mathcal{M}},\omega_{3-j})\in\Gamma_{\mathcal{M}}^{\mathrm{(per)}}\times\Gamma_{3-j}^{\mathrm{(per)}}$

$\displaystyle\begin{split}&\frac{1}{(2N+1)^{2}}\sum_{R_{j}\in{\mathcal{R}}_{j}(N)}h(\left\\{R_{j}+\omega_{\mathcal{M}}\right\\}_{\mathcal{M}},\left\\{R_{j}+\omega_{3-j}\right\\}_{3-j})\,=\\\
&\sum_{R_{j}\in{\mathcal{R}}_{j}(N)}\sum_{\begin{subarray}{c}G_{3-j}\in{\mathcal{R}}_{3-j}^{*},\\\
G_{\mathcal{M}}\in{\mathcal{R}}^{*}_{\mathcal{M}}\end{subarray}}e^{i(G_{\mathcal{M}}\cdot\left\\{R_{j}+\omega_{\mathcal{M}}\right\\}_{\mathcal{M}}+G_{3-j}\cdot\left\\{R_{j}+\omega_{3-j}\right\\}_{3-j})}\hat{h}(G_{\mathcal{M}},G_{3-j})\,.\end{split}$
(3.43)

Realizing that ${\mathcal{R}}^{*}_{\mathcal{M}}\cdot\left\lfloor
R\right\rfloor_{\mathcal{M}}\subseteq 2\pi{\mathbb{Z}}$ for any
$x\in{\mathbb{R}}^{2}$, we have that

$e^{iG_{\mathcal{M}}\cdot\left\\{x\right\\}_{\mathcal{M}}}\,=\,e^{iG_{\mathcal{M}}\cdot
x}$ (3.44)

for any $x\in{\mathbb{R}}^{2}$. A straight-forward calculation analogous to
the computation of the Dirichlet kernel yields

$\begin{split}\mathds{1}_{{\mathcal{R}}_{j}^{*}}^{(N)}(G)\,&:=\,\frac{1}{(2N+1)^{2}}\sum_{R_{j}\in{\mathcal{R}}_{j}(N)}e^{iG\cdot
R_{j}}\\\
&=\,\prod_{\ell=1}^{2}\frac{\sin\big{(}(2N+1)(A_{j}^{T}G)_{\ell}/2\big{)}}{(2N+1)\sin\big{(}(A_{j}^{T}G)_{\ell}/2\big{)}}\,,\end{split}$
(3.45)

where $y_{\ell}$ denotes the $\ell^{\text{th}}$ Euclidean coordinate of $y$.
Using this notation, we obtain that

$\displaystyle\begin{split}&\frac{1}{(2N+1)^{2}}\sum_{R_{j}\in{\mathcal{R}}_{j}(N)}h(\left\\{R_{j}+\omega_{\mathcal{M}}\right\\}_{\mathcal{M}},\left\\{R_{j}+\omega_{3-j}\right\\}_{3-j})\,=\\\
&\sum_{\begin{subarray}{c}G_{3-j}\in{\mathcal{R}}_{3-j}^{*},\\\
G_{\mathcal{M}}\in{\mathcal{R}}^{*}_{\mathcal{M}}\end{subarray}}\mathds{1}_{{\mathcal{R}}_{j}^{*}}^{(N)}(G_{3-j}+G_{\mathcal{M}})\,\hat{h}(G_{\mathcal{M}},G_{3-j})e^{i(G_{\mathcal{M}}\cdot\omega_{\mathcal{M}}+G_{3-j}\cdot\omega_{3-j})}\,.\end{split}$
(3.46)

If $f\in L^{p}(\Gamma_{\mathcal{M}})$, analogous steps yield

$\displaystyle\frac{1}{(2N+1)^{2}}\sum_{R_{j}\in{\mathcal{R}}_{j}(N)}f(\left\\{R_{j}\right\\}_{\mathcal{M}})\,=$
$\displaystyle\,\sum_{G_{\mathcal{M}}\in{\mathcal{R}}^{*}_{\mathcal{M}}}\mathds{1}_{{\mathcal{R}}_{j}^{*}}^{(N)}(G_{\mathcal{M}})\,\hat{f}(G_{\mathcal{M}})\,.$
(3.47)

Next, we will study the convergence properties of
$\mathds{1}_{{\mathcal{R}}_{j}^{*}}^{(N)}$.

####  3.4.4 Properties of the Dirichlet kernel

For all $G_{j}\in{\mathcal{R}}_{j}^{*}$, we have that

$\displaystyle\mathds{1}_{{\mathcal{R}}_{j}^{*}}^{(N)}(G_{j})\,$
$\displaystyle=\,1\,.$ (3.48)

Moreover, for all $G\in{\mathbb{R}}^{2}\setminus{\mathcal{R}}_{j}^{*}$, it
holds that

$|\mathds{1}_{{\mathcal{R}}_{j}^{*}}^{(N)}(G)|\,\leq\,\frac{1}{2N+1}\min_{\ell=1,2}\frac{1}{\big{|}\sin\big{(}(A_{j}^{T}G)_{\ell}/2\big{)}\big{|}}\,.$
(3.49)

In particular, we obtain that for all $G\in{\mathbb{R}}^{2}$

$\lim_{N\to\infty}\mathds{1}_{{\mathcal{R}}_{j}^{*}}^{(N)}(G)\,=\,\mathds{1}_{{\mathcal{R}}_{j}^{*}}(G)\,.$
(3.50)

In particular, if we can interchange the order of taking the limit
$N\to\infty$ and summing over $G_{3-j}$ and $G_{\mathcal{M}}$, (3.46) implies
that

$\displaystyle\begin{split}&\frac{1}{(2N+1)^{2}}\sum_{R_{j}\in{\mathcal{R}}_{j}(N)}h(\left\\{R_{j}+\omega_{\mathcal{M}}\right\\}_{\mathcal{M}},\left\\{R_{j}+\omega_{3-j}\right\\}_{3-j})\,=\\\
&\xrightarrow[]{N\to\infty}\sum_{\begin{subarray}{c}G_{3-j}\in{\mathcal{R}}_{3-j}^{*},\\\
G_{\mathcal{M}}\in{\mathcal{R}}^{*}_{\mathcal{M}}\end{subarray}}\mathds{1}_{{\mathcal{R}}_{j}^{*}}(G_{3-j}+G_{\mathcal{M}})\,\hat{h}(G_{\mathcal{M}},G_{3-j})e^{i(G_{\mathcal{M}}\cdot\omega_{\mathcal{M}}+G_{3-j}\cdot\omega_{3-j})}\,.\end{split}$
(3.51)

Again, assuming interchangeability of involved sums and integrals, we arrive,
after a straight-forward calculation, at

$\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{A_{j}^{-1}\Gamma_{\mathcal{M}}}\mathrm{d}x\
h\big{(}A_{j}x+\omega_{\mathcal{M}},(A_{j}-A_{3-j})x+\omega_{3-j}\big{)}\,,$
(3.52)

as stated in Proposition 3.10. In order to interchange limits, sums, and
integrals, we use the fact that we study the convergence in
$L^{1}(\Gamma_{\mathcal{M}}^{\mathrm{(per)}}\times\Gamma_{3-j}^{\mathrm{(per)}})$,
in order to restrict to $C^{\infty}$-smooth functions.

In order to extract a rate of convergence in (3.46), we need to analyze the
tail-behavior of $\mathds{1}_{{\mathcal{R}}_{j}^{*}}^{(N)}$. For general
irrational angles, the rate of convergence can be arbitrarily slow. However,
we show in the proof of Proposition 3.12, in Appendix C, that for Diophantine
2D rotations $\theta$, see Definition 2.4, and all
$G_{\mathcal{M}}\in{\mathcal{R}}^{*}_{\mathcal{M}}\setminus\\{0\\}$, we have
that

$|\mathds{1}_{{\mathcal{R}}_{j}^{*}}^{(N)}(G_{\mathcal{M}})|\,\leq\,\frac{C_{\sigma,q,A,\theta}|G_{\mathcal{M}}|^{2\sigma}}{2N+1}$
(3.53)

for any $\sigma>\frac{1433}{1248}$, where $C_{\sigma,q,A,\theta}$ depends only
on $\sigma$, $q$, $A$, and $\theta$. In particular, employing (3.53), (3.47)
implies that

$\displaystyle\begin{split}\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle\bigg{|}\frac{1}{(2N+1)^{2}}\sum_{R_{j}\in{\mathcal{R}}_{j}(N)}f(\left\\{R_{j}\right\\}_{\mathcal{M}})\,-\,\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
f(x)\bigg{|}$\mbox{}\hfil\\\
&\leq\,C_{\sigma,q,A,\theta}\sum_{G_{\mathcal{M}}\in{\mathcal{R}}^{*}_{\mathcal{M}}\setminus\\{0\\}}|G_{\mathcal{M}}|^{2\sigma}|\hat{f}(G_{\mathcal{M}})|\frac{1}{2N+1}\,.\end{split}$
(3.54)

Lemma A.1 then implies for any $s>1$ that

$\displaystyle\begin{split}\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle\bigg{|}\frac{1}{(2N+1)^{2}}\sum_{R_{j}\in{\mathcal{R}}_{j}(N)}f(\left\\{R_{j}\right\\}_{\mathcal{M}})\,-\,\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
f(x)\bigg{|}$\mbox{}\hfil\\\ &\leq
C_{\sigma,q,A,\theta}\,4\Big{(}\frac{\rho_{\mathcal{M}}\|A\|_{2}}{2\pi}\Big{)}^{2s}\Big{(}\frac{2s}{2s-1}\,+\,2^{-s}\big{(}\frac{s}{s-1}\big{)}^{2}\Big{)}\\\
&\,\sup_{G_{\mathcal{M}}\in{\mathcal{R}}^{*}_{\mathcal{M}}}|G_{\mathcal{M}}|^{2(\sigma+s)}|\hat{f}(G_{\mathcal{M}})|\frac{1}{2N+1}\,,\end{split}$
(3.55)

where the constant $C_{\sigma,q,A,\theta}$ is the same as in (3.53).

##  4 Statement of results

We are ready to formulate and prove our main results.

###  4.1 Convergence without a rate

Recall from (2.48) that

$\displaystyle\begin{split}&\Phi^{\mathrm{(mono)}}_{j}[u_{j}](x)\,=\\\
&\frac{1}{|{\mathcal{A}}_{j}||\Gamma_{j}|}\sum_{\alpha_{j}\in{\mathcal{A}}_{j}}V^{\mathrm{(mono)}}_{j,\alpha_{j}}\bigg{(}\Big{(}\big{(}u_{j}(x+R_{j}^{\prime},\alpha_{j}^{\prime})\,-u_{j}(x,\alpha_{j})\big{)}_{\alpha_{j}^{\prime}\in{\mathcal{A}}_{j}}\Big{)}_{R_{j}^{\prime}\in{\mathcal{R}}_{j}}\bigg{)}\,,\end{split}$
(4.1)

from (2.49) that

$\displaystyle\begin{split}e^{\mathrm{(mono)}}_{j,N,\gamma_{j}}(u_{j})\,=\,\frac{1}{(2N+1)^{2}}\sum_{R_{j}\in{\mathcal{R}}_{j}(N)}\Phi^{\mathrm{(mono)}}_{j}[u_{j}](R_{j}+\gamma_{j})\,,\end{split}$
(4.2)

and from (2.50) that

$\displaystyle e^{\mathrm{(mono)}}_{j}(u_{j})\,=$ (4.3)
$\displaystyle\frac{1}{|{\mathcal{A}}_{j}||\Gamma_{j}|}\sum_{\alpha_{j}\in{\mathcal{A}}_{j}}\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{A_{j}^{-1}\Gamma_{\mathcal{M}}}\mathrm{d}x\
V^{\mathrm{(mono)}}_{j,\alpha_{j}}\Big{(}\big{(}u_{j}(A_{j}x+R_{j}^{\prime},\alpha_{j}^{\prime})\,-\,u_{j}(A_{j}x,\alpha_{j})\big{)}_{\begin{subarray}{c}R_{j}^{\prime}\in{\mathcal{R}}_{j},\\\
\alpha_{j}^{\prime}\in{\mathcal{A}}_{j}\end{subarray}}\Big{)}$ (4.4)
$\displaystyle=\,\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
\Phi^{\mathrm{(mono)}}_{j}[u_{j}](x)\,,$ (4.5)

where we substituted $x\to A_{j}x$.

As a direct consequence of Proposition 2.7, we obtain the following result.

######  Theorem 4.1 (Monolayer potential).

Assume that

$\Phi^{\mathrm{(mono)}}_{j}[u_{j}]\in L^{1}(\Gamma_{\mathcal{M}})\,.$ (4.6)

Then we have for almost all $\gamma_{j}\in\Gamma_{j}$ that

$\displaystyle\begin{split}\lim_{N\to\infty}e^{\mathrm{(mono)}}_{j,N,\gamma_{j}}(u_{j})\,=\,e^{\mathrm{(mono)}}_{j}(u_{j})\,.\end{split}$
(4.7)

Similarly, recall (2.51) that

$\displaystyle\begin{split}&\Psi^{\mathrm{(inter)}}_{j}[{\bf u}](x,y)\,=\\\
&\frac{1}{|{\mathcal{A}}_{j}||\Gamma_{j}|}\sum_{\alpha_{j}\in{\mathcal{A}}_{j}}V^{\mathrm{(inter)}}_{j,\alpha_{j}}\bigg{(}\Big{(}\big{(}y-R_{3-j}+\tau_{j}^{(\alpha_{j})}-\tau_{3-j}^{(\alpha_{3-j})}+u_{j}(x,\alpha_{j})\\\
&-\,u_{3-j}(x-y+R_{3-j},\alpha_{3-j})\big{)}_{\alpha_{3-j}\in{\mathcal{A}}_{3-j}}\Big{)}_{R_{3-j}\in{\mathcal{R}}_{3-j}}\bigg{)}\,,\end{split}$
(4.8)

and from (2.52) that

$\displaystyle\begin{split}&e^{\mathrm{(inter)}}_{j,N,{\bm{\gamma}}}({\bf
u})\,=\,\frac{1}{(2N+1)^{2}}\sum_{R_{j}\in{\mathcal{R}}_{j}(N)}\Psi^{\mathrm{(inter)}}_{j}[{\bf
u}](R_{j}+\gamma_{j},R_{j}+\gamma_{j}-\gamma_{3-j})\,.\end{split}$ (4.9)

Moreover, define

$\Phi^{\mathrm{(inter)}}_{j,{\bm{\gamma}}}[{\bf
u}](x)\,:=\,\Psi^{\mathrm{(inter)}}_{j}[{\bf u}](x,D_{j\to
3-j}x+A_{3-j}A_{j}^{-1}\gamma_{j}-\gamma_{3-j})$ (4.10)

and from (2.54) that

$\displaystyle e^{\mathrm{(inter)}}_{j,{\bm{\gamma}}}({\bf u})\,$
$\displaystyle=\,\frac{1}{|{\mathcal{A}}_{j}||\Gamma_{j}|}\sum_{\alpha_{j}\in{\mathcal{A}}_{j}}\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{A_{j}^{-1}\Gamma_{\mathcal{M}}}\mathrm{d}x\
V^{\mathrm{(inter)}}_{j,\alpha_{j}}\Big{(}\big{(}Y_{j}(A_{j}x+\gamma_{j},\alpha_{j})$
(4.11) $\displaystyle\quad-
Y_{3-j}(A_{3-j}x+\gamma_{3-j}+R_{3-j},\alpha_{3-j})\big{)}_{\begin{subarray}{c}R_{3-j}\in{\mathcal{R}}_{3-j},\\\
\alpha_{3-j}\in{\mathcal{A}}_{3-j}\end{subarray}}\Big{)}$ (4.12)
$\displaystyle=\,\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
\Phi^{\mathrm{(inter)}}_{j,{\bm{\gamma}}}[{\bf u}](x)\,,$ (4.13)

where we substituted $x\to A_{j}x+\gamma_{j}$

Then Proposition 3.10 implies the following result.

######  Theorem 4.2 (Interlayer many-body potentials).

Assume that for any
${\bm{\gamma}}=(\gamma_{1},\gamma_{2})\in\Gamma_{1}\times\Gamma_{2}$,

$\Psi^{\mathrm{(inter)}}_{j}[{\bf u}]\in
L^{1}(\Gamma_{\mathcal{M}}\times\Gamma_{3-j})\,,\quad\Phi^{\mathrm{(inter)}}_{j,{\bm{\gamma}}}[{\bf
u}]\in L^{1}(\Gamma_{\mathcal{M}})\,.$ (4.14)

We then have for almost all ${\bm{\gamma}}\in\Gamma_{1}\times\Gamma_{2}$ that

$\displaystyle\begin{split}\lim_{N\to\infty}e^{\mathrm{(inter)}}_{j,N,{\bm{\gamma}}}({\bf
u})\,=\,e^{\mathrm{(inter)}}_{j,{\bm{\gamma}}}({\bf u})\,.\end{split}$ (4.15)

For the next statement, recall from (3.18) the definition of the weighted
Lebesgue space, from (3.23) that

$d_{{\mathcal{A}}_{1},{\mathcal{A}}_{2}}\,=\,\max_{\begin{subarray}{c}\alpha_{\ell}\in{\mathcal{A}}_{\ell}\\\
\ell=1,2\end{subarray}}|\tau_{1}^{(\alpha_{1})}-\tau_{2}^{(\alpha_{2})}|\,,$
(4.16)

and from (3.24) that

$L_{{\bf
u}}^{z}\,=\,d_{{\mathcal{A}}_{1},{\mathcal{A}}_{2}}\,+\,\|u_{1,z}\|_{\infty}+\|u_{2,z}\|_{\infty}\,,$
(4.17)

where $f_{z}$ refers to the $z$-component of $f$.

In addition, (4.10) and (3.31) imply

$\displaystyle\begin{split}&\Phi^{\mathrm{(inter)}}_{j,{\bm{\gamma}}}[{\bf
u}](x)\,=\,\Psi^{\mathrm{(inter)}}_{j}[{\bf u}](x,D_{j\to
3-j}x-\gamma_{3-j})\,=\\\
&\frac{1}{2|{\mathcal{A}}_{j}||\Gamma_{j}|}\sum_{\begin{subarray}{c}\alpha_{k}\in{\mathcal{A}}_{k}\\\
k=1,2\end{subarray}}\sum_{R_{3-j}\in{\mathcal{R}}_{3-j}}v_{{\bm{\alpha}}}\big{(}D_{j\to
3-j}x-R_{3-j}-\gamma_{3-j}+\tau_{j}^{(\alpha_{j})}-\tau_{3-j}^{(\alpha_{3-j})}\\\
&+\,u_{j}(x,\alpha_{j})-u_{3-j}(R_{3-j}+\gamma_{3-j}+(I-D_{j\to
3-j})x,\alpha_{3-j})\big{)}\,.\end{split}$ (4.18)

######  Theorem 4.3 (Interlayer pair potentials).

Let $r>1$ and $\gamma_{3-j}\in\Gamma_{3-j}$. Assume that, for
$\alpha_{k}\in{\mathcal{A}}_{k}$, $k=1,2$, $u_{k}(\cdot,\alpha_{k})\in
L^{\infty}({\mathbb{R}}^{2})$, and that

$\displaystyle v_{{\bm{\alpha}}}\in
L^{\infty}_{2r}({\mathbb{R}}^{2}\times[-L_{{\bf u}}^{z},L_{{\bf u}}^{z}])\,,$
(4.19)

We then have for almost all ${\bm{\gamma}}\in\Gamma_{1}\times\Gamma_{2}$

$\displaystyle\begin{split}\lim_{N\to\infty}e^{\mathrm{(inter)}}_{j,N,{\bm{\gamma}}}({\bf
u})\,=\,e^{\mathrm{(inter)}}_{j,{\bm{\gamma}}}(u)\,.\end{split}$ (4.20)

###### Proof.

Lemma 3.7 implies that $\Psi^{\mathrm{(inter)}}_{j}[{\bf u}]\in
L^{1}(\Gamma_{\mathcal{M}}\times\Gamma_{3-j})$. Moreover, Lemma 3.6 implies
that $\Phi^{\mathrm{(inter)}}_{1}[u]\in L^{1}(\Gamma_{\mathcal{M}})$. Theorem
4.2 then yields the claim. ∎

###  4.2 Pointwise convergence with rate

We abbreviate the ergodicity error

$\displaystyle\begin{split}\textnormal{err}^{\textnormal{(mono)}}_{j}\,:=&\,\frac{2\sqrt{2}}{K}\Big{(}\frac{\rho_{\mathcal{M}}\|A\|_{2}}{2\pi}\Big{)}^{2(\sigma+s)}\Big{(}\frac{2s}{2s-1}\,+\,2^{-s}\big{(}\frac{s}{s-1}\big{)}^{2}\Big{)}\\\
&\,\sup_{G_{\mathcal{M}}\in{\mathcal{R}}^{*}_{\mathcal{M}}}|G_{\mathcal{M}}|^{2(\sigma+s)}|{\mathcal{F}}_{\Gamma_{\mathcal{M}}}(\Phi^{\mathrm{(mono)}}_{j}[u_{j}])(G_{\mathcal{M}})|\,.\end{split}$
(4.21)

Then the following statement is a direct consequence of Proposition 3.12.

######  Theorem 4.4 (Monolayer potential).

Let $s>1$, $\sigma>\frac{1433}{1248}$, and $\theta\in[0,2\pi)$ be a
$(q,K,\sigma)$-Diophantine 2D rotation for some $K>0$. Assume that

$\Phi^{\mathrm{(mono)}}_{j}[u_{j}]\in
L^{1}(\Gamma_{\mathcal{M}})\,,\,{\mathcal{F}}_{\Gamma_{\mathcal{M}}}\big{(}|\nabla|^{2(\sigma+s)}\Phi^{\mathrm{(mono)}}_{j}[u_{j}]\big{)}\in\ell^{\infty}({\mathcal{R}}^{*}_{\mathcal{M}})\,.$
(4.22)

Then we have that

$\displaystyle\begin{split}&\big{|}e^{\mathrm{(mono)}}_{j,N,0}(u_{j})-e^{\mathrm{(mono)}}_{j}(u_{j})\big{|}\,\leq\,\frac{\textnormal{err}^{\textnormal{(mono)}}}{2N+1}\,.\end{split}$
(4.23)

Similarly, we abbreviate the ergodicity error

$\displaystyle\begin{split}\textnormal{err}^{\textnormal{(MB)}}_{j}\,:=&\,\frac{2\sqrt{2}}{K}\Big{(}\frac{\rho_{\mathcal{M}}\|A\|_{2}}{2\pi}\Big{)}^{2(\sigma+s)}\Big{(}\frac{2s}{2s-1}\,+\,2^{-s}\big{(}\frac{s}{s-1}\big{)}^{2}\Big{)}\\\
&\,\sup_{G_{\mathcal{M}}\in{\mathcal{R}}^{*}_{\mathcal{M}}}|G_{\mathcal{M}}|^{2(\sigma+s)}|{\mathcal{F}}_{\Gamma_{\mathcal{M}}}(\Phi^{\mathrm{(inter)}}_{j,0}[{\bf
u}])(G_{\mathcal{M}})|\,.\end{split}$ (4.24)

Again, the subsequent result follows directly from Proposition 3.12.

######  Theorem 4.5 (Many-body potentials).

Let $s>1$, $\sigma>\frac{1433}{1248}$, and $\theta\in[0,2\pi)$ be a
$(q,K,\sigma)$-Diophantine 2D rotation for some $K>0$. Assume that

$\Phi^{\mathrm{(inter)}}_{j,0}[{\bf u}]\in
L^{1}(\Gamma_{\mathcal{M}})\,,\quad{\mathcal{F}}_{\Gamma_{\mathcal{M}}}\big{(}|\nabla|^{2(\sigma+s)}\Phi^{\mathrm{(inter)}}_{j,0}[{\bf
u}]\big{)}\in\ell^{\infty}({\mathcal{R}}^{*}_{\mathcal{M}})\,.$ (4.25)

Then we have that

$\displaystyle\begin{split}&\big{|}e^{\mathrm{(inter)}}_{j,N,0}({\bf
u})-e^{\mathrm{(inter)}}_{j,0}({\bf
u})\big{|}\,\leq\,\frac{\textnormal{err}^{\textnormal{(MB)}}}{2N+1}\,.\end{split}$
(4.26)

Let $\theta\in{\mathbb{R}}$ be a 2D Diophantine rotation and
$\sigma>\frac{1433}{1248}$. Abbreviate

$\displaystyle\begin{split}&\kappa^{\textnormal{(pair)}}(\theta,\sigma)\,:=\\\
\sup&\bigg{(}\bigcup_{N\in{\mathbb{N}}}\bigcap_{\begin{subarray}{c}n\in{\mathbb{Z}}^{2}:\\\
|n|^{2}\geq
N\end{subarray}}\bigg{\\{}K>0\mid\operatorname{dist}\big{(}A^{T}R_{\pm\theta}A^{-T}n,{\mathbb{Z}}^{2}\big{)}\geq\frac{K}{|n|^{2\sigma}}\bigg{\\}}\bigg{)}\,,\end{split}$
(4.27)

and define

$M^{\textnormal{(pair)}}(\theta)\,:=\,\inf_{\sigma\in\big{(}\frac{1433}{1248},2\big{)}}\frac{1}{\kappa^{\textnormal{(pair)}}(\theta,\sigma)}\Big{(}\frac{6-2\sigma}{5-2\sigma}\,+\,2^{\sigma-3}\big{(}\frac{3-\sigma}{2-\sigma}\big{)}^{2}\Big{)}\,.$
(4.28)

Again, we abbreviate the ergodicity error

$\displaystyle\begin{split}&\textnormal{err}^{\textnormal{(pair)}}_{j}\,:=\,\\\
&\frac{203\sqrt{2}}{|{\mathcal{A}}_{j}||\Gamma_{1}||\Gamma_{2}|}\Big{(}\frac{\rho_{\mathcal{M}}\|A\|_{2}}{2\pi}\Big{)}^{6}M^{\textnormal{(pair)}}(\theta)(q^{6}+q^{-6}+1)\frac{5^{r-1}\pi}{r-1}\\\
&\,\big{(}1+d_{{\mathcal{A}}_{1},{\mathcal{A}}_{2}}+\|u_{1}\|_{W^{6,\infty}}+\|u_{2}\|_{W^{6,\infty}}\big{)}^{6+2r}\\\
&\,\sum_{\begin{subarray}{c}\alpha_{k}\in{\mathcal{A}}_{k}\\\
k=1,2\end{subarray}}\|v_{{\bm{\alpha}}}\|_{W^{6,\infty}_{2r}({\mathbb{R}}^{2}\times[-L_{{\bf
u}}^{z},L_{{\bf u}}^{z}])}\,.\end{split}$ (4.29)

As a consequence of Theorem 4.5, we obtain the following result.

######  Theorem 4.6 (Pair potentials).

Let $r>1$, $\sigma\in\big{(}\frac{1433}{1248},2\big{)}$, and
$\theta\in[0,2\pi)$ be a $(q,K,\sigma)$-Diophantine 2D rotation for some
$K>0$. Assume that, for $\alpha_{k}\in{\mathcal{A}}_{k}$, $k=1,2$,
$u_{k}(\cdot,\alpha_{k})\in W^{6,\infty}({\mathbb{R}}^{2})$, and that

$\displaystyle v_{{\bm{\alpha}}}\in
W^{6,\infty}_{2r}({\mathbb{R}}^{2}\times[-L_{{\bf u}}^{z},L_{{\bf
u}}^{z}])\,,$ (4.30)

see (3.17). Then we have that

$\displaystyle\begin{split}&|e^{\mathrm{(inter)}}_{j,N,0}({\bf
u})-e^{\mathrm{(inter)}}_{j,0}({\bf
u})|\,\leq\,\frac{\textnormal{err}^{\textnormal{(pair)}}}{2N+1}\,.\end{split}$
(4.31)

######  Remark 4.7.

One could reduce the regularity assumption

${\mathcal{F}}_{\Gamma_{\mathcal{M}}}\big{(}|\nabla|^{2(\sigma+s)}\Phi^{\mathrm{(mono)}}_{j}[u_{j}]\big{)}\in\ell^{\infty}({\mathcal{R}}^{*}_{\mathcal{M}})$
(4.32)

in Theorem 4.4 to assumptions on $V^{\mathrm{(mono)}}_{\alpha_{j}}$ and
$u_{j}$ individually, in analogy to Theorem 4.6, and similarly for
$\Phi^{\mathrm{(inter)}}_{j,{\bm{\gamma}}}[{\bf u}]$ in Theorem 4.5. However,
in order to maintain a clear presentation, we decided to omit such a result,
and leave the details to the interested reader.

###### Proof of Theorem 4.6.

Observe that, due to $\sigma>\frac{1433}{1248}$ and $s>1$ in the assumptions
of Theorem 4.5, we have that $\left\lceil\sigma+s\right\rceil\geq 3$. We
choose $s=3-\sigma$ for some $\sigma\in\big{(}\frac{1433}{1248},2\big{)}$.

We employ Theorem 4.5 and have that

$|e^{\mathrm{(inter)}}_{j,N,0}({\bf u})-e^{\mathrm{(inter)}}_{j,0}({\bf
u})|\,\leq\,\frac{\textnormal{err}^{\textnormal{(MB)}}}{2N+1}$ (4.33)

with $\textnormal{err}^{\textnormal{(MB)}}$ given in (4.24), with the choice
$s=3-\sigma$, by

$\displaystyle\begin{split}\textnormal{err}^{\textnormal{(MB)}}_{j}\,&=\,\frac{2\sqrt{2}}{K}\Big{(}\frac{\rho_{\mathcal{M}}\|A\|_{2}}{2\pi}\Big{)}^{6}\Big{(}\frac{6-2\sigma}{5-2\sigma}\,+\,2^{\sigma-3}\big{(}\frac{3-\sigma}{2-\sigma}\big{)}^{2}\Big{)}\\\
&\quad\sup_{G_{\mathcal{M}}\in{\mathcal{R}}^{*}_{\mathcal{M}}}|G_{\mathcal{M}}|^{6}|{\mathcal{F}}_{\Gamma_{\mathcal{M}}}(\Phi^{\mathrm{(inter)}}_{j,0}[{\bf
u}])(G_{\mathcal{M}})|\,.\end{split}$ (4.34)

Since $\sigma\in\big{(}\frac{1433}{1248},2\big{)}$ was arbitrary, (4.33) and
(4.34) imply

$\displaystyle\begin{split}|e^{\mathrm{(inter)}}_{j,N,0}({\bf
u})-e^{\mathrm{(inter)}}_{j,0}({\bf
u})|\,\leq&\,2\sqrt{2}\Big{(}\frac{\sqrt{3}|\Gamma_{\mathcal{M}}|}{2^{3}\pi^{2}}\Big{)}^{3}M^{\textnormal{(pair)}}(\theta)\\\
&\,\sup_{G_{\mathcal{M}}\in{\mathcal{R}}^{*}_{\mathcal{M}}}|G_{\mathcal{M}}|^{6}|{\mathcal{F}}_{\Gamma_{\mathcal{M}}}(\Phi^{\mathrm{(inter)}}_{j,0}[{\bf
u}])(G_{\mathcal{M}})|\,.\end{split}$ (4.35)

(5.17) in the proof of Lemma 3.6 implies

$\displaystyle\begin{split}&{\mathcal{F}}_{\Gamma_{\mathcal{M}}}(\Phi^{\mathrm{(inter)}}_{j,{\bm{\gamma}}}[{\bf
u}])(G_{\mathcal{M}})\\\
=&\frac{1}{2|{\mathcal{A}}_{j}||\Gamma_{1}||\Gamma_{2}|}\sum_{\begin{subarray}{c}\alpha_{k}\in{\mathcal{A}}_{k}\\\
k=1,2\end{subarray}}\int_{{\mathbb{R}}^{2}}\mathrm{d}y\ e^{-iD_{j\to
3-j}^{-T}G_{\mathcal{M}}\cdot(y+\gamma_{3-j})}v_{{\bm{\alpha}}}\big{(}y+\tau_{j}^{(\alpha_{j})}\\\
&-\tau_{3-j}^{(\alpha_{3-j})}\,+\,u_{j}(D_{j\to
3-j}^{-1}y,\alpha_{j})-u_{3-j}(D_{j\to
3-j}^{-1}y-y,\alpha_{3-j})\big{)}\,.\end{split}$ (4.36)

In particular, we have that
${\mathcal{F}}_{\Gamma_{\mathcal{M}}}(\Phi^{\mathrm{(inter)}}_{j,0}[{\bf
u}])(0)=e^{\mathrm{(inter)}}_{j,0}({\bf u})$. As in (3.25), we define

$\displaystyle g_{j,{\bm{\alpha}}}(y)\,:=$
$\displaystyle\,\tau_{j}^{(\alpha_{j})}-\tau_{3-j}^{(\alpha_{3-j})}\,+\,u_{j}(D_{j\to
3-j}^{-1}y,\alpha_{j})-u_{3-j}(D_{j\to 3-j}^{-1}y-y,\alpha_{3-j})\,.$ (4.37)

Now let $\sigma\in\big{(}\frac{1433}{1248},2\big{)}$ be arbitrary. Using Lemma
3.1 and the definition of the fractional derivative $|\nabla|$ via Fourier
multiplication, we find that

$\displaystyle\begin{split}&|G_{\mathcal{M}}|^{6}{\mathcal{F}}_{\Gamma_{\mathcal{M}}}(\Phi^{\mathrm{(inter)}}_{j,0}[{\bf
u}])(G_{\mathcal{M}})\\\
=&\,\frac{1}{2|{\mathcal{A}}_{j}||\Gamma_{1}||\Gamma_{2}|}\Big{(}\frac{\rho_{\mathcal{M}}}{q^{(-1)^{j+1}1/2}}\Big{)}^{6}\sum_{\begin{subarray}{c}\alpha_{k}\in{\mathcal{A}}_{k}\\\
k=1,2\end{subarray}}\int_{{\mathbb{R}}^{2}}\mathrm{d}y\ e^{-iD_{j\to
3-j}^{-T}G_{\mathcal{M}}\cdot y}\\\
&|\nabla_{y}|^{6}\Big{(}v_{{\bm{\alpha}}}\big{(}y+g_{j,{\bm{\alpha}}}(y)\big{)}\Big{)}\\\
\leq&\,\frac{1}{2|{\mathcal{A}}_{j}||\Gamma_{1}||\Gamma_{2}|}\Big{(}\frac{\rho_{\mathcal{M}}}{q^{(-1)^{j+1}1/2}}\Big{)}^{6}\,\sum_{\begin{subarray}{c}\alpha_{k}\in{\mathcal{A}}_{k}\\\
k=1,2\end{subarray}}\|(-\Delta)^{3}v_{{\bm{\alpha}}}(\cdot+g_{j,{\bm{\alpha}}})\|_{1}\,.\end{split}$
(4.38)

For any $n\in{\mathbb{N}}$, define

$R(n)\,:=\,\\{{\bf
r}_{n}\in{\mathbb{N}}_{0}^{n}\mid\sum_{j=1}^{n}jr_{j}\,=\,n\\}\,.$ (4.39)

Using the Faá di Bruno formula and recalling (4.39), we have that

$\displaystyle\begin{split}&\|(-\Delta)^{3}v_{{\bm{\alpha}}}(\cdot+g_{j,{\bm{\alpha}}})\|_{1}\\\
\leq&\,6!\sum_{{\bf r}_{6}\in R(6)}\big{\|}\big{(}D^{|{\bf
r}_{6}|_{1}}v_{{\bm{\alpha}}}\big{)}(\cdot+g_{j,{\bm{\alpha}}})\big{\|}_{1}\\\
&\,\prod_{k=1}^{6}\frac{1}{r_{k}!}\bigg{(}\frac{\delta_{k,1}+\|D^{k}g_{j,{\bm{\alpha}}}\|_{\infty}}{k!}\bigg{)}^{r_{k}}\\\
\leq&\,6!\sum_{{\bf r}_{6}\in R(6)}\big{\|}\big{(}D^{|{\bf
r}_{6}|_{1}}v_{{\bm{\alpha}}}\big{)}(\cdot+g_{j,{\bm{\alpha}}})\big{\|}_{1}\\\
&\,\Big{(}1+\sum_{k=1}^{6}\|D^{k}g_{j,{\bm{\alpha}}}\|_{\infty}\Big{)}^{|{\bf
r}_{6}|_{1}}\prod_{k=1}^{6}\frac{1}{r_{k}!(k!)^{r_{k}}}\,.\end{split}$ (4.40)

Employing definition (4.39), we obtain that

$1\,\leq\,|{\bf
r}_{6}|_{1}\,=\,\sum_{k=1}^{6}r_{k}\,\leq\,\sum_{k=1}^{6}kr_{k}\,=\,6$ (4.41)

for all ${\bf r}_{6}\in R(6)$. Hence, (4.40) yields

$\displaystyle\|(-\Delta)^{3}v_{{\bm{\alpha}}}(\cdot+g_{{\bm{\alpha}}})\|_{1}\,\leq\,B_{6}\,\sum_{k=1}^{6}\big{\|}\big{(}D^{k}v_{{\bm{\alpha}}}\big{)}(\cdot+g_{j,{\bm{\alpha}}})\big{\|}_{1}\Big{(}1+\sum_{\ell=1}^{6}\|D^{\ell}g_{{\bm{\alpha}}}\|_{\infty}\Big{)}^{6}\,,$
(4.42)

where

$B_{6}\,:=\,6!\sum_{{\bf r}_{6}\in
R(6)}\prod_{k=1}^{6}\frac{1}{r_{k}!(k!)^{r_{k}}}\,=\,\partial_{\lambda}^{6}\Big{|}_{\lambda=0}e^{e^{\lambda}-1}\,=\,203$
(4.43)

denotes the 6th Bell number.

Recalling definition (4.37) of $g_{j,{\bm{\alpha}}}$ and employing Lemma 3.1,
we find that

$\displaystyle\begin{split}\|D^{\ell}g_{j,{\bm{\alpha}}}\|_{\infty}\,\leq&\,\|D^{\ell}u_{j}(D_{j\to
3-j}^{-1}(\cdot),\alpha_{j})\|_{\infty}\\\ &+\,\|D^{\ell}u_{3-j}((D_{j\to
3-j}^{-1}-I)(\cdot),\alpha_{3-j})\|_{\infty}\\\
\leq&\,\Big{(}\frac{q^{(-1)^{j+1}1/2}}{\rho_{\mathcal{M}}}\Big{)}^{\ell}\|D^{\ell}u_{1}(\cdot,\alpha_{1})\|_{\infty}\\\
&\,+\,\Big{(}\frac{q^{(-1)^{j}1/2}}{\rho_{\mathcal{M}}}\Big{)}^{\ell}\|D^{\ell}u_{2}(\cdot,\alpha_{2})\|_{\infty}\\\
\leq&\,\frac{q^{(-1)^{j+1}\ell/2}+q^{(-1)^{j}\ell/2}}{\rho_{\mathcal{M}}^{\ell}}\big{(}\|D^{\ell}u_{1}\|_{\infty}+\|D^{\ell}u_{2}\|_{\infty}\big{)}\,.\end{split}$
(4.44)

Lemma 3.5 implies

$\displaystyle\begin{split}&\big{\|}\big{(}D^{k}v_{{\bm{\alpha}}}\big{)}(\cdot+g_{j,{\bm{\alpha}}})\big{\|}_{1}\\\
\leq&\,\frac{5^{r-1}\pi}{r-1}\,\big{(}1+d_{{\mathcal{A}}_{1},{\mathcal{A}}_{2}}\,+\,\|u_{1}\|_{\infty}+\|u_{2}\|_{\infty}\big{)}^{2r}\\\
&\,\|\langle\cdot\rangle^{2r}\big{(}D^{k}v_{{\bm{\alpha}}}\big{)}\|_{L^{\infty}({\mathbb{R}}^{2}\times[-L_{{\bf
u}}^{z},L_{{\bf u}}^{z}])}\,.\end{split}$ (4.45)

Collecting (4.38), (4.40), (4.42), (4.43), (4.44), and (4.45), and recalling
the weighted Sobolev norm (3.16), we obtain

$\displaystyle\begin{split}&\sup_{G_{\mathcal{M}}}\Big{|}|G_{\mathcal{M}}|^{6}{\mathcal{F}}_{\Gamma_{\mathcal{M}}}(\Phi^{\mathrm{(inter)}}_{j,0}[{\bf
u}])(G_{\mathcal{M}})\Big{|}\,\\\
\leq&\,203\frac{1+q^{(-1)^{j}6}}{2|{\mathcal{A}}_{j}||\Gamma_{1}||\Gamma_{2}|}\frac{5^{r-1}\pi}{r-1}\\\
&\,\big{(}1+d_{{\mathcal{A}}_{1},{\mathcal{A}}_{2}}+\|u_{1}\|_{W^{6,\infty}}+\|u_{2}\|_{W^{6,\infty}}\big{)}^{6+2r}\\\
&\,\sum_{\begin{subarray}{c}\alpha_{k}\in{\mathcal{A}}_{k}\\\
k=1,2\end{subarray}}\|v_{{\bm{\alpha}}}\|_{W^{6,\infty}_{2r}({\mathbb{R}}^{2}\times[-L_{{\bf
u}}^{z},L_{{\bf u}}^{z}])}\,.\end{split}$ (4.46)

Together with (4.35), this concludes the proof. ∎

##  5 Proofs of lattice reduction formulae

###### Proof of Lemma 3.5.

Recalling definition 3.15 of $\langle\cdot\rangle$, we have that

$\displaystyle\begin{split}&\int_{{\mathbb{R}}^{2}}\mathrm{d}x\
|f(x+g_{j,{\bm{\alpha}},\gamma_{3-j}}(x))|\\\
&\leq\,\|\langle\cdot\rangle^{2r}f(\cdot+g_{j,{\bm{\alpha}},\gamma_{3-j}})\|_{L^{\infty}({\mathbb{R}}^{2})}\int_{{\mathbb{R}}^{2}}\mathrm{d}x\
\langle x\rangle^{-2r}\,.\end{split}$ (5.1)

Observe that

$\displaystyle\begin{split}\langle x+y\rangle^{2r}\,=&\,(1+|x+y|^{2})^{r}\\\
\leq&\,\big{(}1+2(|x|^{2}+|y|^{2})\big{)}^{r}\,,\end{split}$ (5.2)

by Cauchy-Schwarz and monotonicity of $t\mapsto t^{r}$. Using convexity of
$t\mapsto t^{r}$, we thus find that

$\displaystyle\begin{split}\langle
x+y\rangle^{2r}\,\leq&\,5^{r}\Big{(}\frac{1}{5}+\frac{2}{5}\big{(}|x|^{2}+|y|^{2}\big{)}\Big{)}^{r}\\\
\leq&2\cdot 5^{r-1}\big{(}1+|x|^{2r}+|y|^{2r}\big{)}\,.\end{split}$ (5.3)

Recall from (3.23) that

$d_{{\mathcal{A}}_{1},{\mathcal{A}}_{2}}\,=\,\max_{\begin{subarray}{c}\alpha_{j}\in{\mathcal{A}}_{j}\\\
j=1,2\end{subarray}}|\tau_{1}^{(\alpha_{1})}-\tau_{2}^{(\alpha_{2})}|\,.$
(5.4)

The definition (3.25) of $g_{j,{\bm{\alpha}},\gamma_{3-j}}$ thus implies

$\displaystyle\begin{split}&\|g_{j,{\bm{\alpha}},\gamma_{3-j}}\|_{\infty}\,\leq\,d_{{\mathcal{A}}_{1},{\mathcal{A}}_{2}}\,+\,\|u_{1}\|_{\infty}+\|u_{2}\|_{\infty}\,,\end{split}$
(5.5)

and that

$\displaystyle\begin{split}&\|(g_{j,{\bm{\alpha}},\gamma_{3-j}})_{z}\|_{\infty}\,\leq\,d_{{\mathcal{A}}_{1},{\mathcal{A}}_{2}}\,+\,\|u_{1,z}\|_{\infty}+\|u_{2,z}\|_{\infty}\,=\,L_{{\bf
u}}^{z}\,,\end{split}$ (5.6)

see definition (3.24) of $L_{{\bf u}}^{z}$. A straight-forward computation
yields

$\int_{{\mathbb{R}}^{2}}\mathrm{d}x\ \langle
x\rangle^{-2r}\,=\,\frac{\pi}{2(r-1)}\,.$ (5.7)

In addition, observe that for any $x\in\ell^{1}({\mathbb{Z}})$ we have that

$\|x\|_{\ell^{r}}\,\leq\,\|x\|_{\ell^{\infty}}^{\frac{r-1}{r}}\|x\|_{\ell^{1}}^{\frac{1}{r}}\,\leq\,\|x\|_{\ell^{1}}\,.$
(5.8)

Collecting (5.3), (5.5), (5.6), and (5.7), (5.1) implies

$\displaystyle\begin{split}&\int_{{\mathbb{R}}^{2}}\mathrm{d}x\
|f(x+g_{j,{\bm{\alpha}},\gamma_{3-j}}(x))|\\\
\leq&\,\frac{5^{r-1}\pi}{r-1}\Big{(}\|\big{(}1+|\cdot+g_{j,{\bm{\alpha}},\gamma_{3-j}}|^{2r}\big{)}f(\cdot+g_{j,{\bm{\alpha}},\gamma_{3-j}})\|_{L^{\infty}({\mathbb{R}}^{2})}\\\
&\,+\,\||g_{j,{\bm{\alpha}},\gamma_{3-j}}|^{2r}f(\cdot+g_{j,{\bm{\alpha}},\gamma_{3-j}})\|_{L^{\infty}({\mathbb{R}}^{2})}\Big{)}\\\
\leq&\,\frac{5^{r-1}\pi}{r-1}\,\big{(}1+d_{{\mathcal{A}}_{1},{\mathcal{A}}_{2}}\,+\,\|u_{1}\|_{\infty}+\|u_{2}\|_{\infty}\big{)}^{2r}\\\
&\,\|\langle\cdot\rangle^{2r}f\|_{L^{\infty}({\mathbb{R}}^{2}\times[-L_{{\bf
u}}^{z},L_{{\bf u}}^{z}])}\,,\end{split}$ (5.9)

where, in the last step, we applied (5.8). This concludes the proof. ∎

###### Proof of Lemma 3.6.

Assume that $v_{{\bm{\alpha}}}$ and $u_{j}$ are simple functions.
Interchanging integration and summation, we then have that

$\displaystyle\begin{split}&{\mathcal{F}}_{\Gamma_{\mathcal{M}}}(\Phi^{\mathrm{(inter)}}_{j,{\bm{\gamma}}}[{\bf
u}])(G_{\mathcal{M}})\,=\\\
&\,\frac{1}{2|{\mathcal{A}}_{j}||\Gamma_{j}|}\sum_{\begin{subarray}{c}\alpha_{k}\in{\mathcal{A}}_{k}\\\
k=1,2\end{subarray}}\sum_{R_{3-j}\in{\mathcal{R}}_{3-j}}\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
e^{-iG_{\mathcal{M}}\cdot x}v_{{\bm{\alpha}}}\big{(}D_{j\to 3-j}x-R_{3-j}\\\
&+A_{3-j}A_{j}^{-1}\gamma_{j}-\gamma_{3-j}+\tau_{j}^{(\alpha_{j})}-\tau_{3-j}^{(\alpha_{3-j})}\,+\,u_{j}(x,\alpha_{j})\\\
&-\,u_{3-j}(R_{3-j}+A_{3-j}A_{j}^{-1}\gamma_{j}+\gamma_{3-j}+(I-D_{j\to
3-j})x,\alpha_{3-j})\big{)}\,.\end{split}$ (5.10)

Substituting $y:=D_{j\to
3-j}x+A_{3-j}A_{j}^{-1}\gamma_{j}-R_{3-j}-\gamma_{3-j}$ and interchanging
integration and summation again, we obtain

$\displaystyle\begin{split}&{\mathcal{F}}_{\Gamma_{\mathcal{M}}}(\Phi^{\mathrm{(inter)}}_{j,{\bm{\gamma}}}[{\bf
u}])(G_{\mathcal{M}})\,=\\\ &\frac{|\det(D_{j\to
3-j}^{-1})|}{2|{\mathcal{A}}_{j}||\Gamma_{\mathcal{M}}||\Gamma_{j}|}\sum_{\begin{subarray}{c}\alpha_{k}\in{\mathcal{A}}_{k}\\\
k=1,2\end{subarray}}\int_{{\mathbb{R}}^{2}}\mathrm{d}y\
\sum_{R_{3-j}\in{\mathcal{R}}_{3-j}}e^{-iG_{\mathcal{M}}\cdot D_{j\to
3-j}^{-1}(y-A_{3-j}A_{j}^{-1}\gamma_{j}+R_{3-j}+\gamma_{3-j})}\\\
&\mathds{1}_{D_{j\to
3-j}\Gamma_{\mathcal{M}}+A_{3-j}A_{j}^{-1}\gamma_{j}-R_{3-j}-\gamma_{3-j}}(y)\,v_{{\bm{\alpha}}}\Big{(}y+\tau_{j}^{(\alpha_{j})}-\tau_{3-j}^{(\alpha_{3-j})}\\\
&+\,u_{j}(D_{j\to
3-j}^{-1}(y-A_{3-j}A_{j}^{-1}\gamma_{j}+R_{3-j}+\gamma_{3-j}),\alpha_{j})\\\
&\,-\,u_{3-j}(D_{j\to
3-j}^{-1}(y-A_{3-j}A_{j}^{-1}\gamma_{j}+R_{3-j}+\gamma_{3-j})-y,\alpha_{3-j})\Big{)}\,.\end{split}$
(5.11)

Observe that, by (2.45), (2.46), (2.3), and (2.11),

$|\det(D_{j\to
3-j})|\,=\,\frac{|\det(A_{3-j})|}{|\det(A_{\mathcal{M}})|}\,=\,\frac{|\Gamma_{3-j}|}{|\Gamma_{\mathcal{M}}|}\,.$
(5.12)

In addition, by (2.45), (2.46), and (2.6), we have that

$D_{j\to
3-j}^{-T}{\mathcal{R}}^{*}_{\mathcal{M}}\,=\,{\mathcal{R}}_{3-j}^{*}\,.$
(5.13)

Furthermore, Lemma 3.2 implies that

$D_{j\to 3-j}\Gamma_{\mathcal{M}}\,=\,(-1)^{j}\Gamma_{3-j}\,,$ (5.14)

and that

$D_{j\to 3-j}^{-1}{\mathcal{R}}_{3-j}\,=\,{\mathcal{R}}_{\mathcal{M}}\,.$
(5.15)

Collecting (5.12), (5.13), (5.14), (5.15), and using the facts that

$\bigcup_{R_{3-j}\in{\mathcal{R}}_{3-j}}\big{(}(-1)^{j}\Gamma_{3-j}-R_{3-j}-\gamma_{3-j}\big{)}\,=\,{\mathbb{R}}^{2}\,,$
(5.16)

and that $u_{k}(\cdot,\alpha_{k})$ is ${\mathcal{R}}_{\mathcal{M}}$-periodic,
(5.11) yields

$\displaystyle\begin{split}&{\mathcal{F}}_{\Gamma_{\mathcal{M}}}(\Phi^{\mathrm{(inter)}}_{j,{\bm{\gamma}}}[{\bf
u}])(G_{\mathcal{M}})\\\
=&\frac{1}{2|{\mathcal{A}}_{j}||\Gamma_{1}||\Gamma_{2}|}\sum_{\begin{subarray}{c}\alpha_{k}\in{\mathcal{A}}_{k}\\\
k=1,2\end{subarray}}\int_{{\mathbb{R}}^{2}}\mathrm{d}y\
e^{-iG_{\mathcal{M}}\cdot D_{j\to
3-j}^{-1}(y-A_{3-j}A_{j}^{-1}\gamma_{j}+\gamma_{3-j})}\\\
&v_{{\bm{\alpha}}}\Big{(}y+\tau_{j}^{(\alpha_{j})}-\tau_{3-j}^{(\alpha_{3-j})}\\\
&+\,u_{j}(D_{j\to
3-j}^{-1}(y-A_{3-j}A_{j}^{-1}\gamma_{j}+\gamma_{3-j}),\alpha_{j})\\\
&\,-\,u_{3-j}(D_{j\to
3-j}^{-1}(y-A_{3-j}A_{j}^{-1}\gamma_{j}+\gamma_{3-j})-y,\alpha_{3-j})\Big{)}\,.\end{split}$
(5.17)

Using the fact that

$A_{j}-A_{3-j}\,=\,D_{j\to
3-j}A_{j}\,=\,(-1)^{j}A_{3-j}A_{\mathcal{M}}^{-1}A_{j}\,,$ (5.18)

see Lemma 3.1, and substituting $y=(A_{j}-A_{3-j})x+\gamma_{j}-\gamma_{3-j}$,
we obtain

$\displaystyle\begin{split}&{\mathcal{F}}_{\Gamma_{\mathcal{M}}}(\Phi^{\mathrm{(inter)}}_{j,{\bm{\gamma}}}[{\bf
u}])(G_{\mathcal{M}})\\\
=&\,\frac{1}{2|{\mathcal{A}}_{j}||\Gamma_{\mathcal{M}}|}\sum_{\begin{subarray}{c}\alpha_{k}\in{\mathcal{A}}_{k}\\\
k=1,2\end{subarray}}\int_{{\mathbb{R}}^{2}}\mathrm{d}x\
e^{-iG_{\mathcal{M}}\cdot(A_{j}x+\gamma_{j})}v_{{\bm{\alpha}}}\big{(}A_{1}x+\gamma_{1}+\tau_{1}^{(\alpha_{1})}\\\
&+u_{1}(A_{1}x+\gamma_{1},\alpha_{1})-A_{2}x-\gamma_{2}-\tau_{2}^{(\alpha_{2})}-u_{2}(A_{2}x+\gamma_{2},\alpha_{2})\big{)}\,,\end{split}$
(5.19)

where we also employed that $v_{{\bm{\alpha}}}$ is even. Due to Lemma 3.5, we
can extend (5.17), and thus (5.19), to all

$v_{{\bm{\alpha}}}\in L^{\infty}_{2r}({\mathbb{R}}^{2}\times[-L_{{\bf
u}}^{z},L_{{\bf u}}^{z}])\,.$ (5.20)

This finishes the proof. ∎

###### Proof of Lemma 3.7.

The idea of this proof follows the steps in the proofs of the two previous
Lemmata. As above, assume that $v_{{\bm{\alpha}}}$ and $u_{j}$ are simple
functions.

Let

$\psi(x,y)\,:=\,\frac{1}{2|{\mathcal{A}}_{j}||\Gamma_{j}|}\sum_{\begin{subarray}{c}\alpha_{k}\in{\mathcal{A}}_{k}\\\
k=1,2\end{subarray}}v_{{\bm{\alpha}}}\big{(}y+h_{j,{\bm{\alpha}}}(x,y)\big{)}\,.$
(5.21)

Then we have that

$\displaystyle\begin{split}&\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
\int_{\Gamma_{3-j}}\mathrm{d}y\ |\Psi^{\mathrm{(inter)}}_{j}[{\bf
u}](x,y)|\,\leq\\\
&\sum_{R_{3-j}\in{\mathcal{R}}_{3-j}}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
\int_{\Gamma_{3-j}}\mathrm{d}y\ |\psi(x-R_{3-j},y)|\,.\end{split}$ (5.22)

Shifting $x-R_{3-j}\to x$, we obtain

$\displaystyle\begin{split}&\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
\int_{\Gamma_{3-j}}\mathrm{d}y\ |\Psi^{\mathrm{(inter)}}_{j}[{\bf
u}](x,y)|\,\leq\\\
&\sum_{R_{3-j}\in{\mathcal{R}}_{3-j}}\int_{\Gamma_{3-j}-R_{3-j}}\mathrm{d}x\
\int_{\Gamma_{\mathcal{M}}}\mathrm{d}y\ |\psi(x,y)|\,=\\\
&\int_{{\mathbb{R}}^{2}}\mathrm{d}x\ \int_{\Gamma_{\mathcal{M}}}\mathrm{d}y\
|\psi(x,y)|\,,\end{split}$ (5.23)

where, again, we used the fact that

$\bigcup_{R_{3-j}\in{\mathcal{R}}_{3-j}}\big{(}\Gamma_{3-j}-R_{3-j}\big{)}\,=\,{\mathbb{R}}^{2}\,.$
(5.24)

Due to

$\int_{{\mathbb{R}}^{2}}\mathrm{d}x\ \langle x\rangle^{-2r}\,<\,\infty$ (5.25)

for $r>1$, (5.23) implies

$\displaystyle\begin{split}&\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
\int_{\Gamma_{3-j}}\mathrm{d}y\ |\Psi^{\mathrm{(inter)}}_{j}[{\bf
u}](x,y)|\,\lesssim\\\ &\|\langle
x\rangle^{2r}\psi(x,y)\|_{L^{\infty}_{x,y}}\,.\end{split}$ (5.26)

With analogous steps as in the proof of Lemma 3.5, we have that

$\displaystyle\begin{split}\|\langle
x\rangle^{2r}\psi(x,y)\|_{L^{\infty}_{x,y}}\,\lesssim&\,(1+d_{{\mathcal{A}}_{1},{\mathcal{A}}_{2}}+\|u_{1}\|_{\infty}+\|u_{2}\|_{\infty})^{2r}\\\
&\sum_{\begin{subarray}{c}\alpha_{k}\in{\mathcal{A}}_{k}\\\
k=1,2\end{subarray}}\|\langle\cdot\rangle^{2r}v_{{\bm{\alpha}}}\|_{L^{\infty}({\mathbb{R}}^{2}\times[-L_{{\bf
u}}^{z},L_{{\bf u}}^{z}])}\,<\,\infty\end{split}$ (5.27)

by assumption on $v_{{\bm{\alpha}}}$. This concludes the proof. ∎

##  Appendix A Lattice calculus

###### Proof of Lemma 2.1.

By definitions (2.7) and (2.1), we have that

$\displaystyle
A_{\mathcal{M}}\,=\,(A_{1}^{-1}-A_{2}^{-1})^{-1}\,=\,(q^{1/2}R_{\theta/2}-q^{-1/2}R_{-\theta/2})^{-1}A\,.$
(A.1)

Consequently, we obtain that

$\displaystyle
B_{\mathcal{M}}\,=\,(q^{1/2}R_{-\theta/2}-q^{-1/2}R_{\theta/2})2\pi A^{-T}\,.$
(A.2)

Observe that

$q^{1/2}R_{\theta/2}-q^{-1/2}R_{-\theta/2}\,=\,\begin{pmatrix}(q^{1/2}-q^{-1/2})\cos(\theta/2)&-(q^{1/2}+q^{-1/2})\sin(\theta/2)\\\
(q^{1/2}+q^{-1/2})\sin(\theta/2)&(q^{1/2}-q^{-1/2})\cos(\theta/2)\end{pmatrix}\,.$
(A.3)

A straight-forward calculation yields

$\displaystyle|(q^{1/2}R_{\theta/2}-q^{-1/2}R_{-\theta/2})x|^{2}\,=\,[(q^{1/2}-q^{-1/2})^{2}\,+\,4\sin^{2}(\theta/2)]|x|^{2}\,.$
(A.4)

Consequently, we have that

$|(q^{1/2}R_{\theta/2}-q^{-1/2}R_{-\theta/2})^{-1}x|\,=\,[(q^{1/2}-q^{-1/2})^{2}\,+\,4\sin^{2}(\theta/2)]^{-\frac{1}{2}}|x|\,.$
(A.5)

Together with (A.1) resp. (A.2), this concludes the proof. ∎

###### Proof of Proposition 2.9.

By Proposition 3.8, let
$({\mathcal{N}}_{G_{\mathcal{M}}}))_{G_{\mathcal{M}}\in{\mathcal{R}}^{*}_{\mathcal{M}}}$
be a sequence of nullsets such that for all
$G_{\mathcal{M}}\in{\mathcal{R}}^{*}_{\mathcal{M}}$ and all
$\gamma_{j}\in\Gamma_{j}\setminus{\mathcal{N}}_{G_{\mathcal{M}}}$

$\displaystyle\lim_{N\to\infty}\frac{1}{(2N+1)^{2}}\sum_{R_{j}\in{\mathcal{R}}_{j}(N)}e^{-iG_{\mathcal{M}}\cdot(\gamma_{j}+R_{j})}u_{j}(\gamma_{j}+R_{j},\alpha_{j})$
(A.6)
$\displaystyle=\,\,\ThisStyle{\ensurestackMath{\stackinset{c}{.2\LMpt}{c}{.5\LMpt}{\SavedStyle-}{\SavedStyle\phantom{\int}}}\kern-16.56937pt}\int_{\Gamma_{\mathcal{M}}}\mathrm{d}x\
e^{-iG_{\mathcal{M}}\cdot
x}u_{j}(x,\alpha_{j})\,=\,\hat{u}_{j}(G_{\mathcal{M}},\alpha_{j})\,,$ (A.7)

where we used the fact that

$e^{-iG_{\mathcal{M}}\cdot
x}\,=\,e^{-iG_{\mathcal{M}}\cdot\left\\{x\right\\}_{\mathcal{M}}}\,.$ (A.8)

Then

$\displaystyle{\mathcal{N}}\,:=\,\bigcup_{G_{\mathcal{M}}\in{\mathcal{R}}^{*}_{\mathcal{M}}}{\mathcal{N}}_{G_{\mathcal{M}}}$
(A.9)

is a nullset, and for all $\gamma_{j}\in\Gamma_{j}\setminus{\mathcal{N}}$, we
have that

$\displaystyle\lim_{N\to\infty}\frac{1}{(2N+1)^{2}}\sum_{R_{j}\in{\mathcal{R}}_{j}(N)}e^{-iG_{\mathcal{M}}\cdot(\gamma_{j}+R_{j})}u_{j}(\gamma_{j}+R_{j},\alpha_{j})\,=\,\hat{u}_{j}(G_{\mathcal{M}},\alpha_{j})\,.$
(A.10)

This concludes the proof. ∎

######  Lemma A.1.

Let $s>1$. Then we have that

$\sum_{G_{\mathcal{M}}\in{\mathcal{R}}^{*}_{\mathcal{M}}\setminus\\{0\\}}\frac{1}{|G_{\mathcal{M}}|^{2s}}\,\leq\,4\Big{(}\frac{\rho_{\mathcal{M}}\|A\|_{2}}{2\pi}\Big{)}^{2s}\Big{(}\frac{2s}{2s-1}\,+\,2^{-s}\big{(}\frac{s}{s-1}\big{)}^{2}\Big{)}\,.$
(A.11)

###### Proof.

Lemma 2.1 yields

$\displaystyle|B_{\mathcal{M}}n|^{2}\,$
$\displaystyle=\,(2\pi)^{2}\rho_{\mathcal{M}}^{-2}|A^{-T}n|^{2}$
$\displaystyle\geq\,(2\pi)^{2}\rho_{\mathcal{M}}^{-2}\|A\|_{2}^{-2}|n|^{2}$
(A.12a)
$\displaystyle\geq\,2\cdot(2\pi)^{2}\rho_{\mathcal{M}}^{-2}\|A\|_{2}^{-2}|n_{1}|\cdot|n_{2}|\,,$
(A.12b)

where, in the last step, we applied the geometric-quadratic-mean inequality.
We split the sum over ${\mathcal{R}}^{*}_{\mathcal{M}}\setminus\\{0\\}$ into

$\\{B_{\mathcal{M}}n\mid
n_{1}n_{2}=0,(n_{1},n_{2})\neq(0,0)\\}\dot{\cup}\\{B_{\mathcal{M}}n\mid
n_{1},n_{2}\neq 0\\}\,.$ (A.13)

Applying (A.12a) on $G_{\mathcal{M}}$ in the first set and (A.12b) on the
latter, we obtain

$\displaystyle\sum_{G_{\mathcal{M}}\in{\mathcal{R}}^{*}_{\mathcal{M}}\setminus\\{0\\}}\frac{1}{|G_{\mathcal{M}}|^{2s}}\,$
$\displaystyle\leq\,\Big{(}\frac{\rho_{\mathcal{M}}\|A\|_{2}}{2\pi}\Big{)}^{2s}\Big{[}4\sum_{n\in{\mathbb{N}}}\frac{1}{n^{2s}}\,+\,2^{-s}\Big{(}2\sum_{n\in{\mathbb{N}}}\frac{1}{n^{s}}\Big{)}^{2}\Big{]}\,.$
(A.14)

In order to estimate the sums, we use the fact that, for any $\sigma>1$

$\displaystyle\sum_{n\in{\mathbb{N}}}\frac{1}{n^{\sigma}}\,$
$\displaystyle\leq\,1\,+\,\sum_{n=2}^{\infty}\int_{n-1}^{n}\frac{dx}{x^{\sigma}}$
$\displaystyle=\,1\,+\,\frac{1}{\sigma-1}$
$\displaystyle=\,\frac{\sigma}{\sigma-1}\,.$ (A.15)

Plugging (A.15) into (A.14), we obtain

$\displaystyle\sum_{G_{\mathcal{M}}\in{\mathcal{R}}^{*}_{\mathcal{M}}\setminus\\{0\\}}\frac{1}{|G_{\mathcal{M}}|^{2s}}\,$
$\displaystyle\leq\,4\Big{(}\frac{\rho_{\mathcal{M}}\|A\|_{2}}{2\pi}\Big{)}^{2s}\Big{(}\frac{2s}{2s-1}\,+\,2^{-s}\big{(}\frac{s}{s-1}\big{)}^{2}\Big{)}\,.$
(A.16)

This concludes the proof. ∎

##  Appendix B Existence of Diophantine 2D rotations

We now prove Proposition 2.5. It suffices to show that for almost every
$\theta\in{\mathbb{R}}$ and all $\sigma>\frac{1433}{1248}$, there exists $K>0$
such that for all $n\in{\mathbb{Z}}^{2}\setminus\\{0\\}$

$\operatorname{dist}\big{(}qA^{T}R_{\theta}A^{-T}n,{\mathbb{Z}}^{2}\big{)}\geq\frac{K}{|n|^{2\sigma}}$
(B.1)

holds.

Due to $2\pi$-periodicity of $R_{\theta}$, it suffices to show that Lebesgue-
almost every $\theta\in[0,2\pi)$ is a Diophantine 2D rotation. We follow the
classical strategy to apply the Borel-Cantelli Lemma to the complement. In
particular, let

$\Omega\,:=\,\bigcap_{\sigma>\frac{1433}{1248}}\bigcup_{\begin{subarray}{c}K>0,\\\
N\in{\mathbb{N}}\end{subarray}}\bigcap_{\begin{subarray}{c}n\in{\mathbb{Z}}^{2}:\\\
|n|^{2}\geq
N\end{subarray}}\bigcap_{m\in{\mathbb{Z}}^{2}}\Big{\\{}\theta\in[0,2\pi)\mid|qA^{T}R_{\theta}A^{-T}n-m|\geq\frac{K}{|n|^{2\sigma}}\Big{\\}}\,.$
(B.2)

$\Omega$ consists of all $\theta\in[0,2\pi)$ such that for all
$\sigma>\frac{1433}{1248}$ there exists $K>0$ and $N\in{\mathbb{N}}$ s.t. for
all $n\in{\mathbb{Z}}^{2}$ with $|n|^{2}\geq N$ we have that

$\operatorname{dist}\big{(}qA^{T}R_{\theta}A^{-T}n,{\mathbb{Z}}^{2}\big{)}\geq\frac{K}{|n|^{2\sigma}}\,.$
(B.3)

Taking
$\tilde{K}:=\min\big{\\{}K,\min_{|n|^{2}<N}\operatorname{dist}\big{(}qA^{T}R_{\theta}A^{-T}n,{\mathbb{Z}}^{2}\big{)}\big{\\}}$,
we can extend (B.3) to hold for all $n\in{\mathbb{Z}}^{2}\setminus\\{0\\}$ by
lowering $K$ to $\tilde{K}$. Our goal is to show that $|\Omega|\,=\,2\pi$.

Let

$E_{N}(\sigma)\,:=\,\bigcap_{K>0}\bigcup_{\begin{subarray}{c}n\in{\mathbb{Z}}^{2}\\\
|n|^{2}=N\end{subarray}}\bigcup_{m\in{\mathbb{Z}}^{2}}\Big{\\{}\theta\in[0,2\pi)\mid|qA^{T}R_{\theta}A^{-T}n-m|<\frac{K}{|n|^{2\sigma}}\Big{\\}}\,.$
(B.4)

Since $A$ is invertible, we have that $|A^{T}\cdot|$ and $|\cdot|$ are
equivalent norms. More precisely, we have that

$\|A^{T}\|_{2}^{-1}|A^{T}v|\,\leq\,|v|\,\leq\,\|A^{-T}\|_{2}|A^{T}v|\,,$ (B.5)

where $\|\cdot\|_{2}$ denotes the spectral/Hilbert-Schmidt norm. In
particular, we have that

$E_{N}(\sigma)\,\subseteq\,\bigcup_{\begin{subarray}{c}n\in{\mathbb{Z}}^{2}\\\
|n|^{2}=N\end{subarray}}\bigcup_{m\in{\mathbb{Z}}^{2}}\Big{\\{}\theta\in[0,2\pi)\mid|qR_{\theta}A^{-T}n-A^{-T}m|<\frac{1}{|n|^{2\sigma}}\Big{\\}}\,,$
(B.6)

In order to apply Borel-Cantelli, we need to control the asymptotic behavior
of the Lebesgue measures of the sets $E_{N}$ in order to study their
summability. We have that

$\displaystyle\begin{split}\@ADDCLASS{ltx_eqn_lefteqn}$\displaystyle|E_{N}(\sigma)|$\mbox{}\hfil\\\
&\leq\,\sum_{\begin{subarray}{c}n\in{\mathbb{Z}}^{2}\\\
|n|^{2}=N\end{subarray}}\sum_{\begin{subarray}{c}m\in{\mathbb{Z}}^{2}\\\
\big{|}|A^{-T}m|-|qA^{-T}n|\big{|}<\frac{1}{|n|^{2\sigma}}\end{subarray}}\int_{0}^{2\pi}d\theta\,\mathds{1}_{B_{\frac{1}{|n|^{2\sigma}}}(A^{-T}m)}\big{(}qR_{\theta}A^{-T}n\big{)}\,.\end{split}$
(B.7)

Observe that for large $|n|$, we have that
$|A^{-T}m|\approx_{\sigma}|qA^{-T}n|$. In particular, the integrand vanishes
unless $\theta$ is on an arc of length $\sim|n|^{-2\sigma}$ for a circle of
radius $|qA^{-T}n|\sim|n|$, see (B.5), about the angle defined by $A^{-T}n$.
In particular, we obtain that

$\int_{0}^{2\pi}d\theta\,\mathds{1}_{B_{\frac{1}{|n|^{2\sigma}}}(A^{-T}m)}\big{(}R_{\theta}A^{-T}n\big{)}\,\lesssim_{q,\sigma,A}\frac{1}{|n|^{1+2\sigma}}$
(B.8)

for sufficiently large $n$. In addition, for $r>0$, Levitan [60] obtained the
bound

$|B_{r}\cap qA^{-T}{\mathbb{Z}}^{2}|\,=\,\frac{\pi
r^{2}}{|\det(qA^{-T})|}\,+\,O_{r\to\infty}(r^{\frac{2}{3}})\,,$ (B.9)

see also [58] for a related works, and also [44]. As a consequence, we find
that

$\displaystyle\begin{split}&\sum_{m\in{\mathbb{Z}}^{2}}\mathds{1}_{B_{|n|^{-2\sigma}}(qA^{-T}n)}(A^{-T}m)\\\
&\leq\,\Big{|}\Big{(}B_{|qA^{-T}n|+\frac{1}{|n|^{2\sigma}}}\setminus
B_{|qA^{-T}n|-\frac{1}{|n|^{2\sigma}}}\Big{)}\cap
A^{-T}{\mathbb{Z}}^{2}\Big{|}\\\
&=\,\big{(}|qA^{-T}n|+\frac{1}{|n|^{2\sigma}}\big{)}^{2}-\big{(}|qA^{-T}n|-\frac{1}{|n|^{2\sigma}}\big{)}^{2}+O_{|n|\to\infty}(|n|^{\frac{2}{3}})\\\
&\lesssim_{q,\sigma,A}\,|n|^{\frac{2}{3}}\,,\end{split}$ (B.10)

for $2\sigma>1$. In addition, Huxley [48] showed that

$\displaystyle|B_{r}\cap{\mathbb{Z}}^{2}|\,=\,\pi
r^{2}\,+\,O_{r\to\infty}(r^{\frac{131}{208}}\log(r)^{\frac{18627}{8320}})\,,$
(B.11)

see also [11, 18, 57] for recent progress. Consequently, we obtain that

$\displaystyle\begin{split}|\\{n\in{\mathbb{Z}}^{2}\mid|n|^{2}=N\\}|\,&\leq\,\big{|}\big{(}(B_{\sqrt{N}+\delta}\setminus
B_{\sqrt{N}-\delta})\cap{\mathbb{Z}}^{2}\big{)}\big{|}\\\ &\lesssim
N^{\frac{131}{416}+{\varepsilon}}\end{split}$ (B.12)

for any $\delta,{\varepsilon}>0$.

Collecting (B.8), (B.10), and (B.12), (B.7) yields

$|E_{N}(\sigma)|\,\lesssim_{\sigma,A}\,N^{\frac{131}{416}+\frac{1}{3}-\frac{1}{2}-\sigma+{\varepsilon}}\,=\,N^{\frac{185}{1248}-\sigma+{\varepsilon}}$
(B.13)

for any $N\in{\mathbb{N}}$ large enough. In particular, choosing any

$\sigma\,>\,\frac{185}{1248}\,+\,1\,=\,\frac{1433}{1248}$ (B.14)

and any $0<{\varepsilon}<\sigma-\frac{1433}{1248}$, we obtain that

$\sum_{N\in{\mathbb{N}}}|E_{N}(\sigma)|\,<\,\infty\,.$ (B.15)

Then the Borel-Cantelli Lemma yields that

$\Big{|}\bigcap_{N_{0}=1}^{\infty}\bigcup_{N=N_{0}}^{\infty}E_{N}(\sigma)\Big{|}\,=\,0\,.$
(B.16)

Observe that $E_{N}(\sigma)$ is decreasing in $\sigma$, i.e.,

$E_{N}(\sigma^{\prime})\,\subseteq E_{N}(\sigma)$ (B.17)

for $\sigma\leq\sigma^{\prime}$. Thus, and using (B.16), we obtain that

$\begin{split}\Big{|}\bigcup_{\sigma>\frac{1433}{624}}\bigcap_{N_{0}=1}^{\infty}\bigcup_{N=N_{0}}^{\infty}E_{N}(\sigma)\Big{|}\,&\leq\,\sum_{\ell=1}^{\infty}\Big{|}\bigcap_{N_{0}=1}^{\infty}\bigcup_{N=N_{0}}^{\infty}E_{N}\big{(}\frac{1433}{1248}+\frac{1}{\ell}\big{)}\Big{|}\\\
&=\,0\,.\end{split}$ (B.18)

Using the fact that

$\displaystyle\Omega\,\supseteq\,\Big{(}\bigcup_{\sigma>\frac{1433}{1248}}\bigcap_{N_{0}=1}^{\infty}\bigcup_{N=N_{0}}^{\infty}E_{N}(\sigma)\Big{)}^{c}\,,$
(B.19)

we finish the proof.

##  Appendix C Proofs of ergodic theorems

###### Proof of Proposition 3.10.

Proposition 3.8 implies almost sure pointwise convergence. For both, almost
sure and $L^{1}$-convergence, imply convergence in probability, it suffices to
compute the $L^{1}$-limit. Using standard approximation arguments, we may
assume that $h$ is a $C^{\infty}$-smooth function. By dominated convergence,
it suffices the almost sure pointwise limit.

W.l.o.g. we set $j=1$. (3.46) implies that

$\displaystyle\begin{split}&{\mathcal{A}}_{{\mathcal{R}}_{1}(N)}(h)(\omega_{\mathcal{M}},\omega_{2})\\\
&=\,\sum_{\begin{subarray}{c}G_{2}\in{\mathcal{R}}_{2}^{*},\\\
G_{\mathcal{M}}\in{\mathcal{R}}^{*}_{\mathcal{M}}\end{subarray}}\mathds{1}_{{\mathcal{R}}_{1}^{*}}^{(N)}(G_{2}+G_{\mathcal{M}})\,\hat{h}(G_{\mathcal{M}},G_{2})e^{i[G_{\mathcal{M}}\cdot\omega_{\mathcal{M}}+G_{2}\cdot\omega_{2}]}\end{split}$
(C.1)

Using dominated convergence, we obtain that

$\displaystyle\begin{split}&\lim_{N\to\infty}{\mathcal{A}}_{{\mathcal{R}}_{1}(N)}(h)(\omega_{\mathcal{M}},\omega_{2})\\\
&=\,\sum_{\begin{subarray}{c}G_{2}\in{\mathcal{R}}_{2}^{*},\\\