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\def\otto{\,{\kern-1.truept\leftarrow\kern-5.truept\to\kern-1.truept}\,} \def\defin{{\buildrel def\over=}} \def\wt{\widetilde} \def\wh{\widehat} \def\to{\rightarrow} \def\la{\left\langle} \def\ra{\right\rangle} \def\qed{\hfill\raise1pt\hbox{\vrule height5pt width5pt depth0pt}} \def\Val{{\rm Val}} \def\ul#1{{\underline#1}} \def\lis{\overline} \def\V#1{{\bf#1}} \def\be{\begin{equation}} \def\ee{\end{equation}} \def\bp{\begin{pmatrix}} \def\ep{\end{pmatrix}} \def\bea{\begin{eqnarray}} \def\eea{\end{eqnarray}} \def\nn{\nonumber} \def\pref#1{(\ref{#1})} \def\ie{{\it i.e.}} \def\lb{\label} \def\eg{{\it e.g.}} \def\Tr{\mathrm{Tr}} \def\eu{\mathrm{e}} \newtheorem{lemma}{Lemma}[section] \newtheorem{remark}{Remark}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{cor}{Corollary}[section] \newtheorem{oss}{Remark} \begin{document} \title{Incommensurate Twisted Bilayer Graphene: emerging quasi-periodicity and stability} \author{Ian Jauslin} \affiliation{Rutgers University, Department of Mathematics, New Brunswick, USA} \email{ian.jauslin@rutgers.edu} \author{Vieri Mastropietro} \affiliation{Università di Roma ``La Sapienza'', Department of Physics, Rome, Italy } \email{vieri.mastropietro@uniroma1.it} \begin{abstract} We consider a lattice model of Twisted Bilayer Graphene (TBG). The presence of incommensurate angles produces an emerging quasi-periodicity manifesting itself in large momenta Umklapp interactions that almost connect the Dirac points. We rigorously establish the stability of the semimetallic phase via a Renormalization Group analysis combined with number theoretical properties of irrationals, similar to the ones used in Kolmogorov-Arnold-Moser (KAM) theory for the stability of invariant tori. The interlayer hopping is weak and short ranged and the angles are chosen in a large measure set. The result provides a justification, in the above regime, to the effective continuum description of TBG in which large momenta interlayer interactions are neglected. \end{abstract} \maketitle \section{Introduction} The discovery that at certain angles Twisted Bilayer Graphene (TBG) develops superconductivity \cite{a} has generated much interest in such materials both for technological and theoretical reasons \cite{b}. It was predicted, using continuum models obtained by keeping only the dominant harmonics in the lattice model \cite{1}, \cite{2},\cite{3}, that at such angles some strongly correlated behaviour should appear, but not of superconducting type. The mechanism behind the superconductivity remains elusive. Taking the lattice into account breaks several symmetries of the continuum description \cite{1aa1}, \cite{1aa2} leading to effects like the possible shift of Fermi points. More interestingly, for generic angles, excluding a special set \cite{1bb}, \cite{1aaa}, one has an incommensurate structure; in such a case Bloch band theory does not apply and one has an emergent quasi-periodicity \cite{H1}-\cite{H5}, with some feature in common with the one in fermions with quasi-periodic potentials \cite{P0}-\cite{P2}. It is known that electronic quasi-periodic systems have remarkable properties. In 1d they produce a metal-insulator transition \cite{Au},\cite{DS},\cite{FS}. The interplay with a many body interaction produces peculiar phases with anomalous gaps or many body localization \cite{Q0}--\cite{M11}. Quasi-periodicity has been studied also in Weyl semimetals \cite{P4}, \cite{W0},\cite{W1} or in the 2d Ising model \cite{Lu}, \cite{Ca}, \cite{Ga}. It is therefore natural to expect that quasi-periodicity plays an important role in the interacting phases of TBG. Most theoretical analyis are however based on continuum effective description, do not distinguish between commensurate and incommensurate angles, and are based on the assumption that lattice effects preserve the semimetallic phase \cite{1}, \cite{2}, \cite{3}. We consider a lattice model for TBG consisting of two graphene layers one on top of the other and rotated by an angle $\th$. The momenta involved in the two-particle scattering process are of the form $ k_1-k_2+G+G'=0$ with $G=l_1b_1+l_2b_2\equiv lb$, $G'=m_1b'_1+m_2b_2'\equiv mb'$, $l\equiv(l_1,l_2)\in\ZZZ^2$, $m\equiv(m_1,m_2)\in \ZZZ^2$, and $b_1,b_2$ are the vectors of the reciprocal lattice and $b'_i=R^T(\th) b_i$ the reciprocal lattice of the twisted layer in which $R(\th)$ is the rotation matrix; the terms involving non zero $G,G'$ are also known as Umklapp interactions. Note that, apart from special angles, $G'$ is not commensurate with $G$ and the effect of the mismatch of the lattices is very similar to the effect of a quasi-periodic potential. This is quite clear comparing for instance with the conservation law of 1d fermions with Aubry-Andr\'e potential $\cos 2\pi \o x$, which is $k_1-k_1+2l\pi+2\pi \o m=0$ with $\o$ irrational. It is expected that the relevant processes in TBG are the ones connecting the Dirac points as closely as possible, that is the terms that minimize the quantity $| G+G'+ p_{F,i}-p'_{F,j}|$ where $p_{F,i}$ $p'_{F,j}$ are the Dirac points of the two layers. The approximation at the basis of the effective models \cite{1}, \cite{2},\cite{3} consists in taking restricting the interaction to only the terms $G=G'=0$ or $G=b_1, G'=-b'_1$ or $G=b_2, G'=-b'_2$ and taking the continuum limit, based on the fact that larger values of $G$ or $G'$ are exponentially depressed \cite{111}. However in the incommensurate case, Umklapp terms with very large values of $G,G'$ make $| G+G'+ p_{F,i}-p'_{F,j}|$ arbitrarely small, producing almost relevant processes which can destroy the semimetallic behaviour. In the 1d Aubry-Andr\'e model the processes that produce small values for $2 \e p_F+2l\pi+2\pi \o m$, $\e=0,\pm 1$ are indeed the ones producing the insulating behaviour at large coupling, while at weak coupling the metallic regime persists. Similarly the persistence or not of the semimetallic regime in TBG depends on the relevance or irrelevance of the terms involving large $G, G'$ that almost connect the Dirac points. This fact cannot be understood only on the basis of perturbative arguments; it is indeed a non perturbative phenomenon which can be established only by the convergence or divergence of the whole series expansion. Despite the similarity of quasi-periodic potentials and incommensurate TBG, there are crucial differences like the higher dimensionality of TBG and the fact that the frequencies are not independent parameters but are functions of a single parameter, the angle between the layers, and this produces rather different small divisors. The aim of this paper is to investigate when the quasi-metallic phase is stable against the large momentum processes in the incommensurate case. The analysis is based on Renormalization Group methods combined with number theoretical properties of irrationals, similar to the ones used in Kolmogorov-Arnold-Moser (KAM) theory for the stability of invariant tori. Due to the difficulty of getting information on the single particle spectrum, we analyze the behavior of the Euclidean correlations, which provide information on the spectrum close to the Fermi points. Such methods are robust enough to be extended to many-body systems, as it was done for the interacting Aubry-Andr\'e model \cite{M11} or in Weyl semimetals \cite{W1}. Our main result is the proof of the stability of the semimetallic phase in a large measure set of angles in the incommensurate case. The paper is organized in the following way. In Section \ref{sec:model} the lattice model of TBG is presented. In Section \ref{sec:feynman} a perturbative expansion for the correlations is derived. In Section \ref{sec:feynman1} the emerging quasi-periodicity and the small divisor problem is described, together with the required (number theoretical) Diophantine conditions. Section \ref{sec:result} contains a statement of the main result and in Section \ref{sec:renormalized} the Renormalization Group derivation is presented. The Appendices detail the more technical aspects of the analysis. \section{Incommensurate TBG}\label{sec:model} We consider the lattice TBG model introduced in \cite{1}, \cite{2}. We focus on this model for the sake of definiteness but our methods could be applied more generally. We consider two graphene layers rotated with respect to one another by an angle $\theta$ around a point $\xi=(0,1/2)$ (that is, the point between an a and b atom, chosen so that the twisted model preserves the $C_2T$ symmetry in Appendix \ref{sec:symmetry}). The Hamiltonian of the system will be written as \begin{equation} H=H_1+H_2+V \end{equation} where $H_1$ and $H_2$ are hopping Hamiltonians within the layers 1 and 2 respectively and $V$ is an interlayer hopping term. The first graphene layer is defined on the lattice $\mathcal L_1:=\{n_1 A_1+n_2 A_2,\ n_1,n_2\in\mathbb Z\}$ with $A_1={1\over 2}(3,\sqrt{3}) ,\quad A_2={1\over 2}(3,-\sqrt{3})$. We introduce the nearest-neighbor vectors: $\d_1=(1,0)$, $\d_2={1\over 2}(-1, \sqrt{3})$, $\d_2={1\over 2}(-1, -\sqrt{3})$. We will write the Hamiltonian in second quantized form: for $x\in \mathcal L_1$, we introduce {\it annihilation operators} $c_{1,x,a}$ and $c_{1,x,b}$ corresponding respectively to annihilating a fermion located at $x$ and $x+\d_1$. The nearest neighbor hopping Hamiltonian is \be H_1= -t\sum_{x\in \mathcal L_1}\sum_{i=0}^2 (c_{1,x,a}^\dagger c_{1,x+A_i,b}+ c_{1,x+A_i,b}^\dagger c_{1,x,a})\ee where $A_0:=0$ (note that $\d_1-\d_2=A_2, \d_2-\d_3=A_3$). We will do much of the computation in Fourier space, and we here introduce the Fourier transform $\hat c_{1,k,\alpha}$ of $c_{1,x,\alpha}^\pm$ in such a way that, for $\alpha\in\{a,b\}$, \begin{equation} c_{1,x,\alpha} =\frac1{|\hat{\mathcal L}_1|}\int_{\hat{\mathcal L}_1} dk\ e^{-ik(x-\xi)}\hat c_{1,k,\alpha} \end{equation} with $|\hat{\mathcal L}_1|=8 \pi^2/3 \sqrt{3}$, and $ \hat{\mathcal L}_1:=\mathbb R^2/(b_1\mathbb Z+b_2\mathbb Z)$ in which \begin{equation} b_1= {\textstyle{2\pi\over 3}}(1,\sqrt{3}) ,\quad b_2= {\textstyle{2\pi\over 3}}(1,-\sqrt{3}) . \label{bi} \end{equation} In Fourier space, \begin{equation} H_1= \frac t{|\hat{\mathcal L}_1|}\int_{\hat{\mathcal L}_1}dk\ \left(\Omega(k)\hat c_{1,k,a}^\dagger\hat c_{1,k,b}+\Omega^*(k)\hat c_{1,k,b}^\dagger\hat c_{1,k,a}\right) \label{H1k} \end{equation} with $\Omega(k_x,k_y):=1+2e^{-i\frac32k_x}\cos({\textstyle\frac{\sqrt 3}2k_y})$. The second graphene layer is rotated by an angle $\theta$ around the point $\xi=(0,1/2)$, that is, it is defined on the lattice \be \mathcal L_2=\xi+R(\theta)(\mathcal L_1-\xi),\quad R(\th)=\begin{pmatrix} c_\th &-s_\th \\s_\th & c_\th \end {pmatrix}\ee (we use the shorthand throughout this paper that $c_\th\equiv\cos\th, s_\th\equiv\sin \th$). The annihilation operators in the second layer are denoted by $c_{2,x,a}$ and $c_{2,x,b}$. The hopping Hamiltonian of this second layer is \begin{equation} H_2= -t\sum_{x\in \mathcal L_2}\sum_{i=0}^2 (c_{2,x,a}^\dagger c_{2,x+RA_i,b}+ c_{2,x+R A_i,b}^\dagger c_{2,x,a}) \end{equation} where $R\equiv R(\theta)$. We define the Fourier transform in the second layer: if $b'_1:=R b_1$, $b'_2:=R b_2$ and \begin{equation} c_{2,x,\alpha} =\frac1{|\hat{\mathcal L}_1|}\int_{\hat{\mathcal L}_2} dk\ e^{-ik(x-\xi)}\hat c_{2,k,\alpha} \end{equation} we find \begin{equation} \begin{array}{r@{\ }>\displaystyle l} H_2=& \frac t{|\hat{\mathcal L}_1|}\int_{\hat{\mathcal L}_2}d k\ \cdot\\&\cdot\left(\Omega(R^T k)\hat c_{2,k,a}^\dagger\hat c_{2,k,b}+\Omega^*(R^Tk)\hat c_{2,k,b}^\dagger\hat c_{2,k,a}\right) . \label{H2k} \end{array} \end{equation} In the absence of interlayer coupling the two graphene layers are decoupled; the single particle spectrum for layer 1 is $\pm |\Omega(k)|$ and the Fermi points are given by the relation $\O(p_{F,1}^\pm)=0$ with \begin{equation} p_{F,1}^\pm={2\pi\over 3}(1,\pm {\textstyle{1\over \sqrt{3}}}) \label{pF} \end{equation} for momenta close to such points one has $|\Omega(k)| \sim {3\over 2} t |k-p_{F,1}^\pm|$, that is the dispersion relation is almost linear (relativistic) up to quadratic corrections, forming approximate {\it Dirac cones}. In the same way the dispersion relation for layer 2 is $\pm |\Omega(R^T k)|$; the Fermi points are $\O(R^T p_{F,2}^\pm)=0$ with $p_{F,2}^\pm=R(p_{F,1}^\pm)$ and $|\Omega(R^T k)| \sim {3\over 2} t |k-p_{F,2}^\pm|$. We are interested in understanding how these four Dirac cones are modified in the presence of the interlayer hopping. We couple the 2 layers by an interlayer hopping Hamiltonian, which couples atoms of type a to atoms of type b: \begin{equation} \begin{array}{>\displaystyle l} V= \l \sum_{x_1\in \mathcal L_1} \sum_{x'_2\in\mathcal L_2}\sum_{\alpha\in\{a,b\}} \varsigma(x_1+d_\alpha-x'_2-Rd_{\alpha}) \cdot\\ \hfill\cdot(c^\dagger_{1,x_1,\alpha}c_{2,x'_2,\alpha}+c_{2,x_2',\alpha}^\dagger c_{1,x_1,\alpha}) \end{array}\end{equation} $d_a=(0,0), d_b=\d_1$, and $\varsigma(x)=\varsigma(-x)$, \be \varsigma(x_1-x_2)=\int_{\mathbb R^2}\frac{dq}{4\pi^2}\ e^{i q(x_1-x_2)} \hat \varsigma(q) ,\quad |\hat\varsigma(q)|\le e^{-\k |q|} . \label{interlayer} \ee We restrict the interlayer term to hoppings between atoms of type a to atoms of type a and type b to b for technical reasons. This is so that the ``Inversion'' symmetry in Appendix \ref{sec:symmetry} is satisfied. We could just as easily consider a model where the interlayer hopping occurs only between atoms of type a to atoms of type b. We could relax this restriction and allow for all possible hoppings, but we would then need to add extra counterterms (see (\ref{counterterms})) to the model. For the sake of simplicity, we avoid this and only consider these interlayer hoppings. Note that, whereas the Fourier transform for $c$ is defined on $\hat{\mathcal L}_i$, the Fourier transform of $\varsigma$ is defined on all $\mathbb R^2$. We write $V$ in Fourier space: we get, see App. \ref{app:fourierV} \bea &&V= \frac \l{4\pi^2|\hat{\mathcal L}_1|}\sum_{\alpha}\left(\sum_{l\in\mathbb Z^2}\int_{\hat{\mathcal L}_1}d k \ \tau^{(1)}_{l,\alpha}(k+l b) \hat c^\dagger_{1,k,\alpha}\hat c_{2,k+l b,\alpha} \right.\nn\\ &&\left.+\sum_{m\in\mathbb Z^2}\int_{\hat{\mathcal L}_2}d k\ \tau_{m,\alpha}^{(2)}(k+m b') \hat c^\dagger_{2,k,\alpha}\hat c_{1,k+m b',\alpha}\right) \nn\label{11} \eea where we use the notation $lb\equiv l_1b_1+l_2b_2$, $mb'\equiv m_1b_1'+m_2b_2'$, and \begin{equation} \tau^{(1)}_{l,\alpha}(k) :=e^{i \xi lb}e^{-ik(d_\alpha-Rd_{\alpha})} e^{-i\xi \sigma_{k,1}b'}\hat\varsigma^*(k) \label{tau1} \end{equation} \begin{equation} \tau^{(2)}_{m,\alpha}(k):=e^{i\xi mb'}e^{-ik(d_\alpha-Rd_{\alpha})}e^{-i\xi \sigma_{k,2}b}\hat\varsigma(k) \label{tau2} \end{equation} in which $\sigma_{k,i}\in \mathbb Z^2$ is the unique integer vector such that $ k-\sigma_{k,1}b'\in\hat{\mathcal L}_2 ,\ k-\sigma_{k,2}b\in\hat{\mathcal L}_1 $. Note that the difference of the momenta of the two fermions is given by $l b+m b'$. The position of the Dirac points are in general modified (renormalized) by the interlayer hopping. It is conventient to fix the values of the renormalized Dirac points by properly choosing the bare ones. This can be achieved by replacing $\O(k)$ with $\O(k)+\nu_{i,\o}$ close to each Dirac points, that is adding a counterterm has the form \begin{eqnarray} &&M= \sum_{\omega\in\{+,-\}}\sum_{i=1,2}\int_{\hat{\mathcal L}_i} dk\ \chi_{\omega,i}(k)( \nu_{i,\omega} \hat c_{i,k,a}^\dagger\hat c_{i,k,b}\nn\\ &&+ \nu^*_{i,\omega} \hat c_{i,k,b}^\dagger\hat c_{i,k,a}) \label{counterterms} \end{eqnarray} where $\chi_{\o,i}(k)$ is a smooth compactly supported function that is non vanishing for $||k-p_{F,i}^\o||_i\le 1/\gamma$, for some $\gamma>1$, in which $||.||_i$ is the norm on the torus $\hat{\mathcal L}_i$. Our main result concerns the two-point Schwinger function, which we define as follows. We first introduce a Euclidean time component: given an inverse temperature $\beta>0$, we define for $x_0\in[0,\beta)$, \begin{equation} c_{j,x,\alpha}(x_0):=e^{-x_0\bar H}c_{j,x,\alpha}e^{x_0\bar H} . \end{equation} and $\bar H=H+M$. Combining the Euclidean time component with the spatial one, we define $\Lambda_i:=[0,\infty)\times \mathcal L_i$. The corresponding Fourier-space operators are \begin{equation} \hat c_{j,k,\alpha}(k_0)= \int_0^\beta dx_0 e^{-ix_0k_0}\sum_{x\in \mathcal L_j}e^{-i(x-\xi)k}c_{j,x,\alpha}(x_0) \end{equation} which is defined for $(k_0,k)\in\hat \Lambda_j:=\frac{2\pi}\beta \mathbb Z\times \hat{\mathcal L}_j$. Now, given $j,j'\in\{1,2\}$, $\mathbf k=(k_0,k)\in \Lambda_j$, the two-point Schwinger function is defined as the $2\times2$ matrix $S_{j,j'}(\mathbf k)$ whose components are indexed by $\alpha,\alpha'\in\{a,b\}$: \begin{equation}\label{xx} (\hat S_{j,j'}(\mathbf k))_{\alpha,\alpha'}:= \frac{\mathrm{Tr}(e^{-\beta \bar H}T(\hat c_{j,k,\alpha}(k_0),\hat c_{j',k,\alpha'}^\dagger(k_0)))}{\mathrm{Tr}(e^{-\beta\bar H})} \end{equation} where $T$ is the time ordering operator, which is bilinear, and is defined in real-space $ T(c_{j,x,\alpha}(x_0),c_{j',y,\alpha'}^\dagger(y_0))=$ \begin{equation} \left\{\begin{array}{>\displaystyle ll} c_{j,x,\alpha}(x_0)c_{j',y,\alpha'}^\dagger(y_0)&\mathrm{if\ }x_00$, there exists a subset of $[\theta_0,\theta_1]$ whose measure is at least $1-O(C_0/(\theta_0-\theta_1)^2)$ such that \pref{cond} holds, and an $\e_0$ (depending on $C_0,\th_0,\th_1$), such that, for any $|\l|\le \e_0$, for a suitable choice of $\n_{i,\o}$ $ (\hat S_{j,j}(\mathbf k+\mathbf p_{F,j}^\o ))=$ \be \begin{pmatrix}\label{43} -i Z_{j,\o} k_0 & (i v_{j,\o} k_1- w_{j,\o} \o k_2 )\\ (-i v_{j,\o}^* k_1- w_{j,\o}^* \o k_2) & -i Z_{j,\o} k_0 \end{pmatrix}^{-1}(1+O(|k|^{\alpha})) \ee with $0\le \alpha\le 1$, $Z=1+O(\l)$ real and $v_{i,\omega}=3t/2+O(\l)$, $w_{i,\omega}=3t/2 +O(\l)$, $\n_{j,\o}=O(\l)$. } \vskip.2cm This result ensures that, even taking into account the Umklapp processes involving the exchange of very high momenta due to the emerging quasi-periodicity, the Weyl semimetallic phase persists for small interalyer coupling and a large measure set of angles. The interlayer coupling modifies the position of the Dirac points; we have properly chosen the bare Dirac points $p_{F,i}^\pm(\l)$ in absence of interlayer coupling given by $\O(p_{F,i}^\pm(\l))+\n_{i,\pm}=0$) so that their renormalized physical value is $p_{F,i}^\pm$ given by (10). This is essentially equivalent to say that the position of the Dirac points genericaly moves in a way depending on the angle, the layer and the coupling. The velocities $w_{j,\o}, v_{j,\o}$ and the wave function normalization $Z_{j,\o}$, are renormalizated in a way generically dependent on the layer and the angle. Note that a priori several other relevant terms could be present, but they are excluded by symmetry. The singularity of the Schiwnger function is given by $Z^2 k_0^2+R(k)$ with $R(k)\sim (|v|^2 k_1^2+|w|^2 k_2^2) $; the singularity of the 2-point function is therefore the same as in absence of interlayer at weak coupling ensuring the stability of the semimetallic phase. The result holds for irrational twisting angles verifying \pref{cond}. The relative measure of this set can be made arbitrarely close to $1$ by decreasing $C_0$. In the remaining sections we prove the above result by a Renormalization Group analysis, leading to a convergent expansion. \section{The renormalized expansion}\label{sec:renormalized} \subsection{Multiscale decomposition} \label{sec:multiscale} We introduce smooth cut-off functions: for $i=1,2$, $\o=\pm$, $h\in\{-\infty,\cdots,0\}$, let $\chi_{h,i,\o}(\kk)$ be a smooth function that vanishes outside the region $||\kk-\pp_{F,i}^\omega|| \le \g^{h-1}$ and that is equal to 1 for $||k-\pp_{F,i}^\omega||\ge \g^{h-2}$. The constant $\g>1$ will be chosen below to be large enough. Note that, in this way, the supports of $\chi_{0,i,+}$ and $\chi_{0,i,-}$ do not overlap. We define $\hat g_{i,\o}^{(\le 0)}(\kk)=\chi_{0,i,\o}(\kk)\hat g_i(\kk)$ and \be \hat g_i(\kk)= g_i^{(1)}(\kk)+\sum_{\o=\pm} \hat g^{(\le 0)}_{i,\o} (\kk) \ee with $\hat g^{(1)}(\kk)=(1-\sum_\o \chi_{0,i,\o}(k))\hat g_i(\kk)$; this induces the Grassmann variable decomposition $\hat\psi_{i,\kk,\alpha}=\hat\psi_{i,\kk,\alpha}^{(1)}+\sum_{\o=\pm} \hat\psi^{(\le 0)}_{i,\kk,\alpha,\o}$ with propagators given by $\hat g^{(1)}_i$ and $\hat g^{(\le 0)}_{i,\o}$ respectively. Note that $\hat\psi^{(1)}$ correspond to fermions with momenta far from the Fermi points, while $\hat\psi^{(\le 0)}$ with momenta around $\pm \pp_{F,i}$. We further decompose \be \hat g_{i,\o}^{(\le 0)} (\kk)=\sum_{h=-\io}^0 \hat g^{(h)}_{i,\o}( \kk) \ee where $\hat g_{i,\o}^{(h)}(\kk):=f_{h,i,\o}(\kk)\hat g_{i,\o}^{(\le 0)}$ in which $f_{h,i,\o}:=\chi_{h,i,\o}-\chi_{h-1,i,\o}$ is a smooth cutoff function supported in $ \g^{h-3} \le |\kk-\pp^\o_{F,i}|\le \g^{h-1}$ such that $\sum_{h=-\io}^0f_{h,i,\o}=\chi_{0,i,\o}$. The integration is done recursively in the following way: assume that we have integrated the fields $\psi^{(1)},..,\psi^{(h-1)}$ obtaining \be e^{W}=\int\bar P(d\psi^{(\le h)}) e^{V^{(h)}(\psi,\phi)} \ee where $\bar P(d\psi^{(\le h)})$ is Gaussian integration with propagator $\bar g_{i,\omega}^{(\le h)}$ which will be defined inductively in (\ref{prop_ind}), and \bea &&V^{(h)}(\psi,0)= \\ &&\sum_{i,\o,\o',l,\alpha,\alpha'} \int_{\hat\L_i} d\kk W^{(h,\omega,\omega')}_{i,2,l,\alpha,\alpha'}(\kk) \psi_{i,\kk,\alpha,\o}^+\psi_{2,\kk+lb,\alpha',\o'}^-+\nn\\ &&\sum_{i,\o,\o',m,\alpha,\alpha'} \int_{\hat\L_i} d\kk W^{(h,\omega,\omega')}_{i,1,m,\alpha,\alpha'}(\kk) \psi_{i,\kk,\alpha,\o}^+\psi_{1,\kk+mb',\alpha',\o'}^- . \label{eff} \eea According to the RG procedure, we renormalize the relevant and marginal terms; we will see below that the term with $l$ or $m$ non zero are actually irrelevant, due to improvements in the estimates due to the Diophantine condition. We therefore define a localization operation in the following way \bea &&\LL W^{(h,\omega,\omega')}_{i,j,l}(\kk)=\d_ {\o,\o'}\d_{i,j} \d_ {l,0}[W^{(h,\omega,\omega)}_{i,i,0}(0,p_{F,i}^\o)+\nn\\ &&k_0 \partial_0 W^{(h,\omega,\omega)}_{i,i,0}(0,p_{F,i}^\o)+ (k-p_{F,i}^\o)\partial W^{(h,\omega,\omega)}_{i,i,0}(0,p_{F,i}^\o) . \label{loc} \eea The terms for which $\LL=0$ are called {\it non resonant} terms and the ones for which $\LL\not=0$ {\it resonant} terms. The terms containing derivatives are marginal ones and produce wave function or velocities renormalizations, while the terms without derivatives are the relevant terms. The action of $\mathcal L$ on the effective potential $V^{(h)}$ is \begin{equation} \LL V^{(h)}=\LL_1 V^{(h)}+\LL_2 V^{(h)} \end{equation} with \begin{equation} \LL_1 V^{(h)}:=\sum_{i,\omega,\a,\a'}\int_{\hat \Lambda_i} d\kk \g^h\n_{h,\omega,\a,\a',i} \psi^+_{\kk,\o,i,\a}\psi^-_{\kk,\o,i,\a'} \end{equation} and \begin{equation} \LL_2 V^{(h)}:=\sum_{i,j,\omega,\a,\a'}\int_{\hat \Lambda_i} d\kk z_{h,\o,\a,\a',i,j} (\mathbf k-\mathbf p_{F,i}^\omega)_j\psi^+_{\kk,\o,i,\a} \psi^-_{\kk,\o,i,\a'} \end{equation} with \begin{equation}\label{ai} \nu_{h,\omega,\alpha,\alpha',i}:=\gamma^{-h} W_{i,i,0,\alpha,\alpha'}^{h,\omega,\omega}(0,p_{F,i}^\omega) \end{equation} \begin{equation} z_{h,\omega,\alpha,\alpha',i,j}:=-\partial_{\mathbf k_j}W_{i,i,0,\alpha,\alpha'}^{h,\omega,\omega}(0,p_{F,i}^\omega) . \end{equation} The form of the resonant terms is severely constrained by symmetries: as is proved in Appendix \ref{sec:symm}, \bea &&\n_{h,\o,a,a,i}=\n_{h,\o,b,b,i}=0\quad \n_{h,\o,a,b,i}=\n^*_{h,\o,b,a,i}\nn\\ &&z_{h,\o,b,a,i,1}=z_{h,\o,a,b,i,1}^*\quad z_{h,\o,b,a,i,2}=z_{h,\o,a,b,i,2}^* \\ &&z_{h,\o,a,a,i,1}=z_{h,\o,b,b,i,1}= z_{h,\o,a,a,i,2}=z_{h,\o,b,b,i,2}=0 \nn \\ &&z_{h,\o,b,a,i,0}=z_{h,\o,a,b,i,0}=0 \quad z_{h,\o,a,a,i,0}=z_{h,\o,b,b,i,0}\in i \mathbb R \nn \eea The contributions from $\mathcal L_2 V$ are marginal, and are absorbed into the propagator at every step of the integration: \begin{equation} \begin{array}{>\displaystyle l} \bar g_{i,\omega}^{(\le h)}(\mathbf k) := \chi_{h,i,\omega}(\mathbf k) \cdot\\\hfill\cdot \left((\bar g_{i,\omega}^{(\le h+1)}(\mathbf k))^{-1} -\sum_j z_{h,\omega,\cdot,\cdot,i,j}(\mathbf k-\mathbf p_{F,i}^\omega)_j\right)^{-1} \end{array} \label{prop_ind} \end{equation} Thus, \begin{equation} \begin{array}{>\displaystyle l} \bar g_{i,\o}^{(\le h)}(\kk+\pp_{F,i}^\o)= \chi_{h,i,\o}(\kk)(1+O(k)) \cdot\\\hfill\cdot \begin{pmatrix} -i Z_{i,\o,h} k_0 & (i v_{i,\o,h} k_1- w_{i,\o,h} \o k_2)\\ (-i v^*_{i,\o,h} k_1- w^*_{i,\o,h} \o k_2) & -i Z_{i,\o} k_0 \end{pmatrix} ^{-1}\label{prop} \end{array} \end{equation} with \bea &&Z_{i,\omega,h}=Z_{i,\omega,h+1}-iz_{h,\o,a,a,i,0} \nn\\ &&v_{i,\omega,h}= v_{i,\omega,h+1}+i z_{h,\o,a,b,i,1} \nn\\ &&w_{i,\omega,h}= w_{i,\omega,h+1}+ \omega z_{h,\o,a,b,i,2} . \eea After absorbing $\mathcal L_2V^{(h)}$ into the propagator, we are left with integrating $\LL_1 V^{(h)}$ and \begin{equation} \RR V^{(h)}:=(1-\mathcal L)V^{(h)} \end{equation} so \be e^{W}=\int \bar P(d\psi^{(\le h)}) e^{\LL_1 V^{(h)}(\psi)+\RR V^{(h)}(\psi)} . \label{renormalized_expansion} \ee \subsection{Feynman rules for the renormalized expansion} \label{sec:renormfeyn} The renormalized expansion described above has a graphical representation that is similar to the Feynman diagram expansion from Section \ref{sec:feynman}. There are two main differences: first, there are two different types of vertices: ``\emph{$\tau$-vertices}'', coming from $\mathcal R V^{(h)}$ in (\ref{renormalized_expansion}), and ``\emph{$\nu$-vertices}'', coming from $\mathcal L_1 V^{(h)}$. Second, every line has a {\it scale label} $h$, corresponding to a propagator on scale $h$: \begin{equation} \bar g_{i,\omega}^{(h)}(\mathbf k):=f_{h,i,\omega}(\mathbf k)\bar g^{(\le h)}_{i,\omega}(\mathbf k) . \end{equation} The scale labels induce an important structure: given a diagram, we group vertices together into nested \emph{clusters}, which are connected subgraphs in which the scales of the lines leaving the cluster are all smaller than the scales of the lines inside the cluster, see Figure \ref{fig:clusters}. A cluster that is such that $\mathcal L$ applied to the cluster yields $0$ is called {\it non-resonant}, otherwise it is called {\it resonant}. In other words, the action of $\mathcal R=1-\mathcal L$ is trivial on non-resonant clusters, and non-trivial on resonant ones. Some clusters are single vertices (either $\nu$ or $\tau$) and are called \emph{trivial clusters}. The clusters that contain internal lines are called \emph{non-trivial clusters}. As per the construction above, if $h^{ext}_T$ is the largest of the scales of the external lines of a non-trivial cluster $T$, all its internal lines have a scale $h>h^{ext}_T$; $h_T$ is the largest scale of the propagators internal to the cluster $T$. A non-trivial cluster $T$ contains sub-clusters $\tilde{T}\subset T$. We call a cluster $\tilde T\subset T$ a \emph{maximal cluster} if there is no other cluster $\bar T$ such that $\tilde T\subset \bar T\subset T$. For each cluster, there are two external lines that connect the cluster to other ones. In a non-resonant cluster $T$ with external lines of type $i_1=i_2=1$ and momenta $k_1, k_2$ with $k_1$ in the first Brillouin zone and $k_2=k_1+\hat m_T b+l b$ where $l b$ is chosen so that $k_2$ is in the first Brillouin zone, if $A_0,..,A_N$ are the momenta associated to the $\t$ vertices contained in $T$ (the $\nu$ vertices do not change momentum) one has $A_0=k_1+ l_0 b$, $A_1=k_1+l_0 b+m_1 b'$, $A_2=k_1+m_1 b'+l_2 b$, $A_3=k_1+m_3 b'+l_2 b$,..., $A_{N}=k_1+m_{N} b'+l_{N-1} b$ and $m_{N}=\hat m_T$ with $N$ odd. In the same way if the non-resonant cluster $T$ has one external line of type $i_1=1$ and momentum $k_1$, and one of type $i_2=2$ and momentum $k_2$; assume that $k_1$ is in the first Brillouin zone and $k_2=k_1+\hat l_T b+m b'$, where $m b'$ is chosen such that $k_2$ is in the first Brillouin zone. Now with $N$ even the momenta associated to the $\t$ vertices in $T$ are $A_0=k_1+ l_0 b$, $A_1=k_1+l_0 b+m_1 b'$, $A_2=k_1+m_1 b'+l_2 b$, $A_3=k_1+m_3 b'+l_2 b$,...$A_{N-1}=k+m_{N-2} b'+l_{N-1}b$, $l_{N-1}=\hat l_T$. The value associated to a graph $\G$ is denoted by $W_\G(\kk)$ and is given by the product of the propagators and $\n,\t$ factors associated to the vertices, with the $\RR$ operation acting on each non-resonant cluster. The effect of the $\RR$ operation on the non-resonant clusters can be written as \be \RR W^h(\kk+p_{F,i}^\o)=k^2\int_0^1 \partial^2 W(t \kk)\ee \begin{figure} \hfil\includegraphics[width=8cm]{cluster.pdf} \caption{\label{fig:clusters} An example of graph of order $\l^7$ with the associated clusters, denoted by thick rectangles. In this example, $h0$. On the other hand if $\hat m_T=0$ but $\o_1\not=\o_2$ then $l=0$ so $\g^{h^{\mathrm{ext}}_T}\ge\frac12|p_{F,1}^{\omega_1}-p_{F,1}^{\omega_2}|=\frac{2\pi}{3\sqrt3}$ (recalling (\ref{pF})). Thus, this eventuality does not occur provided $\gamma$ is large enough. \item In the case $i_1=1$ and $i_2=2$ and $k_1, k_2$ are the momenta of the external lines; assume that $k_1$ is in the first Brillouin zone and $k_1=:\bar k_1+p_{F,1}^{\o_1}$. Moreover $k_2:=k_1+\bar l b+m b'=:\bar k_2+p_{F,2}^{\o_2}$, with $m$ chosen in such a way that $k_2$ is in the first Brillouin zone; then, if $\bar l\not=0$, by (\ref{cond}), \bea &&2 \g^{h^{ext}_T}\ge |\bar k_1|+|\bar k_2|\ge |\bar k_1-\bar k_2|=\\ &&|p_{1,F}^{\o_1}-p_{2,F}^{\o_2}+\hat l_T b+m b'|\ge {C_0\over |\hat l_T|^\t}\nn\eea so that \be|\hat l_T|\ge ({\textstyle\frac12}C_0 \g^{-h^{\mathrm{ext}}_T})^{\frac1\tau}\label{cond2} \ee and so \begin{equation} |\hat l_Tb|\ge c_1\g^{-h^{\mathrm{ext}}_T/\tau} \end{equation} If $\hat l_T=0$ then $2 \g^{h^{\mathrm{ext}}_T}\ge O(\th)$ for $\o_1=\o_2$ and $2 \g^{h^{\mathrm{ext}}_T}\ge O(1)$ for $\o_1=-\o_2$. Thus, provided $\gamma\gg \theta^{-1}$, these eventualities do not present themselves provided $\gamma$ is large enough. \item A similar analysis holds for $i_1=2, i_1=1$, and $i_1=i_2=2$. \end{enumerate} Thus, \begin{equation} \prod_{i\in T}e^{-\kappa 2^{h_T}|A_i|}\le e^{-\kappa 2^{h_T}(c_1 \gamma^{-h_T^{\mathrm{ext}}/\tau}-\frac{4\pi}3)} \end{equation} which, provided $\gamma$ is large enough, yields \begin{equation} \prod_{i\in T}e^{-\kappa 2^{h_T}|A_i|}\le e^{-c_2\gamma^{-h_T^{\mathrm{ext}}/\tau}} \end{equation} for some constant $c_2$. Therefore, \be L(\underline l,\underline m)\le e^{-c_2 \gamma^{-h/\tau}\mathds 1_{\Gamma\,\mathrm{nonres}}} \prod_i e^{-\kappa |A_i|/2} \prod\limits_{T\ \mathrm{n.t.}} e^{-c_2 {M}_{T} \gamma^{-h_{T}/\tau}}\ee where $\mathds 1_{\G\,\mathrm{nonres}}$ is equal to 1 if the maximal cluster is non resonant and $0$ otherwise. Note that, provided $\gamma$ is chosen to be large enough, $e^{-c_2 \gamma^{-h/\tau}\mathds 1_{\Gamma\,\mathrm{nonres}}}\le \gamma^{3h \mathds 1_{\Gamma\,\mathrm{nonres}}}$, so \be L(\underline l,\underline m)\le \gamma^{3h\mathds 1_{\Gamma\,\mathrm{nonres}}} \prod_i e^{-\kappa |A_i|/2} \prod\limits_{T\ \mathrm{n.t.}} e^{-c_2 {M}_{T} \gamma^{-h_{T}/\tau}}.\ee Furthermore, using the bound $e^{-\a x}\le ({\beta\over \a})^\beta e^{-\beta}x^{-\beta}$ with $\beta= 3\tau M_T$, we find \begin{equation} e^{-c_2M_T \gamma^{-h_T/\tau}} \leq (\frac{c_2 e^1}{3\tau})^{-3\tau M_T} \gamma^{3M_T h_T} . \label{exp3} \end{equation} In addition, $\sum_{T\,\text{n.t.}} M_T \leq q$, since the clusters are nested in each other and for two clusters to be different they must differ by at least one vertex. Now, let us introduce $M_T^\tau$ as the number of maximal non-resonant trivial clusters (i.e. maximal $\tau$-vertices) contained in $T$, and use the trivial bound $3M_T \le 2M_T+M_T^\tau$ along with (\ref{exp3}) to obtain \begin{equation} \prod_{T\ \mathrm{n.t.}} e^{-c_2 M_{T} \gamma^{-h_{T}}/\tau} \le C_3^q .\prod_{T\ \mathrm{n.t.}} \gamma^{h_{T} (2M_{T}+M_T^\tau)} \label{asxq} \end{equation} Thus, plugging this into (\ref{lap1}), we find \bea &&\g^h \sum^*_{\underline h,\atop \underline l, \underline m} [L]^{1\over 2} |\l|^q (CC_3)^q \gamma^{h\mathds 1_{\Gamma\,\mathrm{res}}} \gamma^{3h\mathds 1_{\Gamma\,\mathrm{nonres}}} \nn\\&& [\prod_{T\ \mathrm{res}} \gamma^{(h_{T}^{\text{ext}}-h_{T})}] \prod_{T\, \text{n.t.}} \g^{h_T (2M_T+M^\tau_T)} . \eea In addition, \begin{equation} \prod_{T\ \mathrm{n.t.}} \gamma^{2h_T M_T} = \gamma^{-2h \mathds 1_{\Gamma\,\mathrm{nonres}}}\prod_{T\ \mathrm{nonres}}\gamma^{2h_T^{\mathrm{ext}}} \end{equation} \be \g^h \sum^*_{\underline h,\atop \underline l, \underline m} [L]^{1\over 2} |\l|^q (CC_3)^q [\prod_{T\, \text{n.t.}} \gamma^{(h_{T}^{\text{ext}}-h_{T})}] \prod_{T\,\text{n.t.}} \gamma^{h_{T} M^\tau_{T}} \label{ssa} \ee The crucial point is that the sum over the scales $h$ can be performed summing over all the differences, taking into account that the scale $h$ is fixed. Finally the sum over the $l,m$ is done using the factor $[L(\underline l, \underline m)]^{1\over 2}$. (The gain term $\g^{h_T M_T^\tau}$ is dropped, as it does not lead to any significant gain.) In conclusion the bound on a graph with $q$ vertices is $C_4^q\g^h |\l|^q$ assuming that $|\n_h|,|Z_h-1|,|v_h-1|,|w_h-1| \le C |\l|$. \subsection{Beta function and Schwinger functions} We are left with checking our assumption on $Z_h, \nu_h, w_h,v_h$. We know that $v_{i,\omega,h}= v_{i,\omega,h+1}-i z_{h,\o,a,b,i,1}$ with $z_{h,\o,a,b,i,1}$ expressed by the sum of renormalized Feynman graphs $\G$ such that the maximal scale of the clusters is $h+1$, an extra derivative is applied (which costs a factor $\gamma^{-h}$) and the momenta of the external lines is fixed equal to $p_F^\o$. Moreover by the compact support of the propagator there is at least a $\t$ vertex, as the $k=0$ value of a graph wih only $\n$ vertice is zero; therefore the analogue of (\ref{ssa}) becomes \be \sum^*_{\underline h,\atop \underline l, \underline m} [L]^{1\over 2} |\l|^q (CC_3)^q [\prod_{T\, \text{n.t.}} \gamma^{(h_{T}^{\text{ext}}-h_{T})}] \gamma^{2h_{T^*}} \label{weightzz} \ee where $T^*$ is the non trivial cluster containing a $\t$ vertex whose scale is the largest possible (we now use the gain $\g^{h_T M_T^\tau}$ dropped in the bound \pref{ssa}). In addition, summing the differences $h_T^{\mathrm{ext}}-h_T$ along a sequence of clusters that goes from $h$ to $h_{T^*}$ and discarding the others, we bound \begin{equation} \sum_{T}(h_T^{\mathrm{ext}}-h_T) \leqslant h-h_{T^*} \end{equation} and so (\ref{weightzz}) is bounded by \be \label{weightzz1}\g^{h\over 2} \sum^*_{\underline h,\atop \underline l, \underline m} [L]^{1\over 2} |\l|^q (CC_3)^q \prod_{T\, \text{n.t.}} \gamma^{ {1\over 2}(h_{T}^{\text{ext}}-h_{T})} . \ee Estimating the sum as above, we find that $|z_{h,\o,a,b,i,1}|\le C_5 \l \g^{h\over 2}$, and $v_{i,\omega,h}= v_{i,\omega,0}-i \sum_{h'} z_{h',\o,a,b,i,1}$ hence $v_{i,\omega,h}= v_{i,\omega,0}+O(\l)$; moreover the limiting value is reached exponentially fast $v_{i,\omega,h}=v_{i,\omega,-\infty}+O(\l \g^{h/2})$. A similar argument holds for $Z_h, w_h$. Not It remain to discuss the flow of $\n_h$; we can write, see \pref{ai}, \be W_{i,i,0,\alpha,\alpha'}^{h,\omega,\omega}(0,p_{F,i}^\omega) =\g^{h+1}\n_{h+1}+ \tilde W_{i,i,0,\alpha,\alpha'}^{h,\omega,\omega}(0,p_{F,i}^\omega)\ee where $\tilde W$ is given by the sum of the terms with a number of vertices greater or equal to $2$; therefore \be \nu_{h,\omega,\alpha,\alpha',i}=\g \nu_{h+1,\omega,\alpha,\alpha',i}+\b^h_\n \label{appo2} \ee with $\b^h_\n=\g^{-h} \tilde W_{i,i,0,\alpha,\alpha'}^{h,\omega,\omega}(0,p_{F,i}^\omega)$ is given by the sum with $q\ge 2$ of terms bounded by \pref{weightzz1}. We have to prove that it is possible to choose the counterterms $\n_{\omega,\alpha,\alpha',i}$ so that $\n_{h,\omega,\alpha,\alpha',i}$ is bounded by $C \l$ for any scale $h$. Indeed from \pref{appo2} we get, $h\le -1$ \be \nu_{h,\omega,\alpha,\alpha',i}=\g^{-h}(\nu_{\omega,\alpha,\alpha',i}+ \sum_{i=h}^{-1} \g^{i}\b^i_\n) \label{appo3} \ee and choosing $\nu_{\omega,\alpha,\alpha',i}$ so that $\nu_{-\infty,\omega,\alpha,\alpha',i}=0$ we get \be \nu_{h,\omega,\alpha,\alpha',i}=-\g^{-h}\sum_{i=-\infty}^{h} \g^{i}\b^i_\n \label{appo4} \ee and by using a fixed point argument we can show that there is a sequence such that $|\nu_{h,\omega,\alpha,\alpha',i}|\le C \l\g^{h\over 2}$. The application of the above bounds to the 2-point function, in order to derive \pref{43}, is straightforward. The 2-point function can be written as \be \hat S_{j,j}(\mathbf k+\mathbf p_{F,j}^\o )) =\sum_{h=-\infty}^0 [\hat g^{(h)}((\mathbf k+\mathbf p_{F,j}^\o ))+r^h(\mathbf k) \ee where $r^h(\mathbf k) $ includes the contribution of term withs at least a vertex. We can replace in $g^{(h)}((\mathbf k+\mathbf p_{F,j}^\o ))$ the $v_{i,\omega,h},w_{i,\omega,h},Z_{i,\omega,h}$ with $v_{i,\omega,-\infty},w_{i,\omega,-\infty},Z_{i,\omega,-\infty}$ obtaining the dominant term in \pref{43}; the subdominant term is obtained both from the term containing the difference betwen $v_{i,\omega,h}-v_{i,\omega,-\infty}$, $z_{i,\omega,h}-w_{i,\omega,-\infty}$, $Z_{i,\omega,h}-Z_{i,\omega,-\infty}$, which have an extra factor $O(\l \g^{h/2})$, or the terms with at least a vertex which have at least a $\n$ or a non resonant trivial vertex, with an extra $O(\g^{h/2})$ from the bounds after \pref{weightzz}. \section{Conclusion } Theoretical analyses of TBG are based on the assumption of the stability of the Weyl semimetallic phase, leading to the formulation of continuum effective models. However in lattice TBG models wth generic angles there is an emerging quasi-periodicity manifesting themself in large momenta Umklapp interactions that almost connect the Dirac points, similar to the ones appearing in electronic systems with quasi-periodic potential. Such terms are neglected in the continuum semimetallic approximations. In this paper we have rigorously established the stability of the semimetallic phase in a lattice model, taking into full account the large momenta Umklapp interactions. The analysis is based on number theoretical properties of irrationals combined with a Renormalization Group analysis, and requires that the interlayer hopping is weak and short ranged and the the angles are chosen in a large measure set. The effect of the interaction is to produce a finite renormalization of the Dirac points and velocities. Non perturbative effects are excluded as the series are shown to be convergent. Compared to the Aubry-Andr\'e or similar models, the number theoretical analysis is much more involved due to the peculiar structure of the small divisors. The stability of the Weyl phases provides a justification of the use of continnum models under the above assumptions. In addition, the present analysis paves the way to a more accurate evaluation of the velocities as functions of the angles, talking into account lattice or higher orders effects, and the effect of many body interactions, whose interplay with the emerging quasi-periodicity could lead to interesting phases. \begin{acknowledgements} We thank J. Pixley for many interesting discussions. V.M. acknowledges support from the MUR, PRIN 2022 project MaIQuFi cod. 20223J85K3. I.J. gratefully acknowledges support through NSF Grant DMS-2349077, and the Simons Foundation, Grant Number 825876. \end{acknowledgements} \vfill \pagebreak \widetext \appendix \section{Fourier transform of the interlayer hopping}\label{app:fourierV} We write $V$ in Fourier space: we get \begin{eqnarray} &&V=\frac \lambda{|\hat{\mathcal L}_1|^2}\sum_{x_1\in \hat{\mathcal L}_1} \sum_{x'_2\in\Lambda_2}\sum_{\alpha\in\{a,b\}} \int_{\mathbb R^2} \frac{dq}{4\pi^2} \int_{\hat{\mathcal L}_1}dk_1 \int_{\hat{\mathcal L}_2}dk_2'\nonumber\\ && e^{i(k_1x_1-k_2'x_2'+q(x_1-x_2'))} e^{iq(d_\alpha-Rd_{\alpha})} e^{i \xi (k_2'-k_1)} \hat\varsigma(q)\hat c^\dagger_{1,k_1,\alpha}\hat c_{2,k_2',\alpha}+\nonumber\\ &&e^{-i(k_1x_1-k_2'x_2'-q(x_1-x_2'))} e^{iq(d_\alpha-Rd_{\alpha})} e^{-i\xi(k_2'-k_1)} \hat\varsigma(q)\hat c^\dagger_{2,k_2',\alpha}\hat c_{1,k_1,\alpha}\label{V211} \end{eqnarray} and using the Poisson summation formula \begin{equation} \sum_{x_1\in\mathcal L_1} e^{i (k_1+q)x_1}=|\hat{\mathcal L}_1|\sum_{l\in\mathbb Z^2} \d(k_1+q+l b) \end{equation} where we use the shorthand $lb\equiv l_1b_1+l_2b_2$, and \begin{equation} \sum_{x_2'\in \mathcal L_2} e^{-i (k_2+q) x'_2}=|\hat{\mathcal L}_1|\sum_{m\in\mathbb Z^2} \d(k_2+q+m b') \end{equation} Noting that $\hat c_{2,k_1+lb-mb',\alpha} \equiv \hat c_{2,k_1+lb,\alpha}$, $ \hat c_{1,k_2'+mb'-lb,\alpha} \equiv \hat c_{1,k_2'+mb',\alpha}$ we finally obtain \pref{11}. \bigskip Rewriting (\ref{V211}) in terms of Grassmann variables, with the added imaginary time component, reads \be\begin{array}{>\displaystyle l} V=\frac{\beta\lambda}{|\hat\Lambda_1|^2}\sum_{x_1\in\mathcal L_1} \sum_{x'_2\in\mathcal L_2}\sum_{\alpha\in\{a,b\}} \int_{\mathbb R^2} \frac{dq}{4\pi^2} \int_{\hat\Lambda_1}d\kk_1 \int_{\hat\Lambda_2}d\kk_2'\ \delta(k_{1,0}-k_{2,0}') \cdot\\[0.5cm]\cdot \left( e^{i(k_1x_1-k_2x_2'+q(x_1-x_2'))} e^{iq(d_\alpha-Rd_{\alpha})} e^{i\xi(k_2'-k_1)} \hat\varsigma(q)\hat\psi^+_{1,\kk_1,\alpha}\hat\psi^-_{2,\kk_2',\alpha} \label{V2112}+\right.\\[0.5cm]\indent\left.+ e^{-i(k_1x_1-k_2x_2'-q(x_1-x_2'))} e^{iq(d_\alpha-Rd_{\alpha})} e^{-i\xi(k_2'-k_1)} \hat\varsigma(q)\hat\psi^+_{2,\kk_2',\alpha}\hat\psi^-_{1,\kk_1,\alpha} \right) \end{array}\ee where we use the notation $\mathbf k_1=(k_{1,0},k_1)$ and $\mathbf k_2'=(k_{2,0},k_2)$. Again, using the Poisson formula, we find \pref{112}. \section{Proof of Lemma \ref{lemm:dioph}}\label{sec:dioph} To prove lemma \ref{lemm:dioph}, we will first prove a general result on a Diophantine condition for a generic function from $[0,2\pi)$ to $\mathbb R^2$. We will then apply this result to $|p_{F,i}^\omega,p_{F,j}^{\omega'}+lb+mb'|$, viewed as a function of $\theta$, for the various values of $i,j,\omega,\omega'$. \subsection{Diophantine condition from $\mathbb R$ to $\mathbb R^2$} Let us consider an interval $[\theta_0,\theta_1]\subset[0,2\pi]$, and define, given constants $C_1>0,\tau>4$ that are fixed once and for all, two twice-differentiable functions $x:[0,2\pi)\to \mathbb R,f:[0,2\pi)\to \mathbb R^2$, and a subset $\Omega(x,f)\subset[\theta_0,\theta_1]$, \begin{equation}\label{diophantine} \mathcal D(x,f):= \{\theta\in \Omega(x,f):\ \forall k\in\mathbb Z^2\setminus\{0\}, \ \forall l\in\mathbb Z,\ |x(\theta)+l +k \cdot f(\theta)|\geqslant C_1|k|^{-\tau}\} \end{equation} We will show that, provided $\Omega$ is chosen appropriately, under certain conditions on $f$ and $x$, $\mathcal D$ has a large measure. The novelty of this result is that $f$ takes values in $\mathbb R^2$, but is a function of a single variable; if $f$ were a function from $\mathbb R^n$ to $\mathbb R^n$, then the fact that $\mathcal D$ has a large measure would follow from standard arguments \cite{}. Our result is stated for $\mathbb R^2$, but it could easily be adapted to any other dimension, provided $f$ takes a single real-valued argument. \bigskip In order to make our argument work, we will assume that $f'(\theta)$ (the derivative of $f$) remains inside a cone, that is, we assume that $\exists\xi\in \mathbb R^2$ with $|\xi|=1$ and $\alpha\in[0,\frac\pi4)$ such that, $\forall \theta\in [\theta_0,\theta_1]$, \begin{equation} f'(\theta)\in \mathcal C_\xi(\alpha) :=\{y\in \mathbb R^2,\ |y\cdot\xi|>|y|\cos(\alpha)\} . \label{incone} \end{equation} We take the set $\Omega(x,f)$ in (\ref{diophantine}) to be \begin{equation} \Omega(x,f):= \{\theta\in [\theta_0,\theta_1]:\ \forall k\in \zeta, \ |x'(\theta) +k \cdot f'(\theta)|\ge C_3|f'(\theta)||k|^{-\epsilon}\} \label{Omega} \end{equation} where $C_3>0$ is a constant, $\epsilon\in(1,\tau-3)$, and \begin{equation} \zeta:=\mathbb Z^2\setminus(\{0\}\cup\mathcal C_\xi({\textstyle\frac\pi4})) \label{zeta} \end{equation} (the reason why we choose $\Omega$ in this way will become apparent in the proof of Lemma \ref{lemma:diophantine} below). \begin{lemma}\label{lemma:diophantine} If the following estimates hold: \begin{equation} \min_{\theta\in [\theta_0,\theta_1]}|f'(\theta)|>0 ,\ \min_{\theta\in [\theta_0,\theta_1]}|{\textstyle\frac{\partial}{\partial\theta} (\frac{f'(\theta)}{|f'(\theta)|}})|>0 \label{bound_df} \end{equation} $\forall \theta\in [\theta_0,\theta_1]$, \begin{equation} \min_{0<|k|0 \label{small_dx} \end{equation} with \begin{equation} R_1:= \frac{C_3+\frac{|x'(\theta)|}{|f'(\theta)|}}{\cos(\alpha+\frac\pi4)} \label{R1} \end{equation} and, for some $\eta>0$, \begin{equation} \min_{0<|k|0 \label{small_ddx} \end{equation} with \begin{equation} R_2:= \eta+\frac{|{\textstyle \frac \partial{\partial \theta}(\frac{x'(\theta)}{|f'(\theta)|})}|}{|{\textstyle\frac \partial{\partial \theta}(\frac{f'(\theta)}{|f'(\theta)|})}|\cos(\alpha+{\textstyle\frac\pi4})} \label{R2} \end{equation} then the measure of the complement of $\mathcal D$ is bounded by \begin{equation} |[\theta_0,\theta_1]\setminus\mathcal D(x,f)| \le O(C_3)+O({\textstyle \frac{C_1}{C_3 \beta}}) \end{equation} where the constants in $O(\cdot)$ depend only on $\theta_0$, $\theta_1$, $x$, $f$, $\alpha$, $\epsilon$, $\tau$, $\eta$. In particular, if we choose $C_3\ll \theta_1-\theta_0$ and $C_1\ll (\theta_1-\theta_0)^2$, then $\mathcal D(x,f)$ fills most of $[\theta_0,\theta_1]$. \end{lemma} \begin{remark} The conditions (\ref{small_dx}) and (\ref{small_ddx}) concern a finite number of values of $k$. In the applications of this lemma below, we can make both of these conditions trivial by ensuring that $R_1,R_2<1$, which reduces this finite number of values for $k$ to $0$. \end{remark} {\it Proof} Let $|\mathcal D_\Omega^c(x,f)|$ denote the Lebesgue measure of the complement $\Omega(x,f)\setminus\mathcal D(x,f)$. Let \begin{equation} g_{l,k}(\th)=|x(\theta)+l+k \cdot f(\theta)| \end{equation} in terms of which \begin{equation} |\mathcal D_\Omega^c(x,f)|=\sum_{k,l}^* \int_{-C_1|k|^{-\tau}}^{C_1|k|^{-\tau}} \frac1{g_{l,k}'}dg_{l,k} \end{equation} where $\sum^*_{k,l}$ has the constraint that $\exists \theta\in[\theta_0,\theta_1]$ such that $g_{k,l}(\theta)\in[-C_1|k|^{-\tau},C_1|k|^{-\tau}]$. Therefore \be |\mathcal D_\Omega^c(x,f)|\le \sum^*_{l,k} 2C_1 {|k|^{-\tau}\over \min_{\theta\in \Omega(x,f)} |x'(\th)+k\cdot f'(\theta)|} . \ee In addition, the number of values of $l$ such that $g_{k,l}(\theta)\in[-C_1|k|^{-\tau},C_1|k|^{-\tau}]$ is bounded by $C_2|k|$ for some constant $C_2$ (which depends only on $\theta_0,\theta_1,x,f$), and so \be |\mathcal D_\Omega^c(x,f)|\le 2\sum_{k\in \mathbb Z^2\setminus\{0\}} C_2 C_1 {|k|^{1-\tau}\over \min_{\theta\in \Omega(x,f)} |x'(\th)+k\cdot f'(\theta)|} . \label{bound_Dctmp}\ee In order for this bound to be useful, we must obtain a good lower bound on $|x'+k\cdot f'|$. To do so, $\Omega$ must be chosen appropriately: we wish for $k\cdot f'$ to stay as far away from $-x'$ as possible. Now, it cannot avoid it entirely, as $k\cdot f'$ will cover all possible values as $k$ varies in $\mathbb Z^2\setminus\{0\}$. By choosing $\Omega$ as in (\ref{Omega}), we ensure that $k\cdot f'$ may only approach $-x'$ for large values of $k$. In doing so, we can estimate $\mathcal D_\Omega^c$: we split the sum over $\mathbb Z^2\setminus\{0\}$ into a sum over $\zeta$ and a sum over its complement $\zeta^c\equiv\mathbb Z^2\cap\mathcal C_\xi(\frac\pi4)$, and compute a bound for each case. If $k\in \zeta$, then, by (\ref{Omega}), for $\theta\in \Omega(x,f)$, \begin{equation} |x'(\theta)+k\cdot f'(\theta)|\ge C_3|f'(\theta)||k|^{-\epsilon} . \label{boundin} \end{equation} If, on the other hand, $k\in \zeta^c\equiv\mathbb Z^2\cap\mathcal C_\xi(\frac\pi4)$, \begin{equation} |k\cdot f'(\theta)|\ge|k||f'(\theta)|\cos(\alpha+{\textstyle\frac\pi4}) \end{equation} so \begin{equation} |x'(\theta)+k\cdot f'(\theta)|\ge|k||f'(\theta)|\cos(\alpha+{\textstyle\frac\pi4})-|x'(\theta)| . \end{equation} We distinguish two cases once more: either \begin{equation} |k|\ge \frac{C_3+\frac{|x'(\theta)|}{|f'(\theta)|}}{\cos(\alpha+\frac\pi4)}\equiv R_1 \end{equation} (see (\ref{R1})) in which case (\ref{boundin}) holds true for these $k$'s as well, or $|k| |k||{\textstyle\frac \partial{\partial \theta}(\frac{f'(\theta)}{|f'(\theta)|})}|\cos(\alpha+{\textstyle\frac\pi4}) . \end{equation} Thus, \begin{equation} |{\textstyle\frac \partial{\partial \theta}(\frac{x'(\theta)}{|f'(\theta)|})}+k\cdot {\textstyle\frac \partial{\partial \theta}(\frac{f'(\theta)}{|f'(\theta)|})}| > |k||{\textstyle\frac \partial{\partial \theta}(\frac{f'(\theta)}{|f'(\theta)|})}|\cos(\alpha+{\textstyle\frac\pi4}) -|{\textstyle\frac \partial{\partial \theta}(\frac{x'(\theta)}{|f'(\theta)|})}| . \end{equation} Therefore, if \begin{equation} |k|\ge \eta+\frac{|{\textstyle \frac \partial{\partial \theta}(\frac{x'(\theta)}{|f'(\theta)|})}|}{|{\textstyle\frac \partial{\partial \theta}(\frac{f'(\theta)}{|f'(\theta)|})}|\cos(\alpha+{\textstyle\frac\pi4})} \equiv R_2 \end{equation} then \begin{equation} |{\textstyle\frac \partial{\partial \theta}(\frac{x'(\theta)}{|f'(\theta)|})}+k\cdot {\textstyle\frac \partial{\partial \theta}(\frac{f'(\theta)}{|f'(\theta)|})}| >\eta|k||{\textstyle\frac \partial{\partial \theta}(\frac{f'(\theta)}{|f'(\theta)|})}|\cos(\alpha+{\textstyle\frac\pi4}) . \label{tmpineq} \end{equation} If, on the other hand, $|k|1$. We conclude the proof by combining (\ref{bound_D}) with (\ref{bound_Omega}). \qed \subsection{Applying the Diophantine condition to $|p_{F,i}^\omega-p_{F,j}^{\omega'}+lb+mb'|$} We now apply lemma \ref{lemma:diophantine} repeatedly to prove lemma \ref{lemm:dioph}. Let us first consider the case $i=j$, $\omega=\omega'$, and $y=m$, that is, we wish to find a condition on $\theta$ such that \begin{equation} |p_{F,i}^\omega-p_{F,j}^{\omega'}+lb+mb'| \equiv |lb+mb'| \equiv |M| \ge\frac{C_0}{|m|^\tau} \label{cond11} \end{equation} (recall (\ref{cond}) and (\ref{M})). We recall (\ref{Mineq}): \begin{equation} |M|\ge \frac{2\pi}{\sqrt3}(l_\omega+m\cdot f_\omega) \end{equation} with $l_+\equiv l_1$ and $l_-\equiv l_2$, and \begin{equation} f_\omega(\theta):=({\textstyle \frac{\varphi_1-\omega\varphi_3}2},\ {\textstyle \frac{\varphi_2+\omega\varphi_4}2}) . \end{equation} Now, recalling the definition (\ref{diophantine}), we have that if $\theta\in \mathcal D(0,f_\omega)$, then the inequality (\ref{cond}) with $i=i'$, $\omega=\omega'$, and $y=m$ holds with $C_0:=\frac{2\pi}{\sqrt3}C_1$. We therefore just need to use Lemma \ref{lemma:diophantine} to ensure that the measure of this set is large. Taking $\theta_0,\theta_1$ sufficiently small, it suffices to verify the conditions at $\theta=0$, and conclude by continuity. In particular, when $\theta_0,\theta_1$ are small, $f'(\theta)$ will take values in a small cone $\mathcal C_{f'(0)}(\alpha)$ with $\alpha=O(\theta_1)$. Next, by a straightforward computation, we find \begin{equation} |f'_\omega(0)|=\sqrt{\frac53} ,\quad \left|\frac\partial{\partial \theta}\frac{f'_\omega(0)}{|f'_\omega(0)|}\right|=\frac{2\sqrt3}5 \end{equation} Which are both non-zero, so (\ref{bound_df}) is satisfied for small enough $\theta$. Since $x=0$, the other assumptions trivially hold: we choose $C_3<1/\sqrt2$ and $\eta<1$ such that $R_1,R_2<1$, in which case the minima in (\ref{small_dx}) and (\ref{small_ddx}) are taken over empty sets, so (\ref{small_dx}) and (\ref{small_ddx}) hold trivially. Thus, by Lemma \ref{lemma:diophantine}, choosing $C_1=O(\theta_1^2)$, the set $\mathcal D(0,f_\omega)$ has a large measure. We now repeat the argument for the other values of $i,j$, $y$, and $\omega,\omega'$. First, note that $p_F^+-p_F^-=\frac13(b_1-b_2)$ so the condition (\ref{cond}) holds for $\o\neq \o'$ whenever it holds for $\omega=\omega'$. Next, note that \begin{equation} |p_{F,i}^\omega-p_{F,j}^\omega+lb+mb'| = |R^T(p_{F,i}^\omega-p_{F,j}^\omega+lb+mb')| \end{equation} which corresponds to exchanging $m$ and $l$, and flipping the sign of $\theta$. The arguments made for $\theta$ may be adapted in a straightforward way to the case $-\theta$ so our derivation for $y=m$ also applies to $y=l$. We are thus left with the case $i\neq j$, $\omega=\omega'$, and $y=m$. Without loss of generality, we choose $i=1$, $j=2$, and we wish to bound \begin{equation} |p_{F,1}^\omega-p_{F,2}^{\omega}+lb+mb'| \ge\frac{C_0}{|m|^\tau} \label{cond12} \end{equation} Proceeding as we did above, we bound \begin{equation} |p_{F,1}^\omega-p_{F,2}^{\omega}+lb+mb'| \ge \frac{2\pi}{\sqrt3}\left( x_\omega(\theta) +l_1+m\cdot f_\omega(\theta) \right) \end{equation} with \begin{equation} x_\omega(\theta):=\frac{1}3(1-c_\theta)+\omega\frac{1}{\sqrt3}s_\theta . \end{equation} Therefore, if $\theta\in \mathcal D(x_\omega,f_\omega)$, then (\ref{cond12}) holds with $C_0=\frac{2\pi}{\sqrt3}C_1$. To show that this set has a large measure, we check the assumptions of Lemma \ref{lemma:diophantine}, as we did above. Again, we check the assumptions at $\theta=0$, and argue by continuity. We compute \begin{equation} |x'_\omega(0)|=\frac1{\sqrt3} ,\quad \left|\frac\partial{\partial \theta}\frac{x'_\omega(0)}{|f'_\omega(0)|}\right|=\frac{8}{5\sqrt{15}} \end{equation} We thus find that if $C_3<1/\sqrt2-1/\sqrt5$ and $\eta<1-4\sqrt2/\sqrt{45}$, then $R_1,R_2<1$, so the minima in (\ref{small_dx}) and (\ref{small_ddx}) are taken over empty sets, so (\ref{small_dx}) and (\ref{small_ddx}) hold trivially. \bigskip All in all, we have found that if we restrict the values of $\theta$ to an intersection of Diophantine sets: \begin{equation} \theta\in \bigcap_{\omega=\pm}\bigcap_{\sigma=\pm}\mathcal D(0,f_\omega(\sigma \theta)) \cap \bigcap_{\omega=\pm}\bigcap_{\sigma=\pm}\mathcal D(x_\omega(\sigma \theta),f_\omega(\sigma \theta)) \end{equation} then (\ref{cond}) is satisfied for any value of $i,i'$, $\omega,\omega'$, and $y$ with a constant $C_0=O(\theta_1^2)$. Because each set has an arbitrarily large measure (relative to $[\theta_0,\theta_1]$), their intersection also does. \section{Naive perturbation theory} \label{app:explicit_feynman} The Schwinger function is computed using perturbation theory: formally, \begin{equation} \begin{array}{>\displaystyle l} (S_{1,1}(\mathbf k))_{\alpha',\alpha}= \sum_{N=0}^\infty \sum_{\alpha_0,\cdots,\alpha_{2N+1}} (g_1(\mathbf k))_{\alpha,\alpha_0} \left(\prod_{n=0}^N \left( \sum_{l_{2n}}\tau_{l_{2n},\alpha_{2n}}^{(1)}(k+m_{2n-1}b'+l_{2n}b) (g_2(\mathbf k+l_{2n}b))_{\alpha_{2n},\alpha_{2n+1}} \cdot\right.\right.\\\hfill\cdot\left.\left. \sum_{m_{2n+1}}\tau_{m_{2n+1},\alpha_{2n+1}}^{(2)}(k+l_{2n}b+m_{2n+1}b') ((g_1(\mathbf k+m_{2n+1}b'))_{\alpha_{2n+1},\alpha_{2n+2}})^{\mathds 1_{n\displaystyle l} (S_{2,2}(\mathbf k))_{\alpha',\alpha}= \sum_{N=0}^\infty \sum_{\alpha_0,\cdots,\alpha_{2N+1}} (g_2(\mathbf k))_{\alpha,\alpha_0} \left(\prod_{n=0}^N \left( \sum_{m_{2n}}\tau_{m_{2n},\alpha_{2n}}^{(2)}(k+l_{2n-1}b+m_{2n}b') (g_1(\mathbf k+m_{2n}b'))_{\alpha_{2n},\alpha_{2n+1}} \cdot\right.\right.\\\hfill\cdot\left.\left. \sum_{l_{2n+1}}\tau_{l_{2n+1},\alpha_{2n+1}}^{(1)}(k+m_{2n}b'+l_{2n+1}b) ((g_2(\mathbf k+l_{2n+1}b))_{\alpha_{2n+1},\alpha_{2n+2}})^{\mathds 1_{n\displaystyle l} (S_{2,1}(\mathbf k))_{\alpha',\alpha}= \sum_{N=0}^\infty \sum_{\alpha_0,\cdots,\alpha_{2N}} (g_1(\mathbf k))_{\alpha,\alpha_0} \left(\prod_{n=0}^N \left( \sum_{l_{2n}}\tau_{l_{2n},\alpha_{2n}}^{(1)}(k+m_{2n-1}b'+l_{2n}b) ((g_2(\mathbf k+l_{2n}b))_{\alpha_{2n},\alpha_{2n+1}})^{\mathds 1_{n\displaystyle l} (S_{1,2}(\mathbf k))_{\alpha',\alpha}= \sum_{N=0}^\infty \sum_{\alpha_0,\cdots,\alpha_{2N}} (g_2(\mathbf k))_{\alpha,\alpha_0} \left(\prod_{n=0}^N \left( \sum_{m_{2n}}\tau_{m_{2n},\alpha_{2n}}^{(2)}(k+l_{2n-1}b+m_{2n}b') ((g_1(\mathbf k+m_{2n}b'))_{\alpha_{2n},\alpha_{2n+1}})^{\mathds 1_{n