From 0a58a25b1986dbd64052233f7b9378e74324df3e Mon Sep 17 00:00:00 2001 From: Ian Jauslin Date: Tue, 14 Oct 2025 15:15:25 -0400 Subject: [PATCH] Initial commit --- Jauslin_Mastropietro_2025.tex | 2269 +++++++++++++++++++++++++++++ Makefile | 39 + README | 32 + figs/cluster.fig/Makefile | 23 + figs/cluster.fig/cluster.tikz.tex | 33 + figs/feynman.fig/Makefile | 23 + figs/feynman.fig/feynman.tikz.tex | 41 + 7 files changed, 2460 insertions(+) create mode 100644 Jauslin_Mastropietro_2025.tex create mode 100644 Makefile create mode 100644 README create mode 100644 figs/cluster.fig/Makefile create mode 100644 figs/cluster.fig/cluster.tikz.tex create mode 100644 figs/feynman.fig/Makefile create mode 100644 figs/feynman.fig/feynman.tikz.tex diff --git a/Jauslin_Mastropietro_2025.tex b/Jauslin_Mastropietro_2025.tex new file mode 100644 index 0000000..feafde2 --- /dev/null +++ b/Jauslin_Mastropietro_2025.tex @@ -0,0 +1,2269 @@ +\documentclass[a4paper,preprintnumbers,amsmath,amssymb,twocolumn,10pt]{revtex4} + +\usepackage{graphicx} +\usepackage{dcolumn} +\usepackage{bm} +\usepackage{epstopdf} +\usepackage{dsfont} +\usepackage{color} +\usepackage{amsthm} + + +\newcount\driver +\newcount\bozza + + +\font\cs=cmcsc10 scaled\magstep1 +\font\ottorm=cmr8 scaled\magstep1 \font\msxtw=msbm10 +scaled\magstep1 \font\euftw=eufm10 +scaled\magstep1 \font\msytw=msbm10 scaled\magstep1 +\font\msytww=msbm8 scaled\magstep1 \font\msytwww=msbm7 +scaled\magstep1 \font\indbf=cmbx10 scaled\magstep2 +\font\grbold=cmmib10 scaled\magstep1 +\font\amit=cmmi7 \def\sf{\textfont1=\amit} \font\bigtenrm=cmr10 +scaled \magstep2 \font\bigteni=cmmi10 scaled +\magstep1 + +{\count255=\time\divide\count255 by 60 +\xdef\hourmin{\number\count255} + \multiply\count255 by-60\advance\count255 by\time + \xdef\hourmin{\hourmin:\ifnum\count255<10 0\fi\the\count255}} + + + + +\let\a=\alpha \let\b=\beta \let\g=\gamma \let\d=\delta \let\e=\varepsilon +\let\z=\zeta \let\h=\eta \let\th=\vartheta \let\k=\kappa \let\l=\lambda +\let\m=\mu \let\n=\nu \let\x=\xi \let\p=\pi \let\r=\rho +\let\s=\sigma \let\t=\tau \let\f=\varphi \let\ph=\varphi \let\c=\chi +\let\ps=\psi \let\y=\upsilon \let\o=\omega \let\si=\varsigma +\let\G=\Gamma \let\D=\Delta \let\Th=\Theta \let\L=\Lambda \let\X=\Xi +\let\P=\Pi \let\Si=\Sigma \let\F=\Phi \let\Ps=\Psi +\let\O=\Omega \let\Y=\Upsilon + + +\def\PP{{\cal P}}\def\EE{{\cal E}}\def\MM{{\cal M}}\def\VV{{\cal V}} +\def\FF{{\cal F}}\def\HH{{\cal H}}\def\WW{{\cal W}} +\def\TT{{\cal T}}\def\NN{{\cal N}}\def\BB{{\cal B}}\def\ZZ{{\cal Z}} +\def\RR{{\cal R}}\def\LL{{\cal L}}\def\JJ{{\cal J}}\def\QQ{{\cal Q}} +\def\DD{{\cal D}}\def\AA{{\cal A}}\def\GG{{\cal G}}\def\SS{{\cal S}} +\def\OO{{\cal O}}\def\XXX{{\bf X}}\def\YYY{{\bf Y}}\def\WWW{{\bf W}} +\def\KK{{\cal K}} + + +\def\ggg{{\bf g}}\def\fff{{\bf f}}\def\ff{{\bf f}} + + +\def\pp{{\bf p}}\def\qq{{\bf q}}\def\ii{{\bf i}}\def\xx{{\bf x}} +\def\aaa{{\bf a}} \def\bb{{\bf 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+\def\DD{{\cal D}}\def\AA{{\cal A}}\def\GG{{\cal G}}\def\SS{{\cal S}} +\def\OO{{\cal O}}\def\AAA{{\cal A}} + + +\def\T#1{{#1_{\kern-3pt\lower7pt\hbox{$\widetilde{}$}}\kern3pt}} +\def\VVV#1{{\underline #1}_{\kern-3pt +\lower7pt\hbox{$\widetilde{}$}}\kern3pt\,} +\def\W#1{#1_{\kern-3pt\lower7.5pt\hbox{$\widetilde{}$}}\kern2pt\,} +\def\Re{{\rm Re}\,}\def\Im{{\rm Im}\,} +\def\lis{\overline}\def\tto{\Rightarrow} +\def\etc{{\it etc}} \def\acapo{\hfill\break} +\def\mod{{\rm mod}\,} \def\per{{\rm per}\,} \def\sign{{\rm sign}\,} +\def\indica{\leaders \hbox to 0.5cm{\hss.\hss}\hfill} +\def\guida{\leaders\hbox to 1em{\hss.\hss}\hfill} +\mathchardef\oo= "0521 + + +\def\V#1{{\bf #1}} +\def\pp{{\bf p}}\def\qq{{\bf q}}\def\ii{{\bf i}}\def\xx{{\bf x}} +\def\yy{{\bf y}}\def\kk{{\bf k}}\def\mm{{\bf m}}\def\nn{{\bf n}} +\def\dd{{\bf d}}\def\zz{{\bf z}}\def\uu{{\bf u}}\def\vv{{\bf v}} +\def\xxi{\hbox{\grbold \char24}} \def\bP{{\bf P}}\def\rr{{\bf r}} +\def\tt{{\bf t}} \def\bz{{\bf 0}} +\def\ss{{\underline \sigma}}\def\oo{{\underline \omega}} +\def\xxx{{\underline\xx}} +\def\qed{\raise1pt\hbox{\vrule height5pt width5pt depth0pt}} +\def\barf#1{{\tilde \f_{#1}}} \def\tg#1{{\tilde g_{#1}}} +\def\bq{{\bar q}} \def\bh{{\bar h}} \def\bp{{\bar p}} \def\bpp{{\bar \pp}} +\def\Val{{\rm Val}} +\def\indic{\hbox{\raise-2pt \hbox{\indbf 1}}} +\def\bk#1#2{\bar\kk_{#1#2}} +\def\tdh{{\tilde h}} + + +\def\RRR{\hbox{\msytw R}} \def\rrrr{\hbox{\msytww R}} +\def\rrr{\hbox{\msytwww R}} +\def\NNN{\hbox{\msytw N}} \def\nnnn{\hbox{\msytww N}} +\def\nnn{\hbox{\msytwww N}} \def\ZZZ{\hbox{\msytw Z}} +\def\zzzz{\hbox{\msytww Z}} \def\zzz{\hbox{\msytwww Z}} +\def\TTT{\hbox{\msytw T}} \def\tttt{\hbox{\msytww T}} +\def\ttt{\hbox{\msytwww T}} + + + + +\def\ins#1#2#3{\vbox to0pt{\kern-#2 \hbox{\kern#1 #3}\vss}\nointerlineskip} + + +\newdimen\xshift \newdimen\xwidth \newdimen\yshift + + +\def\insertplot#1#2#3#4#5#6{\xwidth=#1pt \xshift=\hsize \advance\xshift by-\xwidth \divide\xshift by 2\begin{figure}[ht] +\vspace{#2pt} \hspace{\xshift} +\begin{minipage}{#1pt} +#3 \ifnum\driver=1 \griglia=#6 +\ifnum\griglia=1 \openout13=griglia.ps \write13{gsave .2 +setlinewidth} \write13{0 10 #1 {dup 0 moveto #2 lineto } for} +\write13{0 10 #2 {dup 0 exch moveto #1 exch lineto } for} +\write13{stroke} \write13{.5 setlinewidth} \write13{0 50 #1 {dup 0 +moveto #2 lineto } for} \write13{0 50 #2 {dup 0 exch moveto #1 +exch lineto } for} \write13{stroke grestore} \closeout13 +\special{psfile=griglia.ps} \fi +\special{psfile=#4.ps}\fi\ifnum\driver=2 \special{pdf:epdf (#4.pdf)}\fi +\end{minipage} +\caption{#5} +\end{figure} +} + + +\def\gtopl{\hbox{\msxtw \char63}} +\def\ltopg{\hbox{\msxtw \char55}} + + +\newdimen\shift \shift=-1.5truecm +\def\lb#1{\ifnum\bozza=1 +\label{#1}\rlap{\hbox{\hskip\shift$\scriptstyle#1$}} +\else\label{#1} \fi} + + +\def\be{\begin{equation}} +\def\ee{\end{equation}} +\def\bea{\begin{eqnarray}}\def\eea{\end{eqnarray}} +\def\bean{\begin{eqnarray*}}\def\eean{\end{eqnarray*}} +\def\bfr{\begin{flushright}}\def\efr{\end{flushright}} +\def\bc{\begin{center}}\def\ec{\end{center}} +\def\bal{\begin{align}}\def\eal{\end{align}} +\def\ba#1{\begin{array}{#1}} \def\ea{\end{array}} +\def\bd{\begin{description}}\def\ed{\end{description}} +\def\bv{\begin{verbatim}}\def\ev{\end{verbatim}} +\def\nn{\nonumber} +\def\Halmos{\hfill\vrule height10pt width4pt depth2pt \par\hbox to \hsize{}} +\def\pref#1{(\ref{#1})} +\def\Dim{{\bf Dim. -\ \ }} \def\Sol{{\bf Soluzione -\ \ }} +\def\virg{\quad,\quad} +\def\bsl{$\backslash$} + + + + + + +\def\ins#1#2#3{\vbox to0pt{\kern-#2 \hbox{\kern#1 #3}\vss}\nointerlineskip} + + +\newdimen\xshift \newdimen\xwidth \newdimen\yshift +\newcount\griglia + + +\def\insertplot#1#2#3#4#5#6{\xwidth=#1pt \xshift=\hsize \advance\xshift by-\xwidth \divide\xshift by 2\begin{figure}[ht] +\vspace{#2pt} \hspace{\xshift} +\begin{minipage}{#1pt} +#3 \ifnum\driver=1 \griglia=#6 +\ifnum\griglia=1 \openout13=griglia.ps \write13{gsave .2 +setlinewidth} \write13{0 10 #1 {dup 0 moveto #2 lineto } for} +\write13{0 10 #2 {dup 0 exch moveto #1 exch lineto } for} +\write13{stroke} \write13{.5 setlinewidth} \write13{0 50 #1 {dup 0 +moveto #2 lineto } for} \write13{0 50 #2 {dup 0 exch moveto #1 +exch lineto } for} \write13{stroke grestore} \closeout13 +\special{psfile=griglia.ps} \fi +\special{psfile=#4.ps}\fi\ifnum\driver=2 \special{pdf:epdf (#4.pdf)}\fi +\end{minipage} +\caption{#5} +\end{figure} +} + + +\def\gtopl{\hbox{\msxtw \char63}} +\def\ltopg{\hbox{\msxtw \char55}} + + +\newdimen\shift \shift=-1.5truecm +\def\lb#1{\label{#1}\rlap{\hbox{\hskip\shift$\scriptstyle#1$}} +\else\label{#1} \fi} + + +\def\be{\begin{equation}} +\def\ee{\end{equation}} +\def\bea{\begin{eqnarray}}\def\eea{\end{eqnarray}} +\def\bean{\begin{eqnarray*}}\def\eean{\end{eqnarray*}} +\def\bfr{\begin{flushright}}\def\efr{\end{flushright}} +\def\bc{\begin{center}}\def\ec{\end{center}} +\def\bal{\begin{align}}\def\eal{\end{align}} +\def\ba#1{\begin{array}{#1}} \def\ea{\end{array}} +\def\bd{\begin{description}}\def\ed{\end{description}} +\def\bv{\begin{verbatim}}\def\ev{\end{verbatim}} +\def\nn{\nonumber} +\def\Halmos{\hfill\vrule height10pt width4pt depth2pt \par\hbox to \hsize{}} +\def\pref#1{(\ref{#1})} +\def\Dim{{\bf Dim. -\ \ }} \def\Sol{{\bf Soluzione -\ \ }} +\def\virg{\quad,\quad} +\def\bsl{$\backslash$} + + + + +\driver=1 \bozza=0 + +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{epstopdf} + + +\font\msytw=msbm9 scaled\magstep1 \font\msytww=msbm7 +scaled\magstep1 \font\msytwww=msbm5 scaled\magstep1 +\font\cs=cmcsc10 + +\let\a=\alpha \let\b=\beta \let\g=\gamma \let\d=\delta +\let\e=\varepsilon +\let\z=\zeta \let\h=\eta \let\th=\theta \let\k=\kappa \let\l=\lambda +\let\m=\mu \let\n=\nu \let\x=\xi \let\p=\pi \let\r=\rho +\let\s=\sigma \let\t=\tau \let\f=\varphi \let\ph=\varphi\let\c=\chi +\let\ps=\Psi \let\y=\upsilon \let\o=\omega\let\si=\varsigma +\let\G=\Gamma \let\D=\Delta \let\Th=\Theta\let\L=\Lambda \let\X=\Xi +\let\P=\Pi \let\Si=\Sigma \let\F=\Phi \let\Ps=\Psi +\let\O=\Omega \let\Y=\Upsilon + +\def\PPP{{\cal P}}\def\EE{{\cal E}}\def\MM{{\cal M}} \def\VV{{\cal V}} +\def\FF{{\cal F}} \def\HHH{{\cal H}}\def\WW{{\cal W}} +\def\TT{{\cal T}}\def\NN{{\cal N}} \def\BBB{{\cal B}}\def\III{{\cal I}} +\def\RR{{\cal R}}\def\LL{{\cal L}} \def\JJ{{\cal J}} \def\OO{{\cal O}} +\def\DD{{\cal D}}\def\AAA{{\cal A}}\def\GG{{\cal G}} \def\SS{{\cal S}} +\def\KK{{\cal K}}\def\UU{{\cal U}} \def\QQ{{\cal Q}} \def\XXX{{\cal X}} + + +\def\qq{{\bf q}} \def\pp{{\bf p}} +\def\vv{{\bf v}} \def\xx{{\bf x}} \def\yy{{\bf y}} \def\zz{{\bf z}} +\def\aa{{\bf a}}\def\hh{{\bf h}}\def\kk{{\bf k}} +\def\mm{{\bf m}}\def\PP{{\bf P}} + +\def\dd{{\boldsymbol{\delta}}} + +\def\ddd{\boldsymbol{\d}} +\def\TTTT{\mathbf{T}} + +\def\nn{\nonumber} +\def\us{\underset} +\def\os{\overset} + +\def\RRR{\hbox{\msytw R}} \def\rrrr{\hbox{\msytww R}} +\def\rrr{\hbox{\msytwww R}} +\def\NNN{\hbox{\msytw N}} \def\nnnn{\hbox{\msytww N}} +\def\nnn{\hbox{\msytwww N}} \def\ZZZ{\hbox{\msytw Z}} +\def\zzzz{\hbox{\msytww Z}} \def\zzz{\hbox{\msytwww Z}} +\def\TTT{\hbox{\msytw T}} + + + +\def\\{\hfill\break} +\def\={:=} +\let\io=\infty +\let\0=\noindent\def\pagina{{\vfill\eject}} +\def\media#1{{\langle#1\rangle}} +\let\dpr=\partial +\def\sign{{\rm sign}} +\def\const{{\rm const}} +\def\tende#1{\,\vtop{\ialign{##\crcr\rightarrowfill\crcr\noalign{\kern-1pt + \nointerlineskip} \hskip3.pt${\scriptstyle #1}$\hskip3.pt\crcr}}\,} +\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}}}](2*\i-2,0)--(2*\i,0); + \draw[decorate, decoration={snake}](2*\i,2)--++(0,-2); + \fill(2*\i,0)circle(0.1); +} +\draw[postaction=decorate, decoration={markings, mark=at position .5 with {\arrow{>}}}](14,0)--(16,0); + +\draw(1,0.25)node{$h$}; +\draw(3,0.25)node{$h_1$}; +\draw(5,0.25)node{$h_2$}; +\draw(7,0.25)node{$h_3$}; +\draw(9,0.25)node{$h_2$}; +\draw(11,0.25)node{$h_1$}; +\draw(13,0.25)node{$h_1$}; +\draw(15,0.25)node{$h$}; + +\draw[line width=0.25em](1.75,-1)--++(0,4)--++(12.5,0)--++(0,-4)--++(-12.5,0); +\draw[line width=0.25em](3.75,-0.75)--++(0,3.5)--++(6.5,0)--++(0,-3.5)--++(-6.5,0); +\draw[line width=0.25em](5.75,-0.5)--++(0,3)--++(2.5,0)--++(0,-3)--++(-2.5,0); +\draw[line width=0.25em](11.75,-0.5)--++(0,3)--++(0.5,0)--++(0,-3)--++(-0.5,0); + + +\end{tikzpicture} +\end{document} diff --git a/figs/feynman.fig/Makefile b/figs/feynman.fig/Makefile new file mode 100644 index 0000000..8820d7e --- /dev/null +++ b/figs/feynman.fig/Makefile @@ -0,0 +1,23 @@ +PROJECTNAME=$(basename $(basename $(wildcard *.tikz.tex))) + +PDFS=$(addsuffix .pdf, $(PROJECTNAME)) + +all: $(PDFS) + +$(PDFS): + pdflatex -jobname $(basename $@) $(patsubst %.pdf, %.tikz.tex, $@) + + +install: $(PDFS) + cp $^ $(INSTALLDIR)/ + + +clean-aux: + rm -f $(addsuffix .aux, $(PROJECTNAME)) + rm -f $(addsuffix .log, $(PROJECTNAME)) + rm -f $(addsuffix .out, $(PROJECTNAME)) + +clean-tex: + rm -f $(PDFS) $(SYNCTEXS) + +clean: clean-aux clean-tex diff --git a/figs/feynman.fig/feynman.tikz.tex b/figs/feynman.fig/feynman.tikz.tex new file mode 100644 index 0000000..0276b54 --- /dev/null +++ b/figs/feynman.fig/feynman.tikz.tex @@ -0,0 +1,41 @@ +\documentclass{standalone} + +\usepackage{tikz} +\usetikzlibrary{decorations.pathmorphing,decorations.markings} + +\begin{document} +\begin{tikzpicture} + + +\foreach \i in {1,...,2}{ + \draw[postaction=decorate, decoration={markings, mark=at position .5 with {\arrow{>}}}](4*\i-2,0)--(4*\i,0); + \draw[decorate, decoration={snake}](4*\i,3)--++(0,-3); + \fill(4*\i,0)circle(0.1); +} +\foreach \i in {1,...,2}{ + \draw[postaction=decorate, decoration={markings, mark=at position .5 with {\arrow{>}}}](4*\i-4,0)--(4*\i-2,0); + \draw[decorate, decoration={snake}](4*\i-2,2)--++(0,-2); + \fill(4*\i-2,0)circle(0.1); +} +\draw[postaction=decorate, decoration={markings, mark=at position .5 with {\arrow{>}}}](8,0)--(10,0); + +\draw(1,0.25)node{$1$}; +\draw(3,0.25)node{$2$}; +\draw(5,0.25)node{$1$}; +\draw(7,0.25)node{$2$}; +\draw(9,0.25)node{$1$}; + +\draw(1,-0.25)node{$\mathbf k$}; +\draw(3,-0.25)node{$\mathbf k+l_1b$}; +\draw(5,-0.25)node{$\mathbf k+m_2b'$}; +\draw(7,-0.25)node{$\mathbf k+l_3b$}; +\draw(9,-0.25)node{$\mathbf k$}; + +\draw(2,2.25)node{$\tau^{(1)}_{l_1}(\mathbf k+l_1b)$}; +\draw(4,3.25)node{$\tau^{(2)}_{m_2}(\mathbf k+l_1b+m_2b')$}; +\draw(6,2.25)node{$\tau^{(1)}_{l_3}(\mathbf k+m_2b'+l_3b)$}; +\draw(8,3.25)node{$\tau^{(2)}_{0}(\mathbf k+l_3b)$}; + + +\end{tikzpicture} +\end{document}