From c6a2b9ffb580c4f50f416fe4c673322875dd6ded Mon Sep 17 00:00:00 2001
From: Ian Jauslin <ian.jauslin@roma1.infn.it>
Date: Mon, 26 Oct 2015 13:53:12 +0000
Subject: Typos, misformats, update style files, add DOIs in bibliography

---
 Giuliani_Jauslin_2015.tex | 161 +++++++++++++++++++++++-----------------------
 1 file changed, 80 insertions(+), 81 deletions(-)

(limited to 'Giuliani_Jauslin_2015.tex')

diff --git a/Giuliani_Jauslin_2015.tex b/Giuliani_Jauslin_2015.tex
index 3b08656..334f3a1 100644
--- a/Giuliani_Jauslin_2015.tex
+++ b/Giuliani_Jauslin_2015.tex
@@ -36,21 +36,21 @@
 
 \section{Introduction}
 \label{introduction}
-\indent Graphene, a one-atom thick layer of graphite, has captivated a large part of the scientific community for the past decade. With good reason: as was shown by A.~Geim's team, graphene is a stable two-dimensional crystal with very peculiar electronic properties~[\cite{ngeZF}]. The mere fact that a two-dimensional crystal can be synthesized, and manipulated, at room temperature without working inside a vacuum~[\cite{geOZ}] is, in and of itself, quite surprising. But the most interesting features of graphene lay within its electronic properties. Indeed, electrons in graphene were found to have an extremely high mobility~[\cite{ngeZF}], which could make it a good candidate to replace silicon in microelectronics; and they were later found to behave like massless Dirac Fermions~[\cite{ngeZFi}, \cite{zteZFi}], which is of great interest for the study of fundamental Quantum Electro-Dynamics. These are but a few of the intriguing features~[\cite{geiZSe}] that have prompted a lively response from the scientific community.\par
-\indent These peculiar electronic properties stem from the particular energy structure of graphene. It consists of two energy bands, that meet at exactly two points, called the {\it Fermi points}~[\cite{walFSe}]. Graphene is thus classified as a {\it semi-metal}: it is not a {\it semi-conductor} because there is no gap between its energy bands, nor is it a {\it metal} either since the bands do not overlap, so that the density of charge carriers vanishes at the Fermi points. Furthermore, the bands around the Fermi points are approximately conical~[\cite{walFSe}], which explains the masslessness of the electrons in graphene, and in turn their high mobility.\par
+\indent Graphene, a one-atom thick layer of graphite, has captivated a large part of the scientific community for the past decade. With good reason: as was shown by A.~Geim's team, graphene is a stable two-dimensional crystal with very peculiar electronic properties~[\cite{NGe04}]. The mere fact that a two-dimensional crystal can be synthesized, and manipulated, at room temperature without working inside a vacuum~[\cite{Ge11}] is, in and of itself, quite surprising. But the most interesting features of graphene lay within its electronic properties. Indeed, electrons in graphene were found to have an extremely high mobility~[\cite{NGe04}], which could make it a good candidate to replace silicon in microelectronics; and they were later found to behave like massless Dirac Fermions~[\cite{NGe05}, \cite{ZTe05}], which is of great interest for the study of fundamental Quantum Electro-Dynamics. These are but a few of the intriguing features~[\cite{GN07}] that have prompted a lively response from the scientific community.\par
+\indent These peculiar electronic properties stem from the particular energy structure of graphene. It consists of two energy bands, that meet at exactly two points, called the {\it Fermi points}~[\cite{Wa47}]. Graphene is thus classified as a {\it semi-metal}: it is not a {\it semi-conductor} because there is no gap between its energy bands, nor is it a {\it metal} either since the bands do not overlap, so that the density of charge carriers vanishes at the Fermi points. Furthermore, the bands around the Fermi points are approximately conical~[\cite{Wa47}], which explains the masslessness of the electrons in graphene, and in turn their high mobility.\par
 \bigskip
 
-\indent Graphene is also interesting for the mathematical physics community: its free energy and correlation functions, in particular its conductivity, can be computed non-perturbatively using constructive Renormalization Group (RG) techniques~[\cite{giuOZ}, \cite{gmpOO}, \cite{gmpOT}], at least if it is at {\it half-filling}, the interaction is {\it short-range} and its strength is {\it small enough}. This is made possible, again, by the special energy structure of graphene. Indeed, since the {\it propagator} (in the quantum field theory formalism) diverges at the Fermi points, the fact that there are only two such singularities in graphene instead of a whole line of them (which is what one usually finds in two-dimensional theories), greatly simplifies the RG analysis. Furthermore, the fact that the bands are approximately conical around the Fermi points, implies that a short-range interaction between electrons is {\it irrelevant} in the RG sense, which means that one need only worry about the renormalization of the propagator, which can be controlled.\par
-\indent Using these facts, the formalism developed in~[\cite{benNZ}] has been applied in~[\cite{giuOZ}, \cite{gmpOT}] to express the free energy and correlation functions as convergent series.\par
+\indent Graphene is also interesting for the mathematical physics community: its free energy and correlation functions, in particular its conductivity, can be computed non-perturbatively using constructive Renormalization Group (RG) techniques~[\cite{GM10}, \cite{GMP11}, \cite{GMP12}], at least if it is at {\it half-filling}, the interaction is {\it short-range} and its strength is {\it small enough}. This is made possible, again, by the special energy structure of graphene. Indeed, since the {\it propagator} (in the quantum field theory formalism) diverges at the Fermi points, the fact that there are only two such singularities in graphene instead of a whole line of them (which is what one usually finds in two-dimensional theories), greatly simplifies the RG analysis. Furthermore, the fact that the bands are approximately conical around the Fermi points, implies that a short-range interaction between electrons is {\it irrelevant} in the RG sense, which means that one need only worry about the renormalization of the propagator, which can be controlled.\par
+\indent Using these facts, the formalism developed in~[\cite{BG90}] has been applied in~[\cite{GM10}, \cite{GMP12}] to express the free energy and correlation functions as convergent series.\par
 \indent Let us mention that the  
 case of Coulomb interactions is more difficult, in that the effective interaction is marginal in an RG sense. In this case, the theory has been constructed at all orders in renormalized perturbation theory
-[\cite{gmpOZ}, \cite{gmpOOt}], but a non-perturbative construction is still lacking. 
+[\cite{GMP10}, \cite{GMP11b}], but a non-perturbative construction is still lacking. 
 \par
 \bigskip
 
-\indent In the present work, we shall extend the results of~[\cite{giuOZ}] by performing an RG analysis of half-filled {\it bilayer} graphene with short-range interactions. Bilayer graphene consists of two layers of graphene in so-called {\it Bernal} or {\it AB} 
-stacking (see below). Before the works of A.~Geim et al.~[\cite{ngeZF}], graphene was mostly studied in order to understand the properties of graphite, so it was natural to investigate the properties of multiple layers of 
-graphene, starting with the bilayer~[\cite{walFSe}, \cite{sloFiE}, \cite{mccFiSe}]. A common model for hopping electrons on graphene bilayers is the so-called {\it Slonczewski-Weiss-McClure} model,
+\indent In the present work, we shall extend the results of~[\cite{GM10}] by performing an RG analysis of half-filled {\it bilayer} graphene with short-range interactions. Bilayer graphene consists of two layers of graphene in so-called {\it Bernal} or {\it AB} 
+stacking (see below). Before the works of A.~Geim et al.~[\cite{NGe04}], graphene was mostly studied in order to understand the properties of graphite, so it was natural to investigate the properties of multiple layers of 
+graphene, starting with the bilayer~[\cite{Wa47}, \cite{SW58}, \cite{Mc57}]. A common model for hopping electrons on graphene bilayers is the so-called {\it Slonczewski-Weiss-McClure} model,
 which is usually studied by retaining only certain hopping terms, depending on the energy regime one is interested in: including more hopping terms corresponds to probing the system at lower energies.
 The fine structure of the Fermi surface and the behavior of the 
 dispersion relation around it depends on which hoppings are considered or, equivalently, on the range of energies under inspection.
@@ -58,25 +58,25 @@ dispersion relation around it depends on which hoppings are considered or, equiv
 \indent In a first approximation, the energy structure of bilayer graphene is similar to that of the monolayer: there are only two Fermi points, 
 and the dispersion relation is approximately conical around them. This picture is valid for energy scales larger than the transverse hopping between the two layers, referred to in the following as the {\it first regime}.
 At lower energies, the effective dispersion relation around the two Fermi points appears to be approximately {\it parabolic}, instead of conical. This implies that the effective mass of the electrons in bilayer graphene does not vanish, unlike those in the monolayer, which has been observed 
-experimentally~[\cite{novZS}].\par
+experimentally~[\cite{NMe06}].\par
 \indent From an RG point of view, the parabolicity implies that the electron interactions are {\it marginal} in bilayer graphene, thus making the RG analysis non-trivial. The flow of the effective couplings
-has been studied by O.~Vafek~[\cite{vafOZ}, \cite{vayOZ}], who has found that it diverges logarithmically, and has identified the most divergent channels, 
-thus singling out which of the possible quantum instabilities are dominant (see also~\cite{tvOT}). 
+has been studied by O.~Vafek~[\cite{Va10}, \cite{VY10}], who has found that it diverges logarithmically, and has identified the most divergent channels, 
+thus singling out which of the possible quantum instabilities are dominant (see also~[\cite{TV12}]). 
 However, as was mentioned earlier, the assumption of parabolic dispersion relation is only an approximation,
 valid in a range of energies between the scale of the transverse hopping and a second threshold, proportional to the cube of the transverse hopping (asymptotically, as this hopping goes to zero).
 This range will be called the {\it second regime}.\par
-\indent By studying the smaller energies in more detail, one finds~[\cite{mccZS}] that around each of the Fermi points, there are three extra Fermi points, forming a tiny equilateral triangle around the original ones. This is referred to in the literature as {\it trigonal warping}. Furthermore, around each of the now eight Fermi points, the energy bands are approximately conical. This means that, from an RG perspective, the
-logarithmic divergence studied in [\cite{vafOZ}] is cut off at the energy scale where the conical nature of the eight Fermi points becomes observable (i.e. at the end of the second regime). At lower energies
-the electron interaction is irrelevant in the RG sense, which implies that the flow of the effective interactions remains bounded at low energies. Therefore, the analysis of [\cite{vafOZ}] is meaningful 
+\indent By studying the smaller energies in more detail, one finds~[\cite{MF06}] that around each of the Fermi points, there are three extra Fermi points, forming a tiny equilateral triangle around the original ones. This is referred to in the literature as {\it trigonal warping}. Furthermore, around each of the now eight Fermi points, the energy bands are approximately conical. This means that, from an RG perspective, the
+logarithmic divergence studied in [\cite{Va10}] is cut off at the energy scale where the conical nature of the eight Fermi points becomes observable (i.e. at the end of the second regime). At lower energies
+the electron interaction is irrelevant in the RG sense, which implies that the flow of the effective interactions remains bounded at low energies. Therefore, the analysis of [\cite{Va10}] is meaningful 
 only if the flow of the effective constants has grown significantly in the second regime. 
 \par
 \indent However, our analysis shows that the flow of the effective couplings in this regime does not grow at all, due to their smallness after integration over the first regime, 
 which we quantify in terms both of the bare couplings and of the transverse hopping. This puts into question the physical relevance of the ``instabilities'' coming from the logarithmic divergence in the second regime, 
 at least in the case we are treating, namely small interaction strength and small interlayer hopping.\par
 
-\indent The transition from a normal phase to one with broken symmetry as the interaction strength is increased from small to intermediate values was studied in~[\cite{ctvOT}] at second order in perturbation theory. Therein, it was found that while at small bare couplings the infrared flow is convergent, at larger couplings it tends to increase, indicating a transition towards an {\it electronic nematic state}.\par
+\indent The transition from a normal phase to one with broken symmetry as the interaction strength is increased from small to intermediate values was studied in~[\cite{CTV12}] at second order in perturbation theory. Therein, it was found that while at small bare couplings the infrared flow is convergent, at larger couplings it tends to increase, indicating a transition towards an {\it electronic nematic state}.\par
 
-\indent Let us also mention that the third regime is not believed to give an adequate description of the system at arbitrarily small energies: at energies smaller than a third threshold (proportional to the fourth power of the transverse hopping) one finds~\cite{parZS} that the six extra Fermi points around the two original ones, are actually microscopic ellipses. The analysis of the thermodynamic properties of the system in this regime (to be called the fourth regime) requires new ideas and techniques, due to the extended nature of the singularity, and goes beyond the scope of this paper. It may be possible to adapt the ideas of \cite{benZS} to this regime, and we hope to come back to this issue in a future publication. 
+\indent Let us also mention that the third regime is not believed to give an adequate description of the system at arbitrarily small energies: at energies smaller than a third threshold (proportional to the fourth power of the transverse hopping) one finds~[\cite{PP06}] that the six extra Fermi points around the two original ones, are actually microscopic ellipses. The analysis of the thermodynamic properties of the system in this regime (to be called the fourth regime) requires new ideas and techniques, due to the extended nature of the singularity, and goes beyond the scope of this paper. It may be possible to adapt the ideas of~[\cite{BGM06}] to this regime, and we hope to come back to this issue in a future publication. 
 \par
 \bigskip
 
@@ -105,7 +105,7 @@ hopping parameters. The proof depends on a sharp multiscale control of the cross
 where the {\it free Hamiltonian} $\mathcal H_0$ plays the role of a kinetic energy for the electrons, and the {\it interaction Hamiltonian} $\mathcal H_I$ describes the interaction between electrons.\par
 \bigskip
 
-\indent $\mathcal H_0$ is given by a {\it tight-binding} approximation, which models the movement of electrons in terms of {\it hoppings} from one atom to the next. There are four inequivalent types of hoppings which we shall consider here, each of which will be associated a different {\it hopping strength} $\gamma_i$. Namely, the hoppings between neighbors of type $a$ and $b$, as well as $\tilde a$ and $\tilde b$ will be associated a hopping strength $\gamma_0$; $a$ and $\tilde b$ a strength $\gamma_1$; $\tilde a$ and $b$ a strength $\gamma_3$; $\tilde a$ and $a$, and $\tilde b$ and $b$ a strength $\gamma_4$ (see figure~\ref{intergraph}). We can thus express $H_0$ in {\it second quantized} form in {\it momentum space} at {\it zero chemical potential} as~[\cite{walFSe}, \cite{sloFiE}, \cite{mccFiSe}]
+\indent $\mathcal H_0$ is given by a {\it tight-binding} approximation, which models the movement of electrons in terms of {\it hoppings} from one atom to the next. There are four inequivalent types of hoppings which we shall consider here, each of which will be associated a different {\it hopping strength} $\gamma_i$. Namely, the hoppings between neighbors of type $a$ and $b$, as well as $\tilde a$ and $\tilde b$ will be associated a hopping strength $\gamma_0$; $a$ and $\tilde b$ a strength $\gamma_1$; $\tilde a$ and $b$ a strength $\gamma_3$; $\tilde a$ and $a$, and $\tilde b$ and $b$ a strength $\gamma_4$ (see figure~\ref{intergraph}). We can thus express $H_0$ in {\it second quantized} form in {\it momentum space} at {\it zero chemical potential} as~[\cite{Wa47}, \cite{SW58}, \cite{Mc57}]
 \begin{equation}
 \mathcal H_0=\frac{1}{|\hat\Lambda|}\sum_{ k\in\hat \Lambda}\hat A_{ k}^\dagger H_0( k)\hat A_{ k}\label{hamkintro}\end{equation}
 \begin{equation}\hat A_{ k}:=\left(\begin{array}{c}\hat a_{ k}\\\hat{\tilde b}_{ k}\\\hat{\tilde a}_{ k}\\\hat b_{ k}\end
@@ -117,16 +117,16 @@ where the {\it free Hamiltonian} $\mathcal H_0$ plays the role of a kinetic ener
 \gamma_0\Omega(k)&\gamma_4\Omega(k)&\gamma_3\Omega^*(k)e^{-3ik_x}&0
 \end{array}\right)\label{hmatintro}\end{equation}
 in which $\hat a_k$, $\hat{\tilde b}_k$, $\hat{\tilde a}_k$ and $\hat b_k$ are {\it annihilation operators} associated to atoms of type $a$, $\tilde b$, $\tilde a$ and $b$, $k\equiv(k_x,k_y)$, $\hat\Lambda$ is the {\it first Brillouin zone}, and $\Omega(k):=1+2e^{-i\frac32k_x}\cos\left(\frac{\sqrt3}2k_y\right)$.
-These objects will be properly defined in section~\ref{modelsec}. The $\Delta$ parameter in $H_0$ models a shift in the chemical potential around atoms of type $a$ and $\tilde b$~[\cite{sloFiE}, \cite{mccFiSe}]. We choose the energy unit in such a way that $\gamma_0=1$. The hopping strengths have been measured experimentally in graphite~[\cite{dreZT}, \cite{toySeSe}, \cite{misSeN}, \cite{doeSeN}] and in bilayer graphene samples~[\cite{zhaZE}, \cite{malZSe}]; their values are given in the following table:
+These objects will be properly defined in section~\ref{modelsec}. The $\Delta$ parameter in $H_0$ models a shift in the chemical potential around atoms of type $a$ and $\tilde b$~[\cite{SW58}, \cite{Mc57}]. We choose the energy unit in such a way that $\gamma_0=1$. The hopping strengths have been measured experimentally in graphite~[\cite{DD02}, \cite{TDD77}, \cite{MMD79}, \cite{DDe79}] and in bilayer graphene samples~[\cite{ZLe08}, \cite{MNe07}]; their values are given in the following table:
 \begin{equation}
 \begin{array}{|c|c|c|}
 \cline{2-3}
-\multicolumn1{c|}{}&\mathrm{bilayer\ graphene~[\cite{malZSe}]}&\mathrm{graphite~[\cite{dreZT}]}\\
+\multicolumn1{c|}{}&\mathrm{bilayer\ graphene~[\cite{MNe07}]}&\mathrm{graphite~[\cite{DD02}]}\\
 \hline
 \gamma_1&0.10&0.12\\
 \gamma_3&0.034&0.10\\
 \gamma_4&0.041&0.014\\
-\Delta&0.006\ \mathrm{[\cite{zhaZE}]}&-0.003\\
+\Delta&0.006\ \mathrm{[\cite{ZLe08}]}&-0.003\\
 \hline
 \end{array}\label{tabgamma}\end{equation}
 We notice that the relative order of magnitude of $\gamma_3$ and $\gamma_4$ is quite different in graphite and in bilayer graphene. In the latter, $\gamma_1$ is somewhat small, and $\gamma_3$ and $\gamma_4$ are of the same order, whereas $\Delta$ is of the order of $\gamma_1^2$. We will take advantage of the smallness of the hopping strengths and treat $\gamma_1=:\epsilon$ as a small parameter: we fix
@@ -136,7 +136,7 @@ and assume that $\epsilon$ is as small as needed.
 \bigskip
 
 {\bf Remark:} The symbols used for the hopping parameters are standard. The reason why $\gamma_2$ was omitted is that it refers to next-to-nearest layer hopping in graphite. 
-In addition, for simplicity, we have neglected the intra-layer next-to-nearest neighbor hopping $\gamma_0'\approx 0.1\gamma_1$, which is known to play an analogous role to $\gamma_4$ and $\Delta$ [\cite{zhaZE}].
+In addition, for simplicity, we have neglected the intra-layer next-to-nearest neighbor hopping $\gamma_0'\approx 0.1\gamma_1$, which is known to play an analogous role to $\gamma_4$ and $\Delta$ [\cite{ZLe08}].
 \par\bigskip
 
 \begin{figure}
@@ -205,7 +205,7 @@ in which the error term in~(\ref{freeschwino}) is {\it small}. In this first reg
 \bigskip
 
 \begin{figure}
-\hfil\includegraphics{Figs/bands.pdf}
+\hfil\includegraphics{Figs/bands.pdf}\par
 \caption{Eigenvalues of $H_0(k)$. The sub-figures b,c,d are finer and finer zooms around one of the Fermi points.}
 \label{figbands}
 \end{figure}
@@ -266,7 +266,7 @@ and, $\forall\mathbf k\in\mathcal B_\infty\setminus\{\tilde{\mathbf p}_{F,j}^\om
 \begin{itemize}
 \item The theorem requires $\gamma_4=\Delta=0$. As we saw above, those quantities play a negligible role in the non-interacting theory as long as we do not move beyond the third regime. This suggests that the theorem should hold with $\gamma_4,\Delta\neq0$ under the condition that $\beta$ is not too large,
 i.e., smaller than $(\mathrm{const}.)\ \epsilon^{-4}$. However, that case presents a number of extra technical complications, which we will spare the reader.
-\item The conditions that $|U|<U_0$ and $\epsilon<\epsilon_0$ are independent, in that we do not require any condition on the relative values of $|U|$ and $\epsilon$. Such a result calls for tight bounds on the integration over the first regime. If we were to assume that $|U|\ll\epsilon$, then the discussion would be greatly simplified, but such a condition would be artificial, and we will not require it be satisfied. L.~Lu~[\cite{luOTh}] sketched 
+\item The conditions that $|U|<U_0$ and $\epsilon<\epsilon_0$ are independent, in that we do not require any condition on the relative values of $|U|$ and $\epsilon$. Such a result calls for tight bounds on the integration over the first regime. If we were to assume that $|U|\ll\epsilon$, then the discussion would be greatly simplified, but such a condition would be artificial, and we will not require it be satisfied. L.~Lu~[\cite{Lu13}] sketched 
 the proof of a result similar to our Main Theorem, without discussing the first two regimes, which requires such an artificial condition on $U/\epsilon$. 
 The renormalization of the secondary Fermi points is also ignored in that reference.
 \end{itemize}
@@ -467,12 +467,12 @@ where $T(k_0,k_x,k_y)$ denotes the rotation of the $k_x$ and $k_y$ components by
 \point{\bf Multiscale decomposition.} The proof relies on a {\it multiscale} analysis of the model, in which the free energy and Schwinger function are expressed as successive integrations over individual scales. 
 Each scale is defined as a set of $\mathbf k$'s contained inside an annulus at a distance of $2^h$ for $h\in\mathbb Z$ around the singularities located at $\mathbf p_{F,j}^\omega$.
 The positive scales correspond to the ultraviolet regime, which we analyze in a multiscale fashion because of the (very mild) singularity of the free propagator at equal imaginary times. 
-It may be possible to avoid the decomposition by employing ideas in the spirit of [\cite{psZE}]. The negative scales are treated differently, depending on the regimes they belong to (see below),
+It may be possible to avoid the decomposition by employing ideas in the spirit of [\cite{SS08}]. The negative scales are treated differently, depending on the regimes they belong to (see below),
 and they contain the essential difficulties of the problem, whose nature is intrinsically infrared. 
 \par
 \bigskip
 
-\point{\bf First regime.} In the first regime, i.e. for $-1\gg h\gg h_\epsilon:=\log_2\epsilon$, the system behaves like two uncoupled graphene layers, so the analysis carried out in~[\cite{giuOZ}] holds. From a renormalization group perspective, this regime is {\it super-renormalizable}: the scaling dimension of diagrams with $2l$ external legs is $3-2l$, so that only the two-legged diagrams are relevant whereas all of the others are irrelevant (see section~\ref{powercountingsec} for precise definitions of scaling dimensions, relevance and irrelevance). This allows us to compute a strong bound on four-legged contributions:
+\point{\bf First regime.} In the first regime, i.e. for $-1\gg h\gg h_\epsilon:=\log_2\epsilon$, the system behaves like two uncoupled graphene layers, so the analysis carried out in~[\cite{GM10}] holds. From a renormalization group perspective, this regime is {\it super-renormalizable}: the scaling dimension of diagrams with $2l$ external legs is $3-2l$, so that only the two-legged diagrams are relevant whereas all of the others are irrelevant (see section~\ref{powercountingsec} for precise definitions of scaling dimensions, relevance and irrelevance). This allows us to compute a strong bound on four-legged contributions:
 $$|\hat W_4^{(h)}(\mathbf k)|\leqslant (\mathrm{const.})\ |U|2^{2h}$$
 whereas a naive power counting argument would give $|U|2^{h}$ (recall that with our conventions $h$ is negative).\par
 \bigskip
@@ -480,7 +480,7 @@ whereas a naive power counting argument would give $|U|2^{h}$ (recall that with
 \indent The super-renormalizability in the first regime stems from the fact that the Fermi surface is 0-dimensional and that $H_0$ is linear around the Fermi points. While performing the multiscale integration, we deal with the two-legged terms by incorporating them into $H_0$, and one must therefore prove that by doing so, the Fermi surface remains 0-dimensional and that the singularity remains linear. This is guaranteed by a symmetry argument, which in particular shows the invariance of the Fermi surface.\par
 \bigskip
 
-\point{\bf Second regime.} In the second regime, i.e. for $3h_\epsilon\ll h\ll h_\epsilon$, the singularities of $H_0$ are quadratic around the Fermi points, which changes the {\it power counting} of the renormalization group analysis: the scaling dimension of $2l$-legged diagrams becomes $2-l$ so that the two-legged diagrams are still relevant, but the four-legged ones become marginal. One can then check~[\cite{vafOZ}] that they are actually marginally relevant, which means that their contribution increases proportionally to $|h|$. This turns out not to matter: since the second regime is only valid for $h\gg 3h_\epsilon$, $|\hat W_4^{(h)}|$ may only increase by $3|h_\epsilon|$, and since the theory is super-renormalizable in the first regime, there is an extra factor $2^{h_\epsilon}$ in $\hat W_4^{(h_\epsilon)}$, so that $\hat W_4^{(h)}$ actually increases from $2^{h_\epsilon}$ to $3|h_\epsilon|2^{h_\epsilon}$, that is to say it barely increases at all if $\epsilon$ is small enough.\par
+\point{\bf Second regime.} In the second regime, i.e. for $3h_\epsilon\ll h\ll h_\epsilon$, the singularities of $H_0$ are quadratic around the Fermi points, which changes the {\it power counting} of the renormalization group analysis: the scaling dimension of $2l$-legged diagrams becomes $2-l$ so that the two-legged diagrams are still relevant, but the four-legged ones become marginal. One can then check~[\cite{Va10}] that they are actually marginally relevant, which means that their contribution increases proportionally to $|h|$. This turns out not to matter: since the second regime is only valid for $h\gg 3h_\epsilon$, $|\hat W_4^{(h)}|$ may only increase by $3|h_\epsilon|$, and since the theory is super-renormalizable in the first regime, there is an extra factor $2^{h_\epsilon}$ in $\hat W_4^{(h_\epsilon)}$, so that $\hat W_4^{(h)}$ actually increases from $2^{h_\epsilon}$ to $3|h_\epsilon|2^{h_\epsilon}$, that is to say it barely increases at all if $\epsilon$ is small enough.\par
 \bigskip
 
 \indent Once this essential fact has been taken into account, the renormalization group analysis can be carried out without major difficulties. As in the first regime, the invariance of the Fermi surface is guaranteed by a symmetry argument.\par
@@ -493,10 +493,10 @@ whereas a naive power counting argument would give $|U|2^{h}$ (recall that with
 \indent The rest of this paper is devoted to the proof of the Main Theorem and of Theorems~\ref{theoo}, \ref{theot} and~\ref{theoth}. The sections are organized as follows.
 
 \begin{itemize}
-\item In section~\ref{themodelsec}, we define the model in a more explicit way than what has been done so far; then we show how to compute the free energy and Schwinger function using a Fermionic path integral formulation and a {\it determinant expansion}, due to Battle, Brydges and Federbush~[\cite{brySeE}, \cite{batEF}], see also [\cite{bkESe}, \cite{arNE}]; and finally we discuss the symmetries of the system.
+\item In section~\ref{themodelsec}, we define the model in a more explicit way than what has been done so far; then we show how to compute the free energy and Schwinger function using a Fermionic path integral formulation and a {\it determinant expansion}, due to Battle, Brydges and Federbush~[\cite{BF78}, \cite{BF84}], see also [\cite{BK87}, \cite{AR98}]; and finally we discuss the symmetries of the system.
 \item In section~\ref{proppropsec}, we discuss the non-interacting system. In particular, we derive detailed formulae for the Fermi points and for the asymptotic behavior of the propagator around its singularities. 
 \item In section~\ref{schemesec}, we describe the multiscale decomposition used to compute the free energy and Schwinger function.
-\item In section~\ref{treeexpsec}, we state and prove a {\it power counting} lemma, which will allow us to compute bounds for the effective potential in each regime. The lemma is based on the Gallavotti-Nicol\`o tree expansion~[\cite{galEFi}], and follows~[\cite{benNZ}, \cite{genZO}, \cite{giuOZh}]. We conclude this section by showing how to compute the two-point Schwinger function from the effective potentials.
+\item In section~\ref{treeexpsec}, we state and prove a {\it power counting} lemma, which will allow us to compute bounds for the effective potential in each regime. The lemma is based on the Gallavotti-Nicol\`o tree expansion~[\cite{GN85}], and follows~[\cite{BG90}, \cite{GM01}, \cite{Gi10}]. We conclude this section by showing how to compute the two-point Schwinger function from the effective potentials.
 \item In section~\ref{uvsec}, we discuss the integration over the {\it ultraviolet regime}, i.e. scales $h>0$.
 \item In sections~\ref{osec}, \ref{tsec} and~\ref{thsec}, we discuss the multiscale integration in the first, second and third regimes,
 and complete the proofs of the Main Theorem, as well as of Theorems \ref{theoo}, \ref{theot}, \ref{theoth}.
@@ -525,7 +525,7 @@ made explicit.
 \label{laeo}\end{equation}
 where we have chosen the unit length to be equal to the distance between two nearest neighbors in a layer (see figure~\ref{cellgraph}). The elementary cell consists of four atoms at the following coordinates
 $$(0,0,0);\mathrm{\ }(0,0,c);\mathrm{\ }(-1,0,c);\mathrm{\ }(1,0,0)$$
-given relatively to the center of the cell. $c$ is the spacing between layers; it can be measured experimentally, and has a value of approximately 2.4~[\cite{triNT}].\par
+given relatively to the center of the cell. $c$ is the spacing between layers; it can be measured experimentally, and has a value of approximately 2.4~[\cite{TMe92}].\par
 \bigskip
 
 \begin{figure}
@@ -623,8 +623,8 @@ in which $(\alpha,\alpha')\in\mathcal A^2:=\{a,\tilde b,\tilde a,b\}^2$; $\mathb
 \indent However, the quantities on the right side of~(\ref{freeen}) and~(\ref{schwindef}) are somewhat difficult to manipulate. In this section, we will re-express $f_\Lambda$ and $\check s$ in terms of {\it Grassmann integrals} and {\it expectations}, and show how such quantities can be computed using a {\it determinant expansion}. This formalism will lay the groundwork for the procedure which will be used in the following to express $f_\Lambda$ and $\check s$ as series, and subsequently prove their convergence.\par
 \bigskip
 
-\point{\bf Grassmann integral formulation.} We first describe how to express~(\ref{freeen}) and~(\ref{schwindef}) as Grassmann integrals. The procedure is well known and details can be found in many references, see e.g. [\cite{giuOZ}, appendix~B] and [\cite{giuOZh}] for a discussion adapted to the case of graphene, or [\cite{genZO}] for a discussion adapted to general low-dimensional Fermi systems, 
-or [\cite{benNFi}] and [\cite{salOTh}] and references therein for an even more general picture. \par
+\point{\bf Grassmann integral formulation.} We first describe how to express~(\ref{freeen}) and~(\ref{schwindef}) as Grassmann integrals. The procedure is well known and details can be found in many references, see e.g. [\cite{GM10}, appendix~B] and [\cite{Gi10}] for a discussion adapted to the case of graphene, or [\cite{GM01}] for a discussion adapted to general low-dimensional Fermi systems, 
+or [\cite{BG95}] and [\cite{Sa13}] and references therein for an even more general picture. \par
 \bigskip
 
 \subpoint{\bf Definition.} We first define a Grassmann algebra and an integration procedure on it. We move to Fourier space: for every $\alpha\in\mathcal A:=\{a,\tilde b,\tilde a,b\}$, the operator $\alpha_{(t,x)}$ is associated
@@ -666,7 +666,7 @@ Gaussian Grassmann integrals satisfy the following {\it addition principle}: giv
 \hat g_{\leqslant M}({\bf k}):=\chi_0(2^{-M}|k_0|) (-ik_0\mathds 1-H_0(k))^{-1}
 \label{freeprop}\end{equation}
 and the Gaussian integration measure $P_{\leqslant M}(d\psi)\equiv P_{\hat g_{\leqslant M}}(d\psi)$.
-One can prove (see e.g. [\cite{giuOZ}, appendix~B]) that if
+One can prove (see e.g. [\cite{GM10}, appendix~B]) that if
 \begin{equation} \frac{1}{\beta|\Lambda|}\log\int P_{\leqslant M}(d\psi)\ e^{-\mathcal V(\psi)}\label{log}\end{equation}
 is analytic in $U$, uniformly as $M\to\infty$, a fact we will check a posteriori, then the finite volume free energy can be written as
 \begin{equation}f_\Lambda=f_{0,\Lambda}-\lim_{M\to\infty}\frac{1}{\beta|\Lambda|}\log\int P_{\leqslant M}(d\psi)\ e^{-\mathcal V(\psi)}\label{freeengrass}\end{equation}
@@ -688,7 +688,7 @@ and equal to $\check s({\bf 0})+1/2$, otherwise. This extra $+1/2$ accounts for
 \begin{equation}
 \frac{\int P_{\leqslant M}(d\psi)\ e^{-\mathcal V(\psi)}\hat \psi_{\mathbf k,\alpha_1}^-\hat \psi_{\mathbf k,\alpha_2}^+}{\int P_{\leqslant M}(d\psi)\ e^{-\mathcal V(\psi)}}\label{sc}
 \end{equation}
-is analytic in $U$ uniformly in $M$, a fact we will also check a posteriori, then one can prove (see e.g. [\cite{giuOZ}, appendix~B]) that the two-point Schwinger function can be written as
+is analytic in $U$ uniformly in $M$, a fact we will also check a posteriori, then one can prove (see e.g. [\cite{GM10}, appendix~B]) that the two-point Schwinger function can be written as
 \begin{equation}
 s_{\alpha_1,\alpha_2}(\mathbf k)=\lim_{M\to\infty}\frac{\int P_{\leqslant M}(d\psi)\ e^{-\mathcal V(\psi)}\hat \psi_{\mathbf k,\alpha_1}^-\hat \psi_{\mathbf k,\alpha_2}^+}{\int P_{\leqslant M}(d\psi)\ e^{-\mathcal V(\psi)}}.
 \label{schwingrass}\end{equation}
@@ -714,11 +714,11 @@ where the {\it truncated expectation} is defined as
 in which $(\mathcal V_1,\cdots,\mathcal V_N)$ is a collection of commuting polynomials and the index $_{\leqslant M}$ refers to the propagator of $P_{\leqslant M}(d\psi)$. A similar formula holds for~(\ref{sc}).\par
 \bigskip
 
-\indent The purpose of this rewriting is that we can compute truncated expectations in terms of a {\it determinant expansion}, also known as the Battle-Brydges-Federbush formula~[\cite{brySeE}, \cite{batEF}], which expresses it as the determinant of a Gram matrix. The advantage of this writing is that, provided we first re-express the propagator $\hat g_{\leqslant M}(\mathbf k)$ in $\mathbf x$-space, the afore-mentioned Gram matrix can be bounded effectively (see section~\ref{powercountingsec}). We therefore first define an $\mathbf x$-space representation for $\hat g(\mathbf k)$:
+\indent The purpose of this rewriting is that we can compute truncated expectations in terms of a {\it determinant expansion}, also known as the Battle-Brydges-Federbush formula~[\cite{BF78}, \cite{BF84}], which expresses it as the determinant of a Gram matrix. The advantage of this writing is that, provided we first re-express the propagator $\hat g_{\leqslant M}(\mathbf k)$ in $\mathbf x$-space, the afore-mentioned Gram matrix can be bounded effectively (see section~\ref{powercountingsec}). We therefore first define an $\mathbf x$-space representation for $\hat g(\mathbf k)$:
 \begin{equation}
 g_{\leqslant M}(\mathbf x):=\frac{1}{\beta|\Lambda|}\sum_{\mathbf k\in\mathcal B^*_{\beta,L}}e^{i\mathbf k\cdot\mathbf x}\hat g_{\leqslant M}(\mathbf k).
 \label{freepropx}\end{equation}
-The determinant expansion is given in the following lemma, the proof of which can be found in [\cite{genZO}, appendix~A.3.2], [\cite{giuOZh}, appendix~B].\par
+The determinant expansion is given in the following lemma, the proof of which can be found in [\cite{GM01}, appendix~A.3.2], [\cite{Gi10}, appendix~B].\par
 \bigskip
 
 \theo{Lemma}\label{detexplemma}
@@ -874,7 +874,7 @@ in which the label $_{\mathrm I}$ stands for ``first regime''. If
 \label{odef}\end{equation}
 for suitable constants $\kappa_1, \bar\kappa_0>0$, then the remainder term in (\ref{detao}) is smaller than the explicit term, so that~(\ref{detao}) is adequate in this regime, which we call the ``first regime''.
 
-We now compute the dominating part of $\hat A^{-1}$ in this regime. The computation is carried out in the following way: we neglect terms of order $\gamma_1$, $\gamma_3$ and $|k'|^2$ in $\hat A$, invert the resulting matrix using~(\ref{invAexpr}), prove that this inverse is bounded by $(\mathrm{const}.)\ \|\mathbf k'\|_{\mathrm{I}}^{-1}$, and deduce a bound on the error terms. We thus find
+\indent We now compute the dominating part of $\hat A^{-1}$ in this regime. The computation is carried out in the following way: we neglect terms of order $\gamma_1$, $\gamma_3$ and $|k'|^2$ in $\hat A$, invert the resulting matrix using~(\ref{invAexpr}), prove that this inverse is bounded by $(\mathrm{const}.)\ \|\mathbf k'\|_{\mathrm{I}}^{-1}$, and deduce a bound on the error terms. We thus find
 \begin{equation}
 \hat A^{-1}(\mathbf p_{F,0}^\omega+\mathbf k')=
 -\frac{1}{k_0^2+|\xi|^2}
@@ -971,7 +971,7 @@ which is identical to the bound at the end of the first regime and at the beginn
 Since we will now investigate the regime in which $|k'|<(\mathrm{const}.)\ \epsilon^2$, we will need to consider all the Fermi points $p_{F,j}^\omega$ with $j\in\{0,1,2,3\}$.\par
 \bigskip
 
-\subpoint{Around $\mathbf p_{F,0}^\omega$} We start with the neighborhood of $\mathbf p_{F,0}^\omega$: 
+\subpoint{\bf Around $\mathbf p_{F,0}^\omega$} We start with the neighborhood of $\mathbf p_{F,0}^\omega$: 
 \begin{equation}
 \det\hat A(\mathbf p_{F,0}^\omega+\mathbf k')=\gamma_1^2(k_0^2+\gamma_3^2|\xi|^2)+O(\epsilon^{-1}\|\mathbf k'\|_{\mathrm{III}}^3)
 \label{detath}\end{equation}
@@ -1009,7 +1009,7 @@ and
 \label{boundfreepropzth}\end{equation} 
 \bigskip
 
-\subpoint{Around $\mathbf p_{F,1}^\omega$} We now discuss the neighborhood of $\mathbf p_{F,1}^\omega$. We define $k'_{1}:=k-p_{F,1}^\omega=(k'_{1,x},k'_{1,y})$ and $\mathbf k'_{1}:=(k_0,k'_{1})$. We have
+\subpoint{\bf Around $\mathbf p_{F,1}^\omega$} We now discuss the neighborhood of $\mathbf p_{F,1}^\omega$. We define $k'_{1}:=k-p_{F,1}^\omega=(k'_{1,x},k'_{1,y})$ and $\mathbf k'_{1}:=(k_0,k'_{1})$. We have
 \begin{equation}\Omega(p_{F,1}^\omega+k'_{1})=\gamma_1\gamma_3+\xi_1+O(\epsilon^2|k'_{1}|)\label{eq:O1}\end{equation}
 where
 $$\xi_1:=\frac{3}{2}(ik'_{1,x}+\omega k'_{1,y}).$$
@@ -1051,7 +1051,7 @@ and
 \label{boundfreepropzthj}\end{equation}
 \bigskip
 
-\subpoint{Around $\mathbf p_{F,j}^\omega$} The behavior of $\hat g(\mathbf k)$ around $p_{F,j}^\omega$ for $j\in\{2,3\}$ can be deduced from~(\ref{freepropoth}) by using the symmetry (\ref{arotation})
+\subpoint{\bf Around $\mathbf p_{F,j}^\omega$} The behavior of $\hat g(\mathbf k)$ around $p_{F,j}^\omega$ for $j\in\{2,3\}$ can be deduced from~(\ref{freepropoth}) by using the symmetry (\ref{arotation})
 under $2\pi/3$ rotations: if we define $k'_{j}:=k-p_{F,j}^\omega=(k'_{j,x},k'_{j,y})$, $\mathbf k'_{j}:=(k_0,k'_{j})$ then, for $j=2,3$ and $\omega\pm$, 
 \begin{equation}
 \hat A^{-1}(\mathbf k'_{j}+\mathbf p_{F,j}^\omega)=
@@ -1297,7 +1297,7 @@ $\psi^{(\leqslant h)}$ in momentum space. In order to get satisfactory bounds on
 \begin{equation}
 -\beta|\Lambda|\mathfrak e_h-\mathcal V^{*(h)}(\Psi):=\sum_{N=1}^\infty\frac{(-1)^N}{N!}\mathcal E_{h+1}^T(\mathcal V^{*(h+1)}(\psi^{(h+1)}+\Psi);N)
 \label{extendedVuv}\end{equation}
-in which $\{\hat\Psi_{\mathbf k,\alpha}\}_{\mathbf k\in\mathcal B_{\beta,L},\alpha\in\mathcal A}$ is a collection of {\it external fields} (in reference to the fact that, contrary to $\psi^{(\leqslant h)}$, they have a non-compact support in momentum space). The use of this specific extension can be justified {\it ab-initio} by re-defining the cutoff function $\chi$ in such a way that its support is $\mathbb{R}$, e.g. using exponential tails that depend on a parameter $\epsilon_\chi$ in such a way that the support tends to be compact as $\epsilon_\chi$ goes to 0. Following this logic, we could first define $\hat W$ using the non-compactly supported cutoff function and then take the $\epsilon_\chi\to0$ limit, thus recovering (\ref{extendedVuv}).  Such an approach is dicussed in~[\cite{bmZT}]. From now on, with some abuse of notation, we shall identify $\mathcal V^{*(h)}$ with $\mathcal V^{(h)}$ and denote them by the same symbol $\mathcal V^{(h)}$, which is justified by the fact that their kernels are (or can be chosen, from what said above, to be) the same.\par
+in which $\{\hat\Psi_{\mathbf k,\alpha}\}_{\mathbf k\in\mathcal B_{\beta,L},\alpha\in\mathcal A}$ is a collection of {\it external fields} (in reference to the fact that, contrary to $\psi^{(\leqslant h)}$, they have a non-compact support in momentum space). The use of this specific extension can be justified {\it ab-initio} by re-defining the cutoff function $\chi$ in such a way that its support is $\mathbb{R}$, e.g. using exponential tails that depend on a parameter $\epsilon_\chi$ in such a way that the support tends to be compact as $\epsilon_\chi$ goes to 0. Following this logic, we could first define $\hat W$ using the non-compactly supported cutoff function and then take the $\epsilon_\chi\to0$ limit, thus recovering (\ref{extendedVuv}).  Such an approach is dicussed in~[\cite{BM02}]. From now on, with some abuse of notation, we shall identify $\mathcal V^{*(h)}$ with $\mathcal V^{(h)}$ and denote them by the same symbol $\mathcal V^{(h)}$, which is justified by the fact that their kernels are (or can be chosen, from what said above, to be) the same.\par
 \bigskip
 
 \point{\bf First and second regimes.} We now discuss the first and second regimes (the third regime is very slightly different in that the index $\omega$ is complemented by an extra index $j$ and the Fermi points are shifted). Similarly to~(\ref{hatWdefo}), we define the {\it kernels} of $\bar{\mathcal V}$:
@@ -1311,7 +1311,7 @@ where $\mathcal B_{\beta,L}^{(\leqslant h,\underline\omega)}=\mathcal B_{\beta,L
 \label{psixdeft}\end{equation}
 \bigskip
 
-{\bf Remark}: Unlike $\hat\psi_{\mathbf k,\alpha,\omega}$, the $_\omega$ index in $\psi_{\mathbf x,\alpha,\omega}^{(\leqslant h)\pm}$ is {\it not} redundant. Keeping track of this dependence is required to prove properties of $\hat W_{2l}(\mathbf k)$ and $\hat{\bar g}_h(\mathbf k)$ close to $\mathbf p_{F,0}^\omega$ while working in $\mathbf x$-space. Such considerations were first discussed in~[\cite{benNZ}] in which $\psi_{\mathbf x,\alpha,\omega}$ were called {\it quasi-particle} fields.\par
+{\bf Remark}: Unlike $\hat\psi_{\mathbf k,\alpha,\omega}$, the $_\omega$ index in $\psi_{\mathbf x,\alpha,\omega}^{(\leqslant h)\pm}$ is {\it not} redundant. Keeping track of this dependence is required to prove properties of $\hat W_{2l}(\mathbf k)$ and $\hat{\bar g}_h(\mathbf k)$ close to $\mathbf p_{F,0}^\omega$ while working in $\mathbf x$-space. Such considerations were first discussed in~[\cite{BG90}] in which $\psi_{\mathbf x,\alpha,\omega}$ were called {\it quasi-particle} fields.\par
 \bigskip
 
 \indent We then define the propagator in $\mathbf x$-space:
@@ -1463,8 +1463,8 @@ $$g_{h,\omega,j}(\mathbf x):=\frac{1}{\beta|\Lambda|}\sum_{\mathbf k\in\mathcal
 
 \section{Tree expansion and constructive bounds}
 \label{treeexpsec}
-\indent In this section, we shall define the Gallavotti-Nicol\`o tree expansion~[\cite{galEFi}], and show how it can be used to compute bounds for the $\mathfrak e_h$, $\mathcal V^{(h)}$,
-$\mathcal Q^{(h)}$ and $\bar{\mathcal V}^{(h)}$ defined above in~(\ref{effpotuv}) and (\ref{effpotoh}), using the estimates~(\ref{estguv}), (\ref{estgo}), (\ref{estgt}) and~(\ref{estgth}). We follow~[\cite{benNZ}, \cite{genZO}, \cite{giuOZ}].
+\indent In this section, we shall define the Gallavotti-Nicol\`o tree expansion~[\cite{GN85}], and show how it can be used to compute bounds for the $\mathfrak e_h$, $\mathcal V^{(h)}$,
+$\mathcal Q^{(h)}$ and $\bar{\mathcal V}^{(h)}$ defined above in~(\ref{effpotuv}) and (\ref{effpotoh}), using the estimates~(\ref{estguv}), (\ref{estgo}), (\ref{estgt}) and~(\ref{estgth}). We follow~[\cite{BG90}, \cite{GM01}, \cite{GM10}].
 We conclude the section by showing how to compute the terms in $\bar{\mathcal W}^{(h)}$ that are quadratic in $\hat J_{\mathbf k,\underline\alpha}$ from $\mathcal V^{(h)}$ and $\hat{\bar g}_h$.\par
 \bigskip
 
@@ -1509,7 +1509,7 @@ We denote the parent-child partial ordering by $v'\prec v$ ($v'$ is the parent o
 \end{itemize}
 \bigskip
 
-{\bf Remark}: Local leaves are called ``local'' because those nodes are usually applied a {\it localization} operation (see e.g. [\cite{benNFi}]). In the present case, such a step is not needed, due to the super-renormalizable nature of the first and third regimes.\par
+{\bf Remark}: Local leaves are called ``local'' because those nodes are usually applied a {\it localization} operation (see e.g. [\cite{BG95}]). In the present case, such a step is not needed, due to the super-renormalizable nature of the first and third regimes.\par
 \bigskip
 
 \begin{figure}
@@ -1571,7 +1571,7 @@ where $\mathbf T(\tau)$ is the set of collections of $(T_v\in\mathbf T(\mathbf R
 \bigskip
 
 \indent\underline{Idea of the proof}: The proof of this lemma can easily be reconstructed from the schematic description below. We do not present it in full detail here because its
-proof has already been discussed in several references, among which~[\cite{benNFi}, \cite{genZO}, \cite{giuOZh}].\par
+proof has already been discussed in several references, among which~[\cite{BG95}, \cite{GM01}, \cite{Gi10}].\par
 \bigskip
 
 \indent The lemma follows from an induction on $h$, in which we write the truncated expectation in the right side of~(\ref{treeindeq}) as
@@ -1589,7 +1589,7 @@ represents a class of labeled
 Feynman diagrams (the labels being the scales attached to the lines, or equivalently to the propagators)
 with similar scaling properties. In fact, given a labeled Feynman diagram, one defines a tree and a set of external field labels by the following procedure. For every $h$, we define the {\it clusters on scale $h$} as the connected components of the diagram one obtains by removing the lines with a scale label that is $< h$. We assign a node with scale label $h$ to every cluster on scale $h$. The set $P_v$ contains the indices of the legs of the Feynman diagram that exit the corresponding cluster. If a cluster on scale $h$ contains a cluster on scale $h+1$, then we draw a branch between the two corresponding nodes. See figure~\ref{feyntreefig} for an example.\par
 \indent Local leaves correspond to clusters that have {\it few} external legs. They are considered as ``black boxes'': the clusters on larger scales contained inside them are discarded.\par
-\indent A more detailed discussion of this correspondence can be found in~[\cite{genZO}, section~5.2] among other references.
+\indent A more detailed discussion of this correspondence can be found in~[\cite{GM01}, section~5.2] among other references.
 
 
 \begin{figure}
@@ -1748,7 +1748,7 @@ the change of variables from $\underline{\mathbf x}_\tau$ to $\{\mathbf x_0,\{\m
 \caption{example of a spanning tree with $s_v=5$ and $|P_{v_1}|=|P_{v_2}|=|P_{v_3}|=|P_{v_4}|=4$, $|P_{v_5}|=6$; whose root is $v_2$.}
 \label{spanningfig}\end{figure}
 
-\point{Decomposing $(\underline{\mathbf x}-\mathbf x_{2l})^m$} We now decompose the $(\underline{\mathbf x}-\mathbf x_{2l})^m$ factor in (\ref{powercountingl}) in the following way (note that  in terms of the indices in $P_{v_0}$, $\mathbf x_{2l}\equiv\mathbf x_{J^{(v_r)}}$):
+\point{\bf Decomposing $(\underline{\mathbf x}-\mathbf x_{2l})^m$} We now decompose the $(\underline{\mathbf x}-\mathbf x_{2l})^m$ factor in (\ref{powercountingl}) in the following way (note that  in terms of the indices in $P_{v_0}$, $\mathbf x_{2l}\equiv\mathbf x_{J^{(v_r)}}$):
 $(\underline{\mathbf x}-\mathbf x_{2l})^m$ is a product of terms of the form $(x_{j,i}-x_{J^{(v_r)},i})$ which we rewrite as a sum of $z_{j',i}$'s for $v\in \mathfrak E(\tau)$ {\it on the path from $J^{(v_r)}$ to $j$}, a concept we will now make more precise. $j$ and $J^{(v_r)}$ are in $I_{v(j)}$ and $I_{v_r}$ respectively, where $v(j)$ is the unique node in $\mathfrak E(\tau)$ such that $j\in I_{v(j)}$. There exists a unique sequence of lines of $T$ that links $v_r$ to $v(j)$, which we denote by $((j_1,j_1'),\cdots,(j_\rho,j_\rho'))$, the convention being that the line 
 $(j,j')$ is oriented from $j$ to $j'$. The {\it path from $J^{(v_r)}$ to $j$} is the sequence $\mathbf z_{j_1}, \mathbf z_{j_1'},\mathbf z_{j_2},\cdots$ and so forth, until $j$ is reached. We can therefore write
 $$x_{j,i}-x_{J^{(v_r)},i}=\sum_{p=1}^{\rho}(z_{j_p,i}+z_{j_p',i}).$$
@@ -1799,9 +1799,9 @@ to find
 \label{boundt}\end{equation}
 \bigskip
 
-\point{\bf Bound on the number of spanning trees.} Finally, the number of choices for $T$ can be bounded (see [\cite{genZO}, lemma~A.5])
+\point{\bf Bound on the number of spanning trees.} Finally, the number of choices for $T$ can be bounded (see [\cite{GM01}, lemma~A.5])
 \begin{equation}\sum_{T\in\mathbf T(\tau)}1\leqslant \prod_{v\in\mathfrak V(\tau)}c_3^{\frac{|R_v|}2}s_v!\label{boundnrspan}\end{equation}
-so that by injecting~(\ref{boundnrspan}) into~(\ref{boundt}), we find~(\ref{powercountingl}), with $C_2=c_2^2c_3$ and $C_3=c_3^{-1}$.\penalty10000\hfill\penalty10000$\square$\par
+so that by injecting~(\ref{boundnrspan}) into~(\ref{boundt}), we find~(\ref{powercountingl}), with $C_2=c_2^2c_3$ and $C_3=c_3^{-1}$.\qed\par
 
 
 \subsection{Schwinger function from the effective potential}
@@ -1972,7 +1972,7 @@ in which $\mathcal T_N^*$ denotes the set of unlabeled rooted trees with $N$ end
 $$\sum_{\tau\in\mathcal T_{N}^{(h)}}\prod_{v\in\mathfrak V(\tau)}2^{-\mu}=\sum_{\tau^*\in\mathcal T_N^*}\sum_{\mathbf h\in\mathbf H_h(\tau^*)}\prod_{v\in\mathfrak V(\tau^*)}2^{-\mu(h_v-h_{p(v)})}$$
 in which $p(v)$ denotes the parent of $v$, so that
 $$\sum_{\tau\in\mathcal T_{N}^{(h)}}\prod_{v\in\mathfrak V(\tau)}2^{-\mu}\leqslant\sum_{\tau^*\in\mathcal T_N^*}\prod_{v\in\mathfrak V(\tau^*)}\sum_{q=1}^\infty2^{-\mu q}\leqslant\sum_{\tau^*\in\mathcal T_N^*}C_{T,1}^N$$
-for some constant $C_{T,1}$, in which we used the fact that $|\mathfrak V(\tau^*)|\leqslant N$. Furthermore, it is a well known fact that $\sum_{\tau^*}1\leqslant 4^N$ (see e.g. [\cite{genZO}, lemma~A.1], the proof is based on constructing an injective map to the set of random walks with $2N$ steps: given a tree, consider a walker that starts at the root, and then travels over branches towards the right until it reaches a leaf, and then travels left until it can go right again on a different branch). Therefore
+for some constant $C_{T,1}$, in which we used the fact that $|\mathfrak V(\tau^*)|\leqslant N$. Furthermore, it is a well known fact that $\sum_{\tau^*}1\leqslant 4^N$ (see e.g. [\cite{GM01}, lemma~A.1], the proof is based on constructing an injective map to the set of random walks with $2N$ steps: given a tree, consider a walker that starts at the root, and then travels over branches towards the right until it reaches a leaf, and then travels left until it can go right again on a different branch). Therefore
 \begin{equation}
 \sum_{\tau\in\mathcal T_{N}^{(h)}}\prod_{v\in\mathfrak V(\tau)}2^{-\mu}\leqslant C_T^N
 \label{boundT}\end{equation}
@@ -2143,7 +2143,7 @@ The action of $\mathcal L$ on functions on $\mathbf k$-space is
 $$c_k-(c_k-c_g)=1$$
 which, roughly, means that $\hat W_2^{(h')}$ is bounded by $2^{(c_k-(c_k-c_g))h'}=2^{h'}$. As was remarked above, this bound is insufficient since it does not constrain $\sum_{h'\geqslant h}\hat W_2^{(h')}$ to be smaller than $2^h\sim\hat g^{-1}$. Note that, while $\hat W_2^{(h')}({\bf k})$ is bounded by $2^{h'}$, irrespective of ${\bf k}$, 
 $(\mathbf k-\mathbf p_{F,0}^\omega)\cdot\partial_{\mathbf k}\hat W_2^{(h')}({\bf k})$ has an improved dimensional bound, proportional to $2^{h-h'}2^{h'}$, where $2^h\sim |{\bf k}-\mathbf p_{F,0}^\omega|$;
-in this sense, we can think of the operator $(\mathbf k-\mathbf p_{F,0}^\omega)\cdot\partial_{\mathbf k}$ as scaling like $2^{h-h'}$. Therefore, the remainder of the first order Taylor expansion is bounded by $2^{2(h-h')}2^{h'}=2^{2h-h'}$ and thereby has a scaling dimension of $-1$ (with respect to $h'$). Thus, by defining $\mathcal L$ as the first order Taylor expansion, we take the focus away from the remainder, which can be bounded easily because it is irrelevant (i.e., it has negative scaling dimension), and concentrate our attention on the relevant and marginal contributions of $\hat W_2^{(h')}$. See~[\cite{benNFi}, chapter~8] for details.\par
+in this sense, we can think of the operator $(\mathbf k-\mathbf p_{F,0}^\omega)\cdot\partial_{\mathbf k}$ as scaling like $2^{h-h'}$. Therefore, the remainder of the first order Taylor expansion is bounded by $2^{2(h-h')}2^{h'}=2^{2h-h'}$ and thereby has a scaling dimension of $-1$ (with respect to $h'$). Thus, by defining $\mathcal L$ as the first order Taylor expansion, we take the focus away from the remainder, which can be bounded easily because it is irrelevant (i.e., it has negative scaling dimension), and concentrate our attention on the relevant and marginal contributions of $\hat W_2^{(h')}$. See~[\cite{BG95}, chapter~8] for details.\par
 \bigskip
 
 \indent We then rewrite~(\ref{gbaroA}) as
@@ -2224,7 +2224,7 @@ By injecting~(\ref{boundzvo}) into~(\ref{summedzvmo}), we find
 \end{array}\label{boundzvtop}\end{equation}
 \bigskip
 
-\subpoint{Dominant part of $\mathcal L\hat{\bar A}_{h,\omega}$} Furthermore, we notice that the terms proportional to $\tilde m_h$ or $\tilde v_h$ are sub-dominant:
+\subpoint{\bf Dominant part of $\mathcal L\hat{\bar A}_{h,\omega}$} Furthermore, we notice that the terms proportional to $\tilde m_h$ or $\tilde v_h$ are sub-dominant:
 \begin{equation}
 \mathcal L\hat{\bar A}_{h,\omega}(\mathbf k'+\mathbf p_{F,0}^\omega)=
 \mathfrak L\hat{\bar A}_{h,\omega}(\mathbf k'+\mathbf p_{F,0}^\omega)(\mathds1+\sigma_1(\mathbf k'))
@@ -2403,7 +2403,7 @@ where $\tilde C_1^{(z)}$, $C_1^{(z)}$ and $C_1^{(v)}$ are constants (independent
 \label{boundsigmapo}\end{equation}
 \bigskip
 
-\point{Proof of Theorem~\ref{theoo}} We now conclude the proof of Theorem~\ref{theoo}, {\it under the assumption} (\ref{sumWo}): we define
+\point{\bf Proof of Theorem~\ref{theoo}} We now conclude the proof of Theorem~\ref{theoo}, {\it under the assumption} (\ref{sumWo}): we define
 $$
 z_1:=z_{\mathfrak h_1},\quad
 \tilde z_1:=\tilde z_{\mathfrak h_1},\quad
@@ -2547,7 +2547,7 @@ the tree expansion (\ref{treeexpT}), thus providing a convergent expansion of $W
 \bigskip
 
 {\bf Remark}: The first of~(\ref{powercountingtt}) exhibits a tendency to grow {\it logarithmically}
-in $2^{-h}$. This is not an artifact of the bounding procedure: indeed the second-order flow, computed in~[\cite{vafOZ}], exhibits the same logarithmic growth. 
+in $2^{-h}$. This is not an artifact of the bounding procedure: indeed the second-order flow, computed in~[\cite{Va10}], exhibits the same logarithmic growth. 
 However, the presence of the $\epsilon$ factor in front of $(h_\epsilon-h)\leqslant2|\log\epsilon|$ ensures this growth is benign: 
 it is cut off before it has a chance to be realized. 
 \par
@@ -2639,7 +2639,7 @@ By injecting~(\ref{boundzvtt}) through~(\ref{boundvot}) into~(\ref{summedzvmt}),
 \end{array}\label{boundzvttp}\end{equation}
 \bigskip
 
-\point{Dominant part of $\mathcal L\hat{\bar A}_{h,\omega}$} Furthermore, we notice that the terms proportional to $\tilde z_h$ or $\tilde v_h$ are sub-dominant:
+\point{\bf Dominant part of $\mathcal L\hat{\bar A}_{h,\omega}$} Furthermore, we notice that the terms proportional to $\tilde z_h$ or $\tilde v_h$ are sub-dominant:
 \begin{equation}
 \mathcal L\hat{\bar A}_{h,\omega}(\mathbf k'+\mathbf p_{F,0}^\omega)=
 \mathfrak L\hat{\bar A}_{h,\omega}(\mathbf k'+\mathbf p_{F,0}^\omega)(\mathds1+\sigma_3(\mathbf k'))
@@ -2771,7 +2771,7 @@ where $\bar M_{{h_{\mathbf k}}}$, $\bar a_{{h_{\mathbf k}}}^{(M)}$ and $\bar a_{
 \label{boundsigmapt}\end{equation}
 \bigskip
 
-\point{Proof of Theorem~\ref{theot}} We now conclude the proof of Theorem~\ref{theot}, {\it under the assumption} (\ref{sumWt}). We define
+\point{\bf Proof of Theorem~\ref{theot}} We now conclude the proof of Theorem~\ref{theot}, {\it under the assumption} (\ref{sumWt}). We define
 $$
 B_{h_{\mathbf k}}(\mathbf k):=(\mathds1+\sigma'(\mathbf k))\left(\hat{\bar g}_{h_{\mathbf k},\omega}(\mathbf k)+\hat{\bar g}_{h_{\mathbf k}-1,\omega}(\mathbf k)\right)^{-1}
 $$
@@ -2798,7 +2798,7 @@ B_{h_{\mathbf k}}^{-1}(\mathbf k)=B_{\mathfrak h_2}^{-1}(\mathbf k)(\mathds1+O(|
 We inject~(\ref{approxBht}) into~(\ref{schwinxprgtp}), which we then combine with~(\ref{schwinexprt}), (\ref{boundsigmat}), (\ref{boundsigmalkt}) and~(\ref{boundsigmapt}), and find an expression for $s_2$ which is similar to the right side of~(\ref{schwinxprgtp}) but with $h_{\mathbf k}$ replaced by $\mathfrak h_2$. This concludes the proof of~(\ref{schwint}). Furthermore, the estimate~(\ref{ineqrcct}) follows from~(\ref{boundzvttp}), which concludes the proof of Theorem~\ref{theot}.\par
 \bigskip
 
-\point{Partial proof of (\ref{sumWo})} Before moving on to the third regime, we bound part of the sum on the left side of~(\ref{sumWo}), which we recall was assumed to be true to prove~(\ref{schwino}) (see section~\ref{schwinosec}). It follows from~(\ref{pct}) that
+\point{\bf Partial proof of (\ref{sumWo})} Before moving on to the third regime, we bound part of the sum on the left side of~(\ref{sumWo}), which we recall was assumed to be true to prove~(\ref{schwino}) (see section~\ref{schwinosec}). It follows from~(\ref{pct}) that
 \begin{equation}
 \left|\sum_{h'=\mathfrak h_2}^{\bar{\mathfrak h}_1}\hat W_2^{(h')}(\mathbf k)\right|\leqslant(\mathrm{const}.)\ 2^{2h_\epsilon}|U|.
 \label{boundtsumW}\end{equation}
@@ -3004,7 +3004,7 @@ where $z_h$, $\tilde z_h$, $m_h$, $v_h$ and $\tilde v_h$ are defined as in~(\ref
 \end{array}\label{boundzvtthp}\end{equation}
 \bigskip
 
-\subpoint{Dominant part of $\mathcal L\hat{\bar A}_{h,\omega,0}$} Furthermore, we notice that the terms proportional to $\tilde z_h$ are sub-dominant:
+\subpoint{\bf Dominant part of $\mathcal L\hat{\bar A}_{h,\omega,0}$} Furthermore, we notice that the terms proportional to $\tilde z_h$ are sub-dominant:
 \begin{equation}
 \mathcal L\hat{\bar A}_{h,\omega,0}(\mathbf k'+\mathbf p_{F,0}^\omega)=\mathfrak L\hat{\bar A}_{h,\omega,0}(\mathbf k'+\mathbf p_{F,0}^\omega)(\mathds1+\sigma_4(\mathbf k'))
 \label{sepLAthdom}\end{equation}
@@ -3245,7 +3245,7 @@ Furthermore, using the bounds (\ref{boundzvthowconst}) through (\ref{boundmzvtho
 \end{array}\label{sboundmzvwthj}\end{equation}
 \bigskip
 
-\subsubpoint{Dominant part of $\hat{\mathcal L}_h\hat{\bar A}_{h,\omega,1}$} Finally, we notice that the terms in~(\ref{LAthj}) that are proportional to $z_{h,1}^{\xi\xi}$, $z_{h,1}^{\xi\phi}$ or $K_{h,1}^{\xi\xi}$ are subdominant:
+\subsubpoint{\bf Dominant part of $\hat{\mathcal L}_h\hat{\bar A}_{h,\omega,1}$} Finally, we notice that the terms in~(\ref{LAthj}) that are proportional to $z_{h,1}^{\xi\xi}$, $z_{h,1}^{\xi\phi}$ or $K_{h,1}^{\xi\xi}$ are subdominant:
 \begin{equation}
 \hat{\mathcal L}_h\hat{\bar A}_{h,\omega,1}(\mathbf k'_{1}+\tilde{\mathbf p}_{F,1}^{(\omega,h)})=\hat{\mathfrak L}_h\hat{\bar A}_{h,\omega,1}(\mathbf k'_{1}+\tilde{\mathbf p}_{F,1}^{(\omega,h)})(\mathds1+\sigma_{4,1}(\mathbf k'_{1}))
 \label{sepLAthjdom}\end{equation}
@@ -3375,7 +3375,7 @@ where $\bar M_{{h_{\mathbf k}},1}$, $\bar a_{{h_{\mathbf k}},1}^{(M)}$ and $\bar
 \subpoint{\bf $j=2,3$ cases.} The cases with $j=2,3$ follow from the $2\pi/3$-rotation symmetry~(\ref{arotation}) (see~(\ref{Wreltht})).\par
 \bigskip
 
-\point{Proof of Theorem~\ref{theoth}} We now conclude the proof of Theorem~\ref{theoth}. We focus our attention on $j=0,1$ since the cases with $j=2,3$ follow by symmetry. Similarly to section~\ref{schwintsec}, we define
+\point{\bf Proof of Theorem~\ref{theoth}} We now conclude the proof of Theorem~\ref{theoth}. We focus our attention on $j=0,1$ since the cases with $j=2,3$ follow by symmetry. Similarly to section~\ref{schwintsec}, we define
 $$
 B_{h_{\mathbf k},j}(\mathbf k):=(\mathds1+\sigma'_j(\mathbf k))\left(\hat{\bar g}_{h_{\mathbf k},\omega,j}(\mathbf k)+\hat{\bar g}_{h_{\mathbf k}-1,\omega,j}(\mathbf k)\right)^{-1}
 $$
@@ -3415,7 +3415,7 @@ We inject~(\ref{approxBhth}) into~(\ref{schwinxprgthp}) and~(\ref{schwinxprgthjp
 
 \bigskip
 
-\point{Proof of (\ref{sumWo}) and (\ref{sumWt})} In order to conclude the proofs of Theorems~\ref{theoo} and~\ref{theot} as well as the Main Theorem, we still have to bound the sums on the left side of~(\ref{sumWo}) and of~(\ref{sumWt}), which we recall were assumed to be true to prove~(\ref{schwino}) and~(\ref{schwint}) (see sections~\ref{schwinosec} and~\ref{schwintsec}). It follows from~(\ref{pcth}) that
+\point{\bf Proof of (\ref{sumWo}) and (\ref{sumWt})} In order to conclude the proofs of Theorems~\ref{theoo} and~\ref{theot} as well as the Main Theorem, we still have to bound the sums on the left side of~(\ref{sumWo}) and of~(\ref{sumWt}), which we recall were assumed to be true to prove~(\ref{schwino}) and~(\ref{schwint}) (see sections~\ref{schwinosec} and~\ref{schwintsec}). It follows from~(\ref{pcth}) that
 \begin{equation}
 \left|\sum_{h'=\mathfrak h_\beta}^{\bar{\mathfrak h}_2}\hat W_2^{(h')}(\mathbf k)\right|\leqslant(\mathrm{const}.)\ 2^{4h_\epsilon}|U|.
 \label{boundthsumW}\end{equation}\par
@@ -3427,10 +3427,10 @@ This, along with~(\ref{boundtsumW}) concludes the proofs of~(\ref{sumWo}) and~(\
 \indent We considered a tight-binding model of bilayer graphene describing spin-less fermions hopping on two hexagonal layers in Bernal stacking, in the presence of short range interactions. We assumed that only three hopping parameters are different from zero (those usually called $\gamma_0,\gamma_1$ and $\gamma_3$ in the literature), in which case the Fermi surface at half-filling degenerates to a collection of 8 Fermi points.  Under a smallness assumption on the interaction strength $U$ and on the transverse hopping $\epsilon$, we proved by rigorous RG methods that the specific ground state energy and correlation functions in the thermodynamic limit are analytic in $U$, uniformly in $\epsilon$. Our proof requires a detailed analysis of the crossover regimes from one in which the two layers are effectively decoupled, to one where the effective dispersion relation is approximately parabolic around the central Fermi points (and the inter-particle interaction is effectively marginal), to the deep infrared one, where the effective dispersion relation is approximately conical around each Fermi points (and the inter-particle interaction is effectively irrelevant).  Such an analysis, in which the influence of the flow of the effective constants in one regime has crucial repercussions in lower regimes, is, to our knowledge, novel.\par
 \bigskip
 
-\indent We expect our proof to be adaptable without substantial efforts to the case where $\gamma_4$ and $\Delta$ are different from zero, as in (\ref{relge}), the intra-layer next-to-nearest neighbor hopping $\gamma_0'$ is $O(\epsilon)$, the chemical potential is $O(\epsilon^3)$, and the temperature is larger than (const.)$\epsilon^4$. At smaller scales, the Fermi set becomes effectively one-dimensional, which thoroughly changes the scaling properties. In particular, the effective inter-particle interaction becomes marginal, again, and its flow tends to grow logarithmically. Perturbative analysis thus breaks down at exponentially small temperatures in $\epsilon$ and in $U$, and it should be possible to rigorously control the system down to such temperatures.  Such an analysis could prove difficult, because it requires fine control on the geometry of the Fermi surface, as in [\cite{benZS}] and in [\cite{fktZFa}, \cite{fktZFb}, \cite{fktZFc}], where the Fermi liquid behavior of a system of interacting electrons was proved, respectively down to exponentially small and zero temperatures, under different physical conditions. We hope to come back to this issue in the future.\par
+\indent We expect our proof to be adaptable without substantial efforts to the case where $\gamma_4$ and $\Delta$ are different from zero, as in (\ref{relge}), the intra-layer next-to-nearest neighbor hopping $\gamma_0'$ is $O(\epsilon)$, the chemical potential is $O(\epsilon^3)$, and the temperature is larger than (const.)$\epsilon^4$. At smaller scales, the Fermi set becomes effectively one-dimensional, which thoroughly changes the scaling properties. In particular, the effective inter-particle interaction becomes marginal, again, and its flow tends to grow logarithmically. Perturbative analysis thus breaks down at exponentially small temperatures in $\epsilon$ and in $U$, and it should be possible to rigorously control the system down to such temperatures.  Such an analysis could prove difficult, because it requires fine control on the geometry of the Fermi surface, as in [\cite{BGM06}] and in [\cite{FKT04}, \cite{FKT04b}, \cite{FKT04c}], where the Fermi liquid behavior of a system of interacting electrons was proved, respectively down to exponentially small and zero temperatures, under different physical conditions. We hope to come back to this issue in the future.\par
 \bigskip
 
-\indent Another possible extension would be the study of crossover effects on other physical observables, such as the conductivity, in the spirit of [\cite{masOO}]. In addition, it would be interesting to study the case of three-dimensional Coulomb interactions, which is physically interesting in describing {\it clean} bilayer graphene samples, i.e.  where screening effects are supposedly negligible. It may be possible to draw inspiration from the analysis of [\cite{gmpOZ}, \cite{gmpOOt}] to construct the ground state, order by order in renormalized perturbation theory.  The construction of the theory in the second and third regimes would pave the way to understanding the universality of the conductivity in the deep infrared, beyond the regime studied in [\cite{masOO}].\par
+\indent Another possible extension would be the study of crossover effects on other physical observables, such as the conductivity, in the spirit of [\cite{Ma11}]. In addition, it would be interesting to study the case of three-dimensional Coulomb interactions, which is physically interesting in describing {\it clean} bilayer graphene samples, i.e.  where screening effects are supposedly negligible. It may be possible to draw inspiration from the analysis of [\cite{GMP10}, \cite{GMP11b}] to construct the ground state, order by order in renormalized perturbation theory.  The construction of the theory in the second and third regimes would pave the way to understanding the universality of the conductivity in the deep infrared, beyond the regime studied in [\cite{Ma11}].\par
 \bigskip
 
 {\bf Acknowledgments} We acknowledge financial support from the ERC Starting Grant CoMBoS (grant agreement No. 239694)
@@ -3505,7 +3505,7 @@ $$
 k_x=\frac{2\pi}{3}-\frac{2}{3}\arccos\left(\frac{\sqrt{1+G}(2-G)}{2}\right),\quad
 k_y=\pm\frac{2}{\sqrt3}\arccos\left(\frac{\sqrt{1+G}}{2}\right).
 $$
-\penalty10000\hfill\penalty10000$\square$
+\qed
 
 
 \section{$4\times4$ matrix inversions}
@@ -3636,7 +3636,7 @@ k_y>\sqrt{\frac{35}{48}}\bar\kappa\bar\epsilon^2,\quad
 $$
 so that
 $$\left|k_y(k_y-D\bar\epsilon^2)\right|-k_x^2>\frac{3\sqrt{105}-1}{48}\bar\kappa^2\bar\epsilon^4$$
-and $l>\alpha((3\sqrt{105}-1)^2/2304)\bar\kappa^4\bar\epsilon^8$ from which~(\ref{ineqdeta}) follows.\penalty10000\hfill\penalty10000$\square$\par
+and $l>\alpha((3\sqrt{105}-1)^2/2304)\bar\kappa^4\bar\epsilon^8$ from which~(\ref{ineqdeta}) follows.\qed\par
 
 
 \section{Symmetries}
@@ -3659,8 +3659,8 @@ where
 $$\hat v_{\alpha,\alpha'}(k):=\sum_{x\in\Lambda}\ e^{ik\cdot x}v(x+d_\alpha-d_{\alpha'}).$$
 
 \point{\bf Global $U(1)$.} Follows immediately from the fact that there are as many $\psi^+$ as $\psi^-$ in $h_0$ and $\mathcal V$.
-\penalty10000\hfill\penalty10000$\square$\par
-
+\qed\par
+\bigskip
 
 \point{\bf  $2\pi/3$ rotation.} We have 
 $$\Omega(e^{i\frac{2\pi}3\sigma_2}k)=e^{il_2 \cdot k}\Omega(k),\quad e^{3i(e^{i\frac{2\pi}3\sigma_2}k)|_x}=e^{-3il_2\cdot k}e^{3ik_x}$$
@@ -3693,14 +3693,14 @@ $$\left(\begin{array}{c}
 e^{il_2(k_1-k_2)}\hat\psi_{\mathbf k_1,\tilde a}^+\hat\psi_{\mathbf k_1,\tilde a}^-\\[0.2cm]
 e^{-il_2(k_1-k_2)}\hat\psi_{\mathbf k_1,b}^+\hat\psi_{\mathbf k_1,b}^-
 \end{array}\right)$$
-from which one easily concludes that $\mathcal V$ is invariant under~(\ref{arotation}).\penalty10000\hfill\penalty10000$\square$\par
-
-
-\point{\bf Complex conjugation.} Follows immediately from $\Omega(-k)=\Omega^*(k)$ and $v(-k)=v^*(k)$.\penalty10000\hfill\penalty10000$\square$\par
-
+from which one easily concludes that $\mathcal V$ is invariant under~(\ref{arotation}).\qed\par
+\bigskip
 
-\point{\bf  Vertical reflection.} Follows immediately from $\Omega(R_vk)=\Omega(k)$ and $v(R_vk)=v(k)$ (since the second component of $d_\alpha$ is 0).\penalty10000\hfill\penalty10000$\square$\par
+\point{\bf Complex conjugation.} Follows immediately from $\Omega(-k)=\Omega^*(k)$ and $v(-k)=v^*(k)$.\qed\par
+\bigskip
 
+\point{\bf  Vertical reflection.} Follows immediately from $\Omega(R_vk)=\Omega(k)$ and $v(R_vk)=v(k)$ (since the second component of $d_\alpha$ is 0).\qed\par
+\bigskip
 
 \point{\bf  Horizontal reflection.} We have $\Omega(R_hk)=\Omega^*(k)$, $\sigma_1A^{\xi\xi}({\bf k})\sigma_1=A^{\xi\xi}({\bf k})$, 
 $$\sigma_1A^{\xi\phi}({\bf k})\sigma_1=\left(\begin{array}{*{2}{c}}0&\Omega(k)\\\Omega^*(k)&0\end{array}\right)
@@ -3708,16 +3708,15 @@ $$\sigma_1A^{\xi\phi}({\bf k})\sigma_1=\left(\begin{array}{*{2}{c}}0&\Omega(k)\\
 $$
 from which the invariance of $h_0$ follows immediately. Furthermore
 $$v_{\alpha,\alpha'}(R_hk)=v_{\pi_h(\alpha),\pi_h(\alpha')}(k)$$
-where $\pi_h$ is the permutation that exchanges $a$ with $\tilde b$ and $\tilde a$ with $b$, from which the invariance of $\mathcal V$ follows immediately.\penalty10000\hfill\penalty10000$\square$\par
-
+where $\pi_h$ is the permutation that exchanges $a$ with $\tilde b$ and $\tilde a$ with $b$, from which the invariance of $\mathcal V$ follows immediately.\qed\par
+\bigskip
 
 \point{\bf  Parity.} We have $\Omega(Pk)=\Omega^*(k)$ so that $\big[A^{\xi\phi }(P\mathbf k)\big]^T=A^{\xi\phi}(\mathbf k)$, $\big[A^{\phi\phi}(P\mathbf k)\big]^T=A^{\phi\phi}(\mathbf k)$, $\big[A^{\xi\xi}(P\mathbf k)\big]^T=A^{\xi\xi}(\mathbf k)$. Therefore $h_0$ is mapped to
 $$
 h_0\longmapsto-\frac{1}{\chi_0(2^{-M}|k_0|)\beta|\Lambda|}\sum_{\mathbf k\in\mathcal B^*_{\beta,L}}\left(\begin{array}{*{2}{c}}\hat\xi_{\mathbf k}^-&\hat\phi_{\mathbf k}^-\end{array}\right)\left(\begin{array}{*{2}{c}}A^{\xi\xi}(\mathbf k)&A^{\xi\phi}(\mathbf k)\\A^{\phi\xi}(\mathbf k)&A^{\phi\phi}(\mathbf k)\end{array}\right)^T\left(\begin{array}{c}\hat\xi^+_{\mathbf k}\\\hat\phi^+_{\mathbf k}\end{array}\right)
 $$
 which is equal to $h_0$ since exchanging $\hat\psi^-$ and $\hat\psi^+$ adds a minus sign. The invariance of $\mathcal V$ follows from the remark that under parity 
-$\hat\psi^+_{{\bf k}_1,\alpha}\hat\psi^-_{{\bf k}_2,\alpha}\mapsto \hat\psi^+_{P{\bf k}_2,\alpha}\hat\psi^-_{P{\bf k}_1,\alpha}$, and $\hat v(k_1-k_2)=\hat v(P(k_2-k_1))$.
-\penalty10000\hfill\penalty10000$\square$\par
+$\hat\psi^+_{{\bf k}_1,\alpha}\hat\psi^-_{{\bf k}_2,\alpha}\mapsto \hat\psi^+_{P{\bf k}_2,\alpha}\hat\psi^-_{P{\bf k}_1,\alpha}$, and $\hat v(k_1-k_2)=\hat v(P(k_2-k_1))$.\qed\par
 \bigskip
 
 \point{\bf  Time inversion.} We have 
@@ -3726,8 +3725,7 @@ $$\begin{array}c
 \sigma_3A^{\xi\phi}(I\mathbf k)\sigma_3=-A^{\xi\phi}(\mathbf k),\\[0.2cm]
 \sigma_3A^{\phi\phi}(I\mathbf k)\sigma_3=-A^{\phi\phi}(\mathbf k)
 \end{array}$$
-from which the invariance of $h_0$ follows immediately. The invariance of $\mathcal V$ is trivial.\penalty10000\hfill\penalty10000$\square$\par
-
+from which the invariance of $h_0$ follows immediately. The invariance of $\mathcal V$ is trivial.\qed\par
 
 \section{Constraints due to the symmetries}
 \label{constWtapp}
@@ -3802,7 +3800,7 @@ We have
 $$
 \partial_{k_2} M(\mathbf p_{F}^\omega)=-(\partial_{k_2} M(\mathbf p_{F}^{-\omega}))^*=-\partial_{k_2} M(\mathbf p_{F}^{-\omega})=\sigma_1\partial_{k_2} M(\mathbf p_{F}^\omega)\sigma_1=-\sigma_3\partial_{k_2} M(\mathbf p_{F}^\omega)\sigma_3.
 $$
-Therefore $t_2^{(\omega)}=y_2^{(\omega)}=z_2^{(\omega)}=0$, $x_2^{(\omega)}=-x_2^{(-\omega)}\in\mathbb{R}$.\penalty10000\hfill\penalty10000$\square$\par
+Therefore $t_2^{(\omega)}=y_2^{(\omega)}=z_2^{(\omega)}=0$, $x_2^{(\omega)}=-x_2^{(-\omega)}\in\mathbb{R}$.\qed\par
 
 
 \theo{Proposition}\label{symfourprop}
@@ -3894,7 +3892,7 @@ $$
 \partial_{k_i}\mathbf M^{\xi\xi}(\mathbf p_{F}^\omega)=\sum_{j=1}^2T_{i,j}\partial_{k_j}\mathbf M^{\xi\xi}(\mathbf p_{F}^\omega)
 $$
 so that
-$\partial_{k_i}\mathbf M^{\xi\xi}(\mathbf p_{F}^\omega)=0$, that is $\nu^{\xi\xi}=\varpi^{\xi\xi}=0$.\penalty10000\hfill\penalty10000$\square$\par
+$\partial_{k_i}\mathbf M^{\xi\xi}(\mathbf p_{F}^\omega)=0$, that is $\nu^{\xi\xi}=\varpi^{\xi\xi}=0$.\qed\par
 
 \vfill
 \eject
@@ -3902,6 +3900,7 @@ $\partial_{k_i}\mathbf M^{\xi\xi}(\mathbf p_{F}^\omega)=0$, that is $\nu^{\xi\xi
 \small
 \BBlography
 
+\vfill
 \eject
 
 \end{document}
-- 
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