Typo in definition of Fourier transform

Giuliani_Jauslin_2015.tex

@@ -23,7 +23,7 @@
 
 \hugeskip
 
-\indent We consider a model of half-filled bilayer graphene, in which the three dominant Slonczewski-Weiss-McClure hopping parameters are retained, in the presence of short range interactions. Under a smallness assumption on the interaction strength $U$ as well as on the inter-layer hopping $\epsilon$, we construct the ground state in the thermodynamic limit, and prove its analyticity in $U$, uniformly in $\epsilon$. The interacting Fermi surface is degenerate, and consists of eight Fermi points, two of which are protected by symmetries, while the locations of the other six are renormalized by the interaction, and the effective dispersion relation at the Fermi points is conical.  The construction reveals the presence of different energy regimes, where the effective behavior of correlation functions changes qualitatively. The analysis of the crossover between regimes plays an important role in the proof of analyticity and in the uniform control of the radius of convergence.  The proof is based on a rigorous implementation of fermionic renormalization group methods, including determinant estimates for the renormalized expansion.\par
+\indent We consider a model of half-filled bilayer graphene, in which the three dominant Slonczewski-Weiss-McClure hopping parameters are retained, in the presence of short range interactions. Under a smallness assumption on the interaction strength $U$ as well as on the inter-layer hopping $\epsilon$, we construct the ground state in the thermodynamic limit, and prove that the pressure and two point Schwinger function, away from its singularities, are analytic in $U$, uniformly in $\epsilon$. The interacting Fermi surface is degenerate, and consists of eight Fermi points, two of which are protected by symmetries, while the locations of the other six are renormalized by the interaction, and the effective dispersion relation at the Fermi points is conical.  The construction reveals the presence of different energy regimes, where the effective behavior of correlation functions changes qualitatively. The analysis of the crossover between regimes plays an important role in the proof of analyticity and in the uniform control of the radius of convergence.  The proof is based on a rigorous implementation of fermionic renormalization group methods, including determinant estimates for the renormalized expansion.\par
 
 \hugeskip
 
@@ -76,7 +76,7 @@ at least in the case we are treating, namely small interaction strength and smal
 
 \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{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. 
+\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, which would allow us to carry out the analysis down to temperatures that are exponentially low in the coupling constant, and we hope to come back to this issue in a future publication.
 \par
 \bigskip
 
@@ -574,8 +574,8 @@ where $\mathcal H_0$ is the {\it free Hamiltonian} and $\mathcal H_I$ is the {\i
 \end{array}\label{hamx}\end{equation}
 Equation~(\ref{hamx}) can be rewritten in Fourier space as follows. We define the Fourier transform of the annihilation operators as
 \begin{equation} \hat a_{k}:=\sum_{x\in\Lambda}e^{ikx}a_{x}\;,\quad 
-\hat{\tilde b}_{k}:=\sum_{x\in\Lambda}e^{ikx}\hat{\tilde b}_{x+\delta_1}\;,\quad
-\hat{\tilde a}_{k}:=\sum_{x\in\Lambda}e^{ikx}\hat{\tilde a}_{x-\delta_1}\;,\quad
+\hat{\tilde b}_{k}:=\sum_{x\in\Lambda}e^{ikx}\tilde b_{x+\delta_1}\;,\quad
+\hat{\tilde a}_{k}:=\sum_{x\in\Lambda}e^{ikx}\tilde a_{x-\delta_1}\;,\quad
 \hat b_{k}:=\sum_{x\in\Lambda}e^{ikx}b_{x+\delta_1}\;\end{equation} 
 in terms of which
 \begin{equation}
@@ -785,7 +785,7 @@ is a symmetry.\par
 and every complex coefficient of $h_0$ and $\mathcal V$ is mapped to its complex conjugate is a symmetry. 
 \bigskip
 
-\point{\bf  Vertical reflection.} Let $R_v\mathbf k=(k_0,k_1,-k_2)$,
+\point{\bf  Vertical reflection.} Let $R_v\mathbf k=(k_0,k_x,-k_y)$,
 \begin{equation}
 \left\{\begin{array}l
 \hat\xi_{\mathbf k}^\pm\longmapsto \hat\xi_{R_v\mathbf k}^\pm\\[0.2cm]
@@ -794,7 +794,7 @@ and every complex coefficient of $h_0$ and $\mathcal V$ is mapped to its complex
 is a symmetry.
 \bigskip
 
-\point{\bf  Horizontal reflection.} Let $R_h\mathbf k=(k_0,-k_1,k_2)$,
+\point{\bf  Horizontal reflection.} Let $R_h\mathbf k=(k_0,-k_x,k_y)$,
 \begin{equation}\left\{\begin{array}l
 \hat\xi_{\mathbf k}^-\longmapsto \sigma_1\hat\xi_{R_h\mathbf k}^-,\ \hat\xi_{\mathbf k}^+\longmapsto\hat\xi_{R_h\mathbf k}^+\sigma_1\\[0.2cm]
 \hat\phi_{\mathbf k}^-\longmapsto \sigma_1\hat\phi_{R_h\mathbf k}^-,\ \hat\phi_{\mathbf k}^+\longmapsto\hat\phi_{R_h\mathbf k}^+\sigma_1
@@ -802,7 +802,7 @@ is a symmetry.
 is a symmetry.\par
 \bigskip
 
-\point{\bf  Parity.} Let $P\mathbf k=(k_0,-k_1,-k_2)$,
+\point{\bf  Parity.} Let $P\mathbf k=(k_0,-k_x,-k_y)$,
 \begin{equation}\left\{\begin{array}l
 \hat\xi_{\mathbf k}^\pm\longmapsto i(\hat\xi_{P\mathbf k}^{\mp})^T\\[0.2cm]
 \hat\phi_{\mathbf k}^\pm\longmapsto i(\hat\phi_{P\mathbf k}^{\mp})^T
@@ -810,7 +810,7 @@ is a symmetry.\par
 is a symmetry.
 \bigskip
 
-\point{\bf  Time inversion.} Let $I\mathbf k=(-k_0,k_1,k_2)$, the mapping
+\point{\bf  Time inversion.} Let $I\mathbf k=(-k_0,k_x,k_y)$, the mapping
 \begin{equation}\left\{\begin{array}l
 \hat\xi_{\mathbf k}^-\longmapsto -\sigma_3\hat\xi_{I\mathbf k}^-,\ \hat\xi_{\mathbf k}^+\longmapsto\hat\xi_{I\mathbf k}^+\sigma_3\\[0.2cm]
 \hat\phi_{\mathbf k}^-\longmapsto-\sigma_3\hat\phi_{I\mathbf k}^-,\ \hat\phi_{\mathbf k}^+\longmapsto\hat\phi_{I\mathbf k}^+\sigma_3
@@ -1383,7 +1383,7 @@ good enough for performing the localization and renormalization procedure descri
 \begin{equation}
 |2^{hm_0}\partial_{k_0}^{m_0}\partial_k^{m_k}\hat g_{h}(\mathbf k)|\leqslant (\mathrm{const.})\ 2^{-h}
 \label{estgkuv}\end{equation}
-in which $\partial_{k_0}$ denotes the discrete derivative with respect to $k_0$ and, with a slightly abusive notation, $\partial_k$ the discrete derivative with respect to either $k_1$ or $k_2$. Indeed the derivatives over $k$ land on $ik_0\hat A^{-1}$, which does not change the previous estimate, and the derivatives over $k_0$ either land on $f_h$, $1/(ik_0)$, or $ik_0\hat A^{-1}$, which yields an extra $2^{-h}$ in the estimate.\par
+in which $\partial_{k_0}$ denotes the discrete derivative with respect to $k_0$ and, with a slightly abusive notation, $\partial_k$ the discrete derivative with respect to either $k_x$ or $k_y$. Indeed the derivatives over $k$ land on $ik_0\hat A^{-1}$, which does not change the previous estimate, and the derivatives over $k_0$ either land on $f_h$, $1/(ik_0)$, or $ik_0\hat A^{-1}$, which yields an extra $2^{-h}$ in the estimate.\par
 \medskip
 
 {\bf Remark}: The previous argument implicitly uses the Leibnitz rule, which must be used carefully since the derivatives are discrete. However, since the estimate is purely dimensional,
@@ -1419,7 +1419,7 @@ and for $m\leqslant7$,
 \begin{equation}
 |2^{mh}\partial_{\mathbf k}^m\hat g_{h,\omega}(\mathbf k)|\leqslant (\mathrm{const.})\ 2^{-h}
 \label{estgko}\end{equation}
-in which we again used the slightly abusive notation of writing $\partial_{\mathbf k}$ to mean any derivative with respect to $k_0$, $k_1$ or $k_2$. Equation~(\ref{estgko}) then follows from similar considerations as those in the ultraviolet regime.\par
+in which we again used the slightly abusive notation of writing $\partial_{\mathbf k}$ to mean any derivative with respect to $k_0$, $k_x$ or $k_y$. Equation~(\ref{estgko}) then follows from similar considerations as those in the ultraviolet regime.\par
 \bigskip
 
 \subpoint{\bf Configuration space bounds.} We estimate the real-space counterpart of $\hat g_{h,\omega}$, 
@@ -1684,7 +1684,7 @@ scaling dimension {\it relevant}.
 \item We will show that in the first and third regimes $c_k=3$ and $c_g=1$, so that the scaling dimension is $3-|P_v|$. Therefore, the nodes with $|P_v|=2$ are relevant whereas all the others are irrelevant. In the second regime,
 $c_k=2$ and $c_g=1$, so that the scaling dimension is $2-|P_v|/2$. Therefore, the nodes with $|P_v|=2$ are relevant, those with $|P_v|=4$ are marginal, and all other nodes are irrelevant.
 
-\item The purpose of the factor $\mathfrak F_h(\underline m)$ is to take into account the dependence of the order of magnitude of the different components $k_0$, $k_1$ and $k_2$ in the different regimes. In other words, as was shown in~(\ref{estguv}), (\ref{estgo}), (\ref{estgt}) and~(\ref{estgth}), the effect of multiplying $g$ by $x_{j,i}$ depends on $i$, which is a fact the lemma must take into account.
+\item The purpose of the factor $\mathfrak F_h(\underline m)$ is to take into account the dependence of the order of magnitude of the different components $k_0$, $k_x$ and $k_y$ in the different regimes. In other words, as was shown in~(\ref{estguv}), (\ref{estgo}), (\ref{estgt}) and~(\ref{estgth}), the effect of multiplying $g$ by $x_{j,i}$ depends on $i$, which is a fact the lemma must take into account.
 
 \item The reason why we have stated this bound in $\mathbf x$-space is because of the estimate of $\det(G^{(h_v,T_v)})$ detailed below, which is very inefficient in $\mathbf k$-space.
 \end{itemize}
@@ -1772,7 +1772,7 @@ Therefore, using the bound~(\ref{boundGt}), the change of variables defined abov
 \begin{equation}
 \begin{array}{>{\displaystyle}r@{\ }>{\displaystyle}l}
 \frac{1}{\beta|\Lambda|}\int d\underline{\mathbf x}\ \left|(\underline{\mathbf x}-\mathbf x_{2l})^mB_{2l,\underline\varpi}^{(h)}(\underline{\mathbf x})\right|
-\leqslant&\sum_{N=1}^\infty\sum_{\tau\in\mathcal T_N^{(h)}}\sum_{T\in\mathbf T(\tau)}\sum_{\underline l_\tau}\sum_{\displaystyle\mathop{\scriptstyle \mathbf P\in\tilde{\mathcal P}_{\tau,\underline l_\tau,\ell_0}}_{|P_{v_0}|=2l}}c_1^N\mathfrak F_h(\underline m)\cdot\\[0.5cm]
+\leqslant&\sum_{N=1}^\infty\sum_{\tau\in\mathcal T_N^{(h)}}\sum_{T\in\mathbf T(\tau)}\sum_{\underline l_\tau}\sum_{\displaystyle\mathop{\scriptstyle \mathbf P\in\tilde{\mathcal P}_{\tau,\underline l_\tau,\ell_0}}_{|P_{v_0}|=2l}}C_1^N\mathfrak F_h(\underline m)\cdot\\[0.5cm]
 &\cdot\prod_{v\in\mathfrak V(\tau)}\frac{1}{s_v!}C_G^{n_v-s_v+1}C_g^{s_v-1}2^{h_v((c_k-c_g)n_v-c_k(s_v-1))}\cdot\\[0.5cm]
 &\cdot\prod_{v\in\mathfrak E(\tau)}c_2^{2l_v}\mathfrak C_{2l_v}|U|^{\max(1,l_v-1)}2^{(h_v-1)(c_k-(c_k-c_g)l_v)}
 \end{array}
@@ -1898,7 +1898,7 @@ one would find a {\it logarithmic} divergence for $\int d\mathbf x|K_{2,(\alpha,
  the dominant terms in $\hat g_{h}(\mathbf k)$ are odd in $k_0$, so they cancel when considering
 $$\sum_{k_0\in\frac{2\pi}\beta(\mathbb Z+\frac12)}\hat g_{h}(\mathbf k).$$
 From this idea, we compute an improved bound for $|g_{h}(\mathbf x)|$ with $x_0=0$:
-$$|g_{h}(0,x_1,x_2)|\leqslant \sum_{k_1,k_2}\left|\sum_{k_0}\hat g_{h}(\mathbf k)\right|\leqslant (\mathrm{const.})\ 2^{-h}.$$
+$$|g_{h}(0,x,y)|\leqslant \sum_{k_x,k_y}\left|\sum_{k_0}\hat g_{h}(\mathbf k)\right|\leqslant (\mathrm{const.})\ 2^{-h}.$$
 All in all, we find
 \begin{equation}
 \int d\mathbf x\ |\mathbf x^mK^{(h)}_{2,(\alpha,\alpha')}(\mathbf x)|\leqslant\mathfrak C_4|U|,\quad
@@ -2128,7 +2128,7 @@ for $h'>h$, which we do not have (and if we tried to prove it by induction, we w
 \begin{equation}
 \mathcal L:\bar A_{h,\omega}(\mathbf x)\longmapsto\delta(\mathbf x)\int d\mathbf y\ \bar A_{h,\omega}(\mathbf y)-\partial_\mathbf x\delta(\mathbf x)\cdot\int d\mathbf y\ \mathbf y \bar A_{h,\omega}(\mathbf y)
 \label{Wreldefo}\end{equation}
-where $\delta(\mathbf x):=\delta(x_0)\delta_{x_1,0}\delta_{x_2,0}$ and in the second term, as usual, the derivative with respect to $x_1$ and $x_2$ is discrete; as well as
+where $\delta(x_0,x_1,x_2):=\delta(x_0)\delta_{x_1,0}\delta_{x_2,0}$ and in the second term, as usual, the derivative with respect to $x_1$ and $x_2$ is discrete; as well as
 the corresponding {\it irrelevator}:
 \begin{equation}
 \mathcal R:=\mathds1-\mathcal L.
@@ -3736,10 +3736,10 @@ from which the invariance of $h_0$ follows immediately. The invariance of $\math
 \indent We recall the definitions of the symmetry transformations from section~\ref{symsec}:
 \begin{equation}\begin{array}c
 T\mathbf k:=(k_0,e^{-i\frac{2\pi}3\sigma_2}k),\quad
-R_v\mathbf k:=(k_0,k_1,-k_2),\quad
-R_h\mathbf k:=(k_0,-k_1,k_2),\\[0.2cm]
-P\mathbf k:=(k_0,-k_1,-k_2),\quad
-I\mathbf k:=(-k_0,k_1,k_2).
+R_v\mathbf k:=(k_0,k_x,-k_y),\quad
+R_h\mathbf k:=(k_0,-k_x,k_y),\\[0.2cm]
+P\mathbf k:=(k_0,-k_x,-k_y),\quad
+I\mathbf k:=(-k_0,k_x,k_y).
 \end{array}\label{symdefs}\end{equation}
 Furthermore, given a $4\times4$ matrix $\mathbf M$ whose components are indexed by $\{a,\tilde b,\tilde a,b\}$, we denote the sub-matrix with components in $\{a,\tilde b\}^2$ by $\mathbf M^{\xi\xi}$, that with $\{\tilde a,b\}^2$ by $\mathbf M^{\phi\phi}$, with $\{a,\tilde b\}\times\{\tilde a,b\}$ by $\mathbf M^{\xi\phi}$ and with $\{\tilde a,b\}\times\{a,\tilde b\}$ by $\mathbf M^{\phi\xi}$. In addition, given a complex matrix $M$, we denote its component-wise complex conjugate by $M^*$ (which is not to be confused with its adjoint $M^\dagger$).\par
 \bigskip 
@@ -3757,8 +3757,8 @@ for $\omega\in\{-,+\}$, then $\exists(\mu,\zeta,\nu,\varpi)\in\mathbb{R}^4$ such
 \begin{equation}\begin{array}c
 M(\mathbf p_{F}^\omega)=\mu\sigma_1,\quad
 \partial_{k_0} M(\mathbf p_{F}^\omega)=i\zeta\mathds1,\\[0.5cm]
-\partial_{k_1} M(\mathbf p_{F}^\omega)=\nu\sigma_2,\quad
-\partial_{k_2} M(\mathbf p_{F}^\omega)=\omega \varpi\sigma_1.
+\partial_{k_x} M(\mathbf p_{F}^\omega)=\nu\sigma_2,\quad
+\partial_{k_y} M(\mathbf p_{F}^\omega)=\omega \varpi\sigma_1.
 \end{array}\label{Apfzt}\end{equation}
 \endtheo
 \bigskip
@@ -3784,21 +3784,21 @@ $$
 Therefore $(t_0,x_0,y_0,z_0)$ are independent of $\omega$, $x_0=y_0=z_0=0$ and $t_0\in i\mathbb{R}$.\par
 \bigskip
 
-\point We now turn our attention to $\partial_{k_1}M$:
-$$\partial_{k_1} M(\mathbf p_{F}^\omega)=:t_1\mathds1+x_1\sigma_1+y_1\sigma_2+z_1\sigma_3.$$
+\point We now turn our attention to $\partial_{k_x}M$:
+$$\partial_{k_x} M(\mathbf p_{F}^\omega)=:t_1\mathds1+x_1\sigma_1+y_1\sigma_2+z_1\sigma_3.$$
 We have
 $$
-\partial_{k_1} M(\mathbf p_{F}^\omega)=-(\partial_{k_1} M(\mathbf p_{F}^{-\omega}))^*=\partial_{k_1} M(\mathbf p_{F}^{-\omega})=-\sigma_1\partial_{k_1} M(\mathbf p_{F}^\omega)\sigma_1
-=-\sigma_3\partial_{k_1} M(\mathbf p_{F}^{\omega})\sigma_3.
+\partial_{k_x} M(\mathbf p_{F}^\omega)=-(\partial_{k_x} M(\mathbf p_{F}^{-\omega}))^*=\partial_{k_x} M(\mathbf p_{F}^{-\omega})=-\sigma_1\partial_{k_x} M(\mathbf p_{F}^\omega)\sigma_1
+=-\sigma_3\partial_{k_x} M(\mathbf p_{F}^{\omega})\sigma_3.
 $$
 Therefore $(t_1,x_1,y_1,z_1)$ are independent of $\omega$, $t_1=x_1=z_1=0$ and $y_1\in\mathbb{R}$.\par
 \bigskip
 
 \point Finally, we consider $\partial_{k_y}M$:
-$$\partial_{k_2} M(\mathbf p_{F}^\omega)=:t_2^{(\omega)}\mathds1+x_2^{(\omega)}\sigma_1+y_2^{(\omega)}\sigma_2+z_2^{(\omega)}\sigma_3.$$
+$$\partial_{k_y} M(\mathbf p_{F}^\omega)=:t_2^{(\omega)}\mathds1+x_2^{(\omega)}\sigma_1+y_2^{(\omega)}\sigma_2+z_2^{(\omega)}\sigma_3.$$
 We have
 $$
-\partial_{k_2} M(\mathbf p_{F}^\omega)=-(\partial_{k_2} M(\mathbf p_{F}^{-\omega}))^*=-\partial_{k_2} M(\mathbf p_{F}^{-\omega})=\sigma_1\partial_{k_2} M(\mathbf p_{F}^\omega)\sigma_1=-\sigma_3\partial_{k_2} M(\mathbf p_{F}^\omega)\sigma_3.
+\partial_{k_y} M(\mathbf p_{F}^\omega)=-(\partial_{k_y} M(\mathbf p_{F}^{-\omega}))^*=-\partial_{k_y} M(\mathbf p_{F}^{-\omega})=\sigma_1\partial_{k_y} M(\mathbf p_{F}^\omega)\sigma_1=-\sigma_3\partial_{k_y} M(\mathbf p_{F}^\omega)\sigma_3.
 $$
 Therefore $t_2^{(\omega)}=y_2^{(\omega)}=z_2^{(\omega)}=0$, $x_2^{(\omega)}=-x_2^{(-\omega)}\in\mathbb{R}$.\qed\par
 
@@ -3808,8 +3808,8 @@ Given a $4\times4$ complex matrix $\mathbf M(\mathbf k)$ parametrized by $\mathb
 \begin{equation}\begin{array}c
 \mathbf M^{ff'}(\mathbf p_{F}^\omega)=\mu^{ff'}\sigma_1,\quad
 \partial_{k_0}\mathbf M^{ff'}(\mathbf p_{F}^\omega)=i\zeta^{ff'}\mathds1,\\[0.5cm]
-\partial_{k_1}\mathbf M^{ff'}(\mathbf p_{F}^\omega)=\nu^{ff'}\sigma_2,\quad
-\partial_{k_2}\mathbf M^{ff'}(\mathbf p_{F}^\omega)=\omega\varpi^{ff'}\sigma_1
+\partial_{k_x}\mathbf M^{ff'}(\mathbf p_{F}^\omega)=\nu^{ff'}\sigma_2,\quad
+\partial_{k_y}\mathbf M^{ff'}(\mathbf p_{F}^\omega)=\omega\varpi^{ff'}\sigma_1
 \end{array}\label{Msymhyprot}\end{equation}
 with $(\mu^{ff'}, \zeta^{ff'}, \nu^{ff'}, \varpi^{ff'})\in\mathbb R^{4}$ independent of $\omega$, 
 and $\forall\mathbf k\in\mathcal B_\infty$
@@ -3857,9 +3857,9 @@ Evaluating this formula at ${\bf k}=\mathbf p_F^\omega$, recalling that $\mathbf
  $$\partial_{k_i}\mathbf M^{\phi\phi}(\mathbf p_{F}^\omega)=\sum_{j=1}^2T_{i,j}\mathcal T_{\mathbf p_{F}^\omega}^\dagger\partial_{k_j}\mathbf M^{\phi\phi}(\mathbf p_{F}^\omega)\mathcal T_{\mathbf p_{F}^\omega}$$
 with
 $$T=\frac{1}{2}\left(\begin{array}{*{2}{c}}-1&-\sqrt3\\[0.2cm]\sqrt3&-1\end{array}\right).$$
-Furthermore, recalling that $\partial_{k_1}\mathbf M^{\phi\phi}=\nu^{\phi\phi}\sigma_2$ and $\partial_{k_2}\mathbf M^{\phi\phi}=\omega\varpi^{\phi\phi}\sigma_1$,
-$$\mathcal T_{\mathbf p_{F}^\omega}^\dagger\partial_{k_1}\mathbf M^{\phi\phi}\mathcal T_{\mathbf p_{F}^\omega}=\nu^{\phi\phi}\Big(-\frac12\sigma_2-\omega\frac{\sqrt3}{2}\sigma_1\Big), \quad 
-\mathcal T_{\mathbf p_{F}^\omega}^\dagger\partial_{k_2}\mathbf M^{\phi\phi}\mathcal T_{\mathbf p_{F}^\omega}=\omega\varpi^{\phi\phi}\Big(-\frac12\sigma_1+\omega\frac{\sqrt3}{2}\sigma_2\Big),$$
+Furthermore, recalling that $\partial_{k_x}\mathbf M^{\phi\phi}=\nu^{\phi\phi}\sigma_2$ and $\partial_{k_y}\mathbf M^{\phi\phi}=\omega\varpi^{\phi\phi}\sigma_1$,
+$$\mathcal T_{\mathbf p_{F}^\omega}^\dagger\partial_{k_x}\mathbf M^{\phi\phi}\mathcal T_{\mathbf p_{F}^\omega}=\nu^{\phi\phi}\Big(-\frac12\sigma_2-\omega\frac{\sqrt3}{2}\sigma_1\Big), \quad 
+\mathcal T_{\mathbf p_{F}^\omega}^\dagger\partial_{k_y}\mathbf M^{\phi\phi}\mathcal T_{\mathbf p_{F}^\omega}=\omega\varpi^{\phi\phi}\Big(-\frac12\sigma_1+\omega\frac{\sqrt3}{2}\sigma_2\Big),$$
 which implies
 $$\left(\begin{array}{c}\nu^{\phi\phi}\sigma_2\\[0.2cm] \omega \varpi^{\phi\phi}\sigma_1\end{array}\right)=\frac{1}{4}\left(\begin{array}{*{2}{c}} \nu^{\phi\phi}-3 \varpi^{\phi\phi}&\omega\sqrt3(\nu^{\phi\phi}+\varpi^{\phi\phi})\\[0.2cm]-\sqrt3(\nu^{\phi\phi}+\varpi^{\phi\phi})&\omega(\varpi^{\phi\phi}-3\nu^{\phi\phi})\end{array}\right)\left(\begin{array}{c}\sigma_2\\[0.2cm]\sigma_1\end{array}\right)$$
 so $\nu^{\phi\phi}=-\varpi^{\phi\phi}$.
@@ -3872,10 +3872,10 @@ $$
 Evaluating this formula and its derivative with respect to $k_0$ at ${\bf k}=\mathbf p_F^\omega$, we obtain 
 $\mu^{\phi\xi}=\zeta^{\phi\xi}=0$.
 Evaluating the derivative of this formula with respect to $k_i$ at ${\bf k}=\mathbf p_F^\omega$, we obtain
-$$\partial_{k_i}\mathbf M^{\phi\xi}(\mathbf p_{F}^\omega)=\sum_{j=1}^2T_{i,j}\mathcal T_{\mathbf p_{F}^\omega}^\dagger\partial_{k_j}\mathbf M^{\phi\xi}(\mathbf p_{F}^\omega).$$
-Furthermore, 
-$$\mathcal T_{\mathbf p_{F}^\omega}^\dagger\partial_{k_1}\mathbf M^{\phi\xi}=\nu^{\phi\xi}\Big(-\frac12\sigma_2+\omega\frac{\sqrt3}{2}\sigma_1\Big), \quad 
-\mathcal T_{\mathbf p_{F}^\omega}^\dagger\partial_{k_2}\mathbf M^{\phi\xi}=\omega\varpi^{\phi\xi}\Big(-\frac12\sigma_1-\omega\frac{\sqrt3}{2}\sigma_2\Big),$$
+$$\partial_{k_i}\mathbf M^{\phi\xi}(\mathbf p_{F}^\omega)=\sum_{j=1}^2T_{i,j}\mathcal T_{\mathbf p_{F}^\omega}^\dagger\partial_{k_j}\mathbf M^{\phi\xi}(\mathbf p_{F}^\omega)$$
+where we used the following notation $k_1\equiv k_x$ and $k_2\equiv k_y$. Furthermore, 
+$$\mathcal T_{\mathbf p_{F}^\omega}^\dagger\partial_{k_x}\mathbf M^{\phi\xi}=\nu^{\phi\xi}\Big(-\frac12\sigma_2+\omega\frac{\sqrt3}{2}\sigma_1\Big), \quad 
+\mathcal T_{\mathbf p_{F}^\omega}^\dagger\partial_{k_y}\mathbf M^{\phi\xi}=\omega\varpi^{\phi\xi}\Big(-\frac12\sigma_1-\omega\frac{\sqrt3}{2}\sigma_2\Big),$$
 which implies
 $$\left(\begin{array}{c}\nu^{\phi\xi}\sigma_2\\[0.2cm] \omega \varpi^{\phi\xi}\sigma_1\end{array}\right)=\frac{1}{4}\left(\begin{array}{*{2}{c}} \nu^{\phi\xi}+3 \varpi^{\phi\xi}&-\omega\sqrt3(\nu^{\phi\xi}-\varpi^{\phi\xi})\\[0.2cm]-\sqrt3(\nu^{\phi\xi}-\varpi^{\phi\xi})&\omega(\varpi^{\phi\xi}+3\nu^{\phi\xi})\end{array}\right)\left(\begin{array}{c}\sigma_2\\[0.2cm]\sigma_1\end{array}\right)$$
 so that $\nu^{\phi\xi}_h=\varpi^{\phi\xi}_h$. 
@@ -3891,7 +3891,7 @@ Therefore for $i\in\{1,2\}$,
 $$
 \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
+where we used the following notation $k_1\equiv k_x$ and $k_2\equiv k_y$, so that
 $\partial_{k_i}\mathbf M^{\xi\xi}(\mathbf p_{F}^\omega)=0$, that is $\nu^{\xi\xi}=\varpi^{\xi\xi}=0$.\qed\par
 
 \vfill