From fddcf91b6a21d7ecd47422e122b13cc2651fed52 Mon Sep 17 00:00:00 2001
From: Ian Jauslin <ian@jauslin.org>
Date: Wed, 15 Jun 2022 22:51:58 +0200
Subject: Fix notations and typos, and shorten proof of lemma A1.1

---
 Jauslin_2022.tex | 106 +++++++++++++++++--------------------------------------
 1 file changed, 33 insertions(+), 73 deletions(-)

(limited to 'Jauslin_2022.tex')

diff --git a/Jauslin_2022.tex b/Jauslin_2022.tex
index f200478..958cb08 100644
--- a/Jauslin_2022.tex
+++ b/Jauslin_2022.tex
@@ -255,8 +255,8 @@ We define the Fourier transform of the annihilation operators as
   \hat a_{k,\sigma}:=\frac1{\sqrt{|\Lambda|}}\sum_{x\in\Lambda}e^{ikx}a_{x,\sigma}
   ,\quad 
   \hat b_{k,\sigma}:=\frac1{\sqrt{|\Lambda|}}\sum_{x\in\Lambda}e^{ikx}b_{x+\delta_1,\sigma}
-  .
 \end{equation} 
+where $|\Lambda|=L^2$.
 Note that, with this choice of normalization, $\hat a_{k,\sigma}$ and $\hat b_{k,\sigma}$ satisfy the canonical anticommutation relations:
 \begin{equation}
   \{a_{k,\sigma},a_{k',\sigma'}^\dagger\}
@@ -276,7 +276,7 @@ We express $\mathcal H_0$ in terms of $\hat a$ and $\hat b$:
   \mathcal H_0=-\sum_{\sigma\in\{\uparrow,\downarrow\}}\sum_{ k\in\hat\Lambda}\hat A_{k,\sigma}^\dagger H_0(k)\hat A_{k,\sigma}
   \label{hamk}
 \end{equation}
-where $|\Lambda|=L^2$, $\hat A_{k,\sigma}$ is a column vector whose transpose is $\hat A_{k,\sigma}^T=(\hat a_{k,\sigma},\hat{b}_{k,\sigma})$, 
+$\hat A_{k,\sigma}$ is a column vector whose transpose is $\hat A_{k,\sigma}^T=(\hat a_{k,\sigma},\hat{b}_{k,\sigma})$, 
 \begin{equation}
   H_0( k):=
   \left(\begin{array}{*{2}{c}}
@@ -440,7 +440,7 @@ The Gaussian Grassmann measure is specified by a {\it propagator}, which is a $2
   \begin{largearray}
     P_{\hat g}(d\psi) := \left(
       \prod_{\mathbf k\in\mathcal B_{\beta,L}^*}
-      (\beta\det\hat g(\mathbf k))^4
+      (\beta^2\det\hat g(\mathbf k))^2
       \left(\prod_{\sigma\in\{\uparrow,\downarrow\}}\prod_{\alpha\in\{a,b\}}d\hat\psi_{\mathbf k,\alpha}^+d\hat\psi_{\mathbf k,\alpha}^-\right)
     \right)
     \cdot\\[0.5cm]\hfill\cdot
@@ -512,7 +512,7 @@ Thus, we define the Gaussian Grassmann integration measure $P_{\leqslant M}(d\ps
 \end{equation}
 where $f_{0,\Lambda}$ is the free energy in the $U=0$ case and
 \begin{equation}
-  \mathcal V(\psi)=U\sum_{\alpha\in\{a,b\}}\frac1{|\Lambda|}\int_{0}^\beta dt \sum_{x\in \Lambda}\psi^+_{\mathbf x,\alpha,\uparrow}\psi^{-}_{\mathbf x,\alpha,\uparrow}
+  \mathcal V(\psi)=U\sum_{\alpha\in\{a,b\}}\int_{0}^\beta dt \sum_{x\in \Lambda}\psi^+_{\mathbf x,\alpha,\uparrow}\psi^{-}_{\mathbf x,\alpha,\uparrow}
   \psi^+_{\mathbf x,\alpha,\downarrow}\psi^{-}_{\mathbf x,\alpha,\downarrow}
   \label{V_grassmann}
 \end{equation}
@@ -549,7 +549,7 @@ The idea is to approach the singularities $p_F^{(\omega)}$ slowly, by defining s
   \Phi_{h}(\mathbf k-\mathbf p_F^{(\omega)}):=(\chi_0(2^{-h}|\mathbf k-\mathbf p_F^{(\omega)}|)-\chi_0(2^{-h+1}|\mathbf k-\mathbf p_F^{(\omega)}|))
   \label{fh}
 \end{equation}
-which is a smooth function that is supported in $|\mathbf k-\mathbf p_F^{(\omega)}|\in[2^h\frac16,2^h\frac23]$, in other words, if localizes $\mathbf k$ to be at a distance from $\mathbf p_F^{(\omega)}$ that is of order $2^h$, see figure\-~\ref{fig:scale}.
+which is a smooth function that is supported in $|\mathbf k-\mathbf p_F^{(\omega)}|\in[2^h\frac16,2^h\frac23]$, in other words, it localizes $\mathbf k$ to be at a distance from $\mathbf p_F^{(\omega)}$ that is of order $2^h$, see figure\-~\ref{fig:scale}.
 Since $|k_0|\geqslant\frac\pi\beta$, we only need to consider
 \begin{equation}
   h\geqslant -N_\beta:=\log_2\frac\pi\beta
@@ -679,7 +679,6 @@ We will take the propagators to be
 \end{equation}
 \begin{equation}
   \int P^{[h]}(d\psi^{[h]})\ \psi_{b,\sigma}^{[h]-}(\Delta)\psi_{a,\sigma'}^{[h]+}(\Delta')=\delta_{\sigma,\sigma'}\delta_{\Delta,\Delta'}
-  .
 \end{equation}
 and all other propagators will be set to 0.
 We can now evaluate how well these propagators approximate the non-hierarchical ones.
@@ -754,7 +753,7 @@ Because there are only four Grassmann fields and their conjugates per cell, $v_h
 In fact, by symmetry considerations, we find that $v_h$ must be of the form
 \begin{equation}
   v_h(\psi)=
-  \sum_{i=0}^6\alpha_i^{(h)}O_i(\psi)
+  \sum_{i=0}^6\ell_i^{(h)}O_i(\psi)
   \label{vh_rcc}
 \end{equation}
 with
@@ -894,19 +893,19 @@ We expand the exponential and use\-~(\ref{vh_rcc}):
   \begin{largearray}
     \beta|\Lambda|c^{[h]}
     +
-    \sum_{i=0}^6\alpha_i^{(h-1)}O_i(\psi^{[\leqslant h-1]}(\bar\Delta))
+    \sum_{i=0}^6\ell_i^{(h-1)}O_i(\psi^{[\leqslant h-1]}(\bar\Delta))
     =\\\hfill=
     2^{d+1}\log
     \int P(d\psi^{[h]}(\Delta))
     \sum_{n=0}^\infty
     \frac1{n!}
-    \left(\sum_{i=0}^6\alpha_i^{(h)}O_i\left(\psi^{[h]}(\Delta)+2^{-\gamma}\psi^{[\leqslant h-1]}(\bar\Delta)\right)\right)^n
+    \left(\sum_{i=0}^6\ell_i^{(h)}O_i\left(\psi^{[h]}(\Delta)+2^{-\gamma}\psi^{[\leqslant h-1]}(\bar\Delta)\right)\right)^n
     .
   \end{largearray}
   \label{betadef}
 \end{equation}
-The computation is thus reduced to computing the map $\alpha^{(h)}\mapsto\alpha^{(h-1)}$ using\-~(\ref{betadef}).
-The coefficients $\alpha_i^{(h)}$ are called {\it running coupling constants}, and the map $\alpha^{(h)}\mapsto\alpha^{(h-1)}$ is called the {\it beta function} of the model.
+The computation is thus reduced to computing the map $\ell^{(h)}\mapsto\ell^{(h-1)}$ using\-~(\ref{betadef}).
+The coefficients $\ell_i^{(h)}$ are called {\it running coupling constants}, and the map $\ell^{(h)}\mapsto\ell^{(h-1)}$ is called the {\it beta function} of the model.
 The running coupling constants play a very important role, as they specify the effective potential on scale $h$, and thereby the physical properties of the system at distances $\sim2^{-h}$.
 \bigskip
 
@@ -914,7 +913,7 @@ The running coupling constants play a very important role, as they specify the e
 Having defined the hierarchical model as we have, the infinite sum in\-~(\ref{betadef}) is actually finite ($n\leqslant 4$), so to compute the beta function, it suffices to compute Gaussian Grassmann integrals of a finite number of Grassmann monomials.
 A convenient way to carry out this computation is to represent each term graphically, using {\it Feynman diagrams}.
 First, let us expand the power $n$ and graphically represent the terms that must be integrated.
-For each $n$, we have $n$ possible choices of $\alpha_iO_i$.
+For each $n$, we have $n$ possible choices of $\ell_iO_i$.
 Now, $O_i$ can be quadratic in $\psi$ ($O_0$), quartic ($O_1$, $O_2$, $O_3$, $O_4$), sextic ($O_5$) or octic ($O_6$).
 We will represent $O_i$ by a vertex with the label $i$, from which two, four, six or eight edges emanate, depending on the degree of $O_i$.
 Each edge corresponds to a factor $\psi^{[h]}+2^{-\gamma}\psi^{[\leqslant h-1]}$.
@@ -964,13 +963,13 @@ In other words, no integrating is taking place.
 Let us denote the number of external edges by $2l$, which can either be 2, 4, 6 or 8.
 The contribution of this graph is (keeping track of the $2^{d+1}$ factor in\-~(\ref{betadef}))
 \begin{equation}
-  2^{d+1-2l\gamma}\alpha_i^{(h)}
+  2^{d+1-2l\gamma}\ell_i^{(h)}
   .
 \end{equation}
-Furthermore, this graph will contribute to the running coupling constant $\alpha_i$, and so, on scale $h-1$, we will have
+Furthermore, this graph will contribute to the running coupling constant $\ell_i$, and so, on scale $h-1$, we will have
 \begin{equation}
-  \alpha_i^{(h-1)}=
-  2^{d+1-2l\gamma}\alpha_i^{(h)}
+  \ell_i^{(h-1)}=
+  2^{d+1-2l\gamma}\ell_i^{(h)}
   +
   \cdots
 \end{equation}
@@ -1004,7 +1003,7 @@ For a more general treatment of power counting in Fermionic models with point-si
 \indent
 In the case of graphene, we have one relevant coupling: $O_0$, which is quadratic in the Grassmann fields.
 This is the only relevant coupling, and all others stay small.
-However, since the relevant coupling is quadratic, it merely shifts the non-interacting system (whose Hamiltonian is quadratic in the Grassmann fields) to another system with a quadratic (that is non-interacting) Hamiltonian.
+However, since the relevant coupling is quadratic, it merely shifts the non-interacting system (whose Hamiltonian is quadratic in the Grassmann fields) to another system with a quadratic (that is, non-interacting) Hamiltonian.
 Thus the relevant coupling does {\it not} imply that the interactions are preponderant, but rather that the interaction terms shifts the system from one non-interacting system to another.
 Since graphene only has one relevant coupling, and that one is quadratic, graphene is called {\it super-renormalizable}.
 \bigskip
@@ -1013,20 +1012,20 @@ Since graphene only has one relevant coupling, and that one is quadratic, graphe
 As was mentioned above, the beta function can be computed {\it explicitly} for the hierarchical model, so the claims in the previous paragraph can be verified rather easily.
 The exact computation involves many terms, but it can be done easily using the {\tt meankondo} software package\-~\cite{mk}.
 The resulting beta function contains 888 terms, and will not be written out here.
-A careful analysis of the beta function shows that there is an equilibrium point at $\alpha_i=0$ for $i=1,2,3,4,5,6$ and
+A careful analysis of the beta function shows that there is an equilibrium point at $\ell_i=0$ for $i=1,2,3,4,5,6$ and
 \begin{equation}
-  \alpha_0\in\{0,1\}
+  \ell_0\in\{0,1\}
   .
 \end{equation}
-The point with $\alpha_0=0$ is unstable, whereas $\alpha_0=1$ is stable.
+The point with $\ell_0=0$ is unstable, whereas $\ell_0=1$ is stable.
 \bigskip
 
 \begin{figure}
 \hfil\includegraphics[width=12cm]{graphene_vector_field.pdf}
 \caption{
-  The projection of the directional vector field of the beta function for hierarchical graphene onto the $(\alpha_0,\alpha_1)$ plane.
+  The projection of the directional vector field of the beta function for hierarchical graphene onto the $(\ell_0,\ell_1)$ plane.
   (Each arrow shows the direction of the vector field, the color corresponds to the logarithm of the amplitude, with red being larger and blue smaller.)
-  The stable equilibrium point at $\alpha_0=1$ and $\alpha_i=0$ is clearly visible.
+  The stable equilibrium point at $\ell_0=1$ and $\ell_i=0$ is clearly visible.
 }
 \label{fig:vector_field}
 \end{figure}
@@ -1509,69 +1508,30 @@ Let us first prove a technical lemma.
     =
     e^{-t\lambda_j}a_j^\dagger\prod_{i\neq j}
     (1+(e^{-t\lambda_i}-1)a_i^\dagger a_i)
-    .
-    \label{fock2}
   \end{equation}
-  Similarly,
+  and since $(a_j^\dagger)^2=0$,
   \begin{equation}
-    e^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}a_j
-    =
-    \left(\prod_{i=1}^n(1+(e^{-t\lambda_i}-1)a_i^\dagger a_i)\right)a_j
-  \end{equation}
-  and so, using $a_i^2=0$, we find
-  \begin{equation}
-    e^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}a_j
+    e^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}a_j^\dagger
     =
-    a_j
-    \prod_{i\neq j}
+    e^{-t\lambda_j}a_j^\dagger\prod_{i}
     (1+(e^{-t\lambda_i}-1)a_i^\dagger a_i)
-    .
-    \label{fock3}
-  \end{equation}
-  Furthermore, taking the $\dagger$ of\-~(\ref{fock3}), we find
-  \begin{equation}
-    a_j^\dagger e^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}
-    =
-    \left(
-      \prod_{i\neq j}
-      (1+(e^{-t\lambda_i}-1)a_i^\dagger a_i)
-    \right)
-    a_j^\dagger
-  \end{equation}
-  and since
-  \begin{equation}
-    (1+(e^{-t\lambda_j}-1)a_j^\dagger a_j)a_j^\dagger
     =
     e^{-t\lambda_j}a_j^\dagger
-  \end{equation}
-  we have
-  \begin{equation}
-    a_j^\dagger e^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}
-    =
-    e^{t\lambda_j}e^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}a_j^\dagger 
+    e^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}
     .
-    \label{fock2'}
-  \end{equation}
-  This implies the first of\-~(\ref{fock4}).
-  Taking the $\dagger$ of\-~(\ref{fock2}) yields
-  \begin{equation}
-    a_je^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}
-    =e^{-t\lambda_j}\prod_{i\neq j}
-    (1+(e^{-t\lambda_i}-1)a_i^\dagger a_i)a_j
-  \end{equation}
-  and since
-  \begin{equation}
-    (1+(e^{-t\lambda_i}-1)a_j^\dagger a_k)a_j
-    =a_j
+    \label{fock2}
   \end{equation}
-  we have
+  Taking the $\dagger$ of\-~(\ref{fock2}), we find
   \begin{equation}
     a_je^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}
-    =e^{-t\lambda_j}e^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}a_j
+    =
+    e^{-t\lambda_j}
+    e^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}
+    a_j
     .
-    \label{fock3'}
+    \label{fock3}
   \end{equation}
-  This implies the second of\-~(\ref{fock4}).
+  Combining\-~(\ref{fock2}) and\-~(\ref{fock3}), we find\-~(\ref{fock4}).
 \qed
 \bigskip
 
-- 
cgit v1.2.3-70-g09d2

