Fix notations and typos, and shorten proof of lemma A1.1
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Jauslin_2022.tex
112
Jauslin_2022.tex
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@ -255,8 +255,8 @@ We define the Fourier transform of the annihilation operators as
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\hat a_{k,\sigma}:=\frac1{\sqrt{|\Lambda|}}\sum_{x\in\Lambda}e^{ikx}a_{x,\sigma}
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,\quad
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\hat b_{k,\sigma}:=\frac1{\sqrt{|\Lambda|}}\sum_{x\in\Lambda}e^{ikx}b_{x+\delta_1,\sigma}
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.
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\end{equation}
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where $|\Lambda|=L^2$.
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Note that, with this choice of normalization, $\hat a_{k,\sigma}$ and $\hat b_{k,\sigma}$ satisfy the canonical anticommutation relations:
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\begin{equation}
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\{a_{k,\sigma},a_{k',\sigma'}^\dagger\}
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@ -276,7 +276,7 @@ We express $\mathcal H_0$ in terms of $\hat a$ and $\hat b$:
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\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}
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\label{hamk}
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\end{equation}
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where $|\Lambda|=L^2$, $\hat A_{k,\sigma}$ is a column vector whose transpose is $\hat A_{k,\sigma}^T=(\hat a_{k,\sigma},\hat{b}_{k,\sigma})$,
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$\hat A_{k,\sigma}$ is a column vector whose transpose is $\hat A_{k,\sigma}^T=(\hat a_{k,\sigma},\hat{b}_{k,\sigma})$,
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\begin{equation}
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H_0( k):=
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\left(\begin{array}{*{2}{c}}
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@ -440,7 +440,7 @@ The Gaussian Grassmann measure is specified by a {\it propagator}, which is a $2
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\begin{largearray}
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P_{\hat g}(d\psi) := \left(
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\prod_{\mathbf k\in\mathcal B_{\beta,L}^*}
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(\beta\det\hat g(\mathbf k))^4
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(\beta^2\det\hat g(\mathbf k))^2
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\left(\prod_{\sigma\in\{\uparrow,\downarrow\}}\prod_{\alpha\in\{a,b\}}d\hat\psi_{\mathbf k,\alpha}^+d\hat\psi_{\mathbf k,\alpha}^-\right)
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\right)
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\cdot\\[0.5cm]\hfill\cdot
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@ -512,7 +512,7 @@ Thus, we define the Gaussian Grassmann integration measure $P_{\leqslant M}(d\ps
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\end{equation}
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where $f_{0,\Lambda}$ is the free energy in the $U=0$ case and
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\begin{equation}
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\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}
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\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}
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\psi^+_{\mathbf x,\alpha,\downarrow}\psi^{-}_{\mathbf x,\alpha,\downarrow}
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\label{V_grassmann}
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\end{equation}
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@ -549,7 +549,7 @@ The idea is to approach the singularities $p_F^{(\omega)}$ slowly, by defining s
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\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)}|))
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\label{fh}
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\end{equation}
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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}.
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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}.
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Since $|k_0|\geqslant\frac\pi\beta$, we only need to consider
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\begin{equation}
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h\geqslant -N_\beta:=\log_2\frac\pi\beta
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@ -679,7 +679,6 @@ We will take the propagators to be
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\end{equation}
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\begin{equation}
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\int P^{[h]}(d\psi^{[h]})\ \psi_{b,\sigma}^{[h]-}(\Delta)\psi_{a,\sigma'}^{[h]+}(\Delta')=\delta_{\sigma,\sigma'}\delta_{\Delta,\Delta'}
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.
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\end{equation}
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and all other propagators will be set to 0.
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We can now evaluate how well these propagators approximate the non-hierarchical ones.
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@ -754,7 +753,7 @@ Because there are only four Grassmann fields and their conjugates per cell, $v_h
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In fact, by symmetry considerations, we find that $v_h$ must be of the form
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\begin{equation}
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v_h(\psi)=
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\sum_{i=0}^6\alpha_i^{(h)}O_i(\psi)
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\sum_{i=0}^6\ell_i^{(h)}O_i(\psi)
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\label{vh_rcc}
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\end{equation}
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with
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@ -894,19 +893,19 @@ We expand the exponential and use\-~(\ref{vh_rcc}):
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\begin{largearray}
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\beta|\Lambda|c^{[h]}
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+
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\sum_{i=0}^6\alpha_i^{(h-1)}O_i(\psi^{[\leqslant h-1]}(\bar\Delta))
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\sum_{i=0}^6\ell_i^{(h-1)}O_i(\psi^{[\leqslant h-1]}(\bar\Delta))
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=\\\hfill=
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2^{d+1}\log
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\int P(d\psi^{[h]}(\Delta))
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\sum_{n=0}^\infty
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\frac1{n!}
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\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
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\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
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.
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\end{largearray}
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\label{betadef}
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\end{equation}
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The computation is thus reduced to computing the map $\alpha^{(h)}\mapsto\alpha^{(h-1)}$ using\-~(\ref{betadef}).
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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.
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The computation is thus reduced to computing the map $\ell^{(h)}\mapsto\ell^{(h-1)}$ using\-~(\ref{betadef}).
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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.
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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 $\sim2^{-h}$.
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\bigskip
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@ -914,7 +913,7 @@ The running coupling constants play a very important role, as they specify the e
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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.
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A convenient way to carry out this computation is to represent each term graphically, using {\it Feynman diagrams}.
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First, let us expand the power $n$ and graphically represent the terms that must be integrated.
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For each $n$, we have $n$ possible choices of $\alpha_iO_i$.
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For each $n$, we have $n$ possible choices of $\ell_iO_i$.
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Now, $O_i$ can be quadratic in $\psi$ ($O_0$), quartic ($O_1$, $O_2$, $O_3$, $O_4$), sextic ($O_5$) or octic ($O_6$).
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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 $O_i$.
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Each edge corresponds to a factor $\psi^{[h]}+2^{-\gamma}\psi^{[\leqslant h-1]}$.
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@ -964,13 +963,13 @@ In other words, no integrating is taking place.
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Let us denote the number of external edges by $2l$, which can either be 2, 4, 6 or 8.
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The contribution of this graph is (keeping track of the $2^{d+1}$ factor in\-~(\ref{betadef}))
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\begin{equation}
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2^{d+1-2l\gamma}\alpha_i^{(h)}
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2^{d+1-2l\gamma}\ell_i^{(h)}
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.
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\end{equation}
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Furthermore, this graph will contribute to the running coupling constant $\alpha_i$, and so, on scale $h-1$, we will have
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Furthermore, this graph will contribute to the running coupling constant $\ell_i$, and so, on scale $h-1$, we will have
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\begin{equation}
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\alpha_i^{(h-1)}=
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2^{d+1-2l\gamma}\alpha_i^{(h)}
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\ell_i^{(h-1)}=
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2^{d+1-2l\gamma}\ell_i^{(h)}
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+
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\cdots
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\end{equation}
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@ -1004,7 +1003,7 @@ For a more general treatment of power counting in Fermionic models with point-si
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\indent
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In the case of graphene, we have one relevant coupling: $O_0$, which is quadratic in the Grassmann fields.
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This is the only relevant coupling, and all others stay small.
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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.
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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.
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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.
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Since graphene only has one relevant coupling, and that one is quadratic, graphene is called {\it super-renormalizable}.
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\bigskip
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@ -1013,20 +1012,20 @@ Since graphene only has one relevant coupling, and that one is quadratic, graphe
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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.
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The exact computation involves many terms, but it can be done easily using the {\tt meankondo} software package\-~\cite{mk}.
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The resulting beta function contains 888 terms, and will not be written out here.
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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
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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
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\begin{equation}
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\alpha_0\in\{0,1\}
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\ell_0\in\{0,1\}
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.
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\end{equation}
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The point with $\alpha_0=0$ is unstable, whereas $\alpha_0=1$ is stable.
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The point with $\ell_0=0$ is unstable, whereas $\ell_0=1$ is stable.
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\bigskip
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\begin{figure}
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\hfil\includegraphics[width=12cm]{graphene_vector_field.pdf}
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\caption{
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The projection of the directional vector field of the beta function for hierarchical graphene onto the $(\alpha_0,\alpha_1)$ plane.
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The projection of the directional vector field of the beta function for hierarchical graphene onto the $(\ell_0,\ell_1)$ plane.
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(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.)
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The stable equilibrium point at $\alpha_0=1$ and $\alpha_i=0$ is clearly visible.
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The stable equilibrium point at $\ell_0=1$ and $\ell_i=0$ is clearly visible.
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}
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\label{fig:vector_field}
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\end{figure}
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@ -1509,69 +1508,30 @@ Let us first prove a technical lemma.
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=
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e^{-t\lambda_j}a_j^\dagger\prod_{i\neq j}
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(1+(e^{-t\lambda_i}-1)a_i^\dagger a_i)
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\end{equation}
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and since $(a_j^\dagger)^2=0$,
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\begin{equation}
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e^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}a_j^\dagger
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=
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e^{-t\lambda_j}a_j^\dagger\prod_{i}
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(1+(e^{-t\lambda_i}-1)a_i^\dagger a_i)
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=
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e^{-t\lambda_j}a_j^\dagger
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e^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}
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.
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\label{fock2}
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\end{equation}
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Similarly,
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Taking the $\dagger$ of\-~(\ref{fock2}), we find
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\begin{equation}
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e^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}a_j
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=
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\left(\prod_{i=1}^n(1+(e^{-t\lambda_i}-1)a_i^\dagger a_i)\right)a_j
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\end{equation}
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and so, using $a_i^2=0$, we find
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\begin{equation}
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e^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}a_j
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a_je^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}
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=
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e^{-t\lambda_j}
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e^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}
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a_j
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\prod_{i\neq j}
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(1+(e^{-t\lambda_i}-1)a_i^\dagger a_i)
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.
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\label{fock3}
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\end{equation}
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Furthermore, taking the $\dagger$ of\-~(\ref{fock3}), we find
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\begin{equation}
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a_j^\dagger e^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}
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=
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\left(
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\prod_{i\neq j}
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(1+(e^{-t\lambda_i}-1)a_i^\dagger a_i)
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\right)
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a_j^\dagger
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\end{equation}
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and since
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\begin{equation}
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(1+(e^{-t\lambda_j}-1)a_j^\dagger a_j)a_j^\dagger
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=
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e^{-t\lambda_j}a_j^\dagger
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\end{equation}
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we have
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\begin{equation}
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a_j^\dagger e^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}
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=
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e^{t\lambda_j}e^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}a_j^\dagger
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.
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\label{fock2'}
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\end{equation}
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This implies the first of\-~(\ref{fock4}).
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Taking the $\dagger$ of\-~(\ref{fock2}) yields
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\begin{equation}
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a_je^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}
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=e^{-t\lambda_j}\prod_{i\neq j}
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(1+(e^{-t\lambda_i}-1)a_i^\dagger a_i)a_j
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\end{equation}
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and since
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\begin{equation}
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(1+(e^{-t\lambda_i}-1)a_j^\dagger a_k)a_j
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=a_j
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\end{equation}
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we have
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\begin{equation}
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a_je^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}
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=e^{-t\lambda_j}e^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}a_j
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.
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\label{fock3'}
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\end{equation}
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This implies the second of\-~(\ref{fock4}).
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Combining\-~(\ref{fock2}) and\-~(\ref{fock3}), we find\-~(\ref{fock4}).
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\qed
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\bigskip
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2
README
2
README
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@ -28,6 +28,8 @@ Some extra functionality is provided in custom style files, located in the
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gnuplot
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meankondo v1.5
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meankondo is available from http://ian.jauslin.org/software/meankondo
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* Files:
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@ -1,5 +1,5 @@
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set ylabel "$\\alpha_1$" norotate
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set xlabel "$\\alpha_0$"
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set ylabel "$\\ell_1$" norotate
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set xlabel "$\\ell_0$"
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# default output canvas size: 12.5cm x 8.75cm
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set term lua tikz size 8,6 standalone
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