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Fix notations and typos, and shorten proof of lemma A1.1

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Ian Jauslin 2022-06-15 22:51:58 +02:00
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@ -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} \hat a_{k,\sigma}:=\frac1{\sqrt{|\Lambda|}}\sum_{x\in\Lambda}e^{ikx}a_{x,\sigma}
,\quad ,\quad
\hat b_{k,\sigma}:=\frac1{\sqrt{|\Lambda|}}\sum_{x\in\Lambda}e^{ikx}b_{x+\delta_1,\sigma} \hat b_{k,\sigma}:=\frac1{\sqrt{|\Lambda|}}\sum_{x\in\Lambda}e^{ikx}b_{x+\delta_1,\sigma}
.
\end{equation} \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: Note that, with this choice of normalization, $\hat a_{k,\sigma}$ and $\hat b_{k,\sigma}$ satisfy the canonical anticommutation relations:
\begin{equation} \begin{equation}
\{a_{k,\sigma},a_{k',\sigma'}^\dagger\} \{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} \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} \label{hamk}
\end{equation} \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} \begin{equation}
H_0( k):= H_0( k):=
\left(\begin{array}{*{2}{c}} \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} \begin{largearray}
P_{\hat g}(d\psi) := \left( P_{\hat g}(d\psi) := \left(
\prod_{\mathbf k\in\mathcal B_{\beta,L}^*} \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) \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) \right)
\cdot\\[0.5cm]\hfill\cdot \cdot\\[0.5cm]\hfill\cdot
@ -512,7 +512,7 @@ Thus, we define the Gaussian Grassmann integration measure $P_{\leqslant M}(d\ps
\end{equation} \end{equation}
where $f_{0,\Lambda}$ is the free energy in the $U=0$ case and where $f_{0,\Lambda}$ is the free energy in the $U=0$ case and
\begin{equation} \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} \psi^+_{\mathbf x,\alpha,\downarrow}\psi^{-}_{\mathbf x,\alpha,\downarrow}
\label{V_grassmann} \label{V_grassmann}
\end{equation} \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)}|)) \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} \label{fh}
\end{equation} \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 Since $|k_0|\geqslant\frac\pi\beta$, we only need to consider
\begin{equation} \begin{equation}
h\geqslant -N_\beta:=\log_2\frac\pi\beta h\geqslant -N_\beta:=\log_2\frac\pi\beta
@ -679,7 +679,6 @@ We will take the propagators to be
\end{equation} \end{equation}
\begin{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'} \int P^{[h]}(d\psi^{[h]})\ \psi_{b,\sigma}^{[h]-}(\Delta)\psi_{a,\sigma'}^{[h]+}(\Delta')=\delta_{\sigma,\sigma'}\delta_{\Delta,\Delta'}
.
\end{equation} \end{equation}
and all other propagators will be set to 0. and all other propagators will be set to 0.
We can now evaluate how well these propagators approximate the non-hierarchical ones. 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 In fact, by symmetry considerations, we find that $v_h$ must be of the form
\begin{equation} \begin{equation}
v_h(\psi)= 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} \label{vh_rcc}
\end{equation} \end{equation}
with with
@ -894,19 +893,19 @@ We expand the exponential and use\-~(\ref{vh_rcc}):
\begin{largearray} \begin{largearray}
\beta|\Lambda|c^{[h]} \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= =\\\hfill=
2^{d+1}\log 2^{d+1}\log
\int P(d\psi^{[h]}(\Delta)) \int P(d\psi^{[h]}(\Delta))
\sum_{n=0}^\infty \sum_{n=0}^\infty
\frac1{n!} \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} \end{largearray}
\label{betadef} \label{betadef}
\end{equation} \end{equation}
The computation is thus reduced to computing the map $\alpha^{(h)}\mapsto\alpha^{(h-1)}$ using\-~(\ref{betadef}). The computation is thus reduced to computing the map $\ell^{(h)}\mapsto\ell^{(h-1)}$ 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 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 $\sim2^{-h}$. 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 \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. 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}. 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. 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 ($O_6$). 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 $O_i$. 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 $\psi^{[h]}+2^{-\gamma}\psi^{[\leqslant h-1]}$. 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. 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})) The contribution of this graph is (keeping track of the $2^{d+1}$ factor in\-~(\ref{betadef}))
\begin{equation} \begin{equation}
2^{d+1-2l\gamma}\alpha_i^{(h)} 2^{d+1-2l\gamma}\ell_i^{(h)}
. .
\end{equation} \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} \begin{equation}
\alpha_i^{(h-1)}= \ell_i^{(h-1)}=
2^{d+1-2l\gamma}\alpha_i^{(h)} 2^{d+1-2l\gamma}\ell_i^{(h)}
+ +
\cdots \cdots
\end{equation} \end{equation}
@ -1004,7 +1003,7 @@ For a more general treatment of power counting in Fermionic models with point-si
\indent \indent
In the case of graphene, we have one relevant coupling: $O_0$, which is quadratic in the Grassmann fields. 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. 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. 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}. Since graphene only has one relevant coupling, and that one is quadratic, graphene is called {\it super-renormalizable}.
\bigskip \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. 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 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. 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} \begin{equation}
\alpha_0\in\{0,1\} \ell_0\in\{0,1\}
. .
\end{equation} \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 \bigskip
\begin{figure} \begin{figure}
\hfil\includegraphics[width=12cm]{graphene_vector_field.pdf} \hfil\includegraphics[width=12cm]{graphene_vector_field.pdf}
\caption{ \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.) (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} \label{fig:vector_field}
\end{figure} \end{figure}
@ -1509,69 +1508,30 @@ Let us first prove a technical lemma.
= =
e^{-t\lambda_j}a_j^\dagger\prod_{i\neq j} e^{-t\lambda_j}a_j^\dagger\prod_{i\neq j}
(1+(e^{-t\lambda_i}-1)a_i^\dagger a_i) (1+(e^{-t\lambda_i}-1)a_i^\dagger a_i)
\end{equation}
and since $(a_j^\dagger)^2=0$,
\begin{equation}
e^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}a_j^\dagger
=
e^{-t\lambda_j}a_j^\dagger\prod_{i}
(1+(e^{-t\lambda_i}-1)a_i^\dagger a_i)
=
e^{-t\lambda_j}a_j^\dagger
e^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}
. .
\label{fock2} \label{fock2}
\end{equation} \end{equation}
Similarly, Taking the $\dagger$ of\-~(\ref{fock2}), we find
\begin{equation} \begin{equation}
e^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}a_j a_je^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}
=
\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\lambda_j}
e^{-t\sum_{i=1}^N\lambda_ia_i^\dagger a_i}
a_j a_j
\prod_{i\neq j}
(1+(e^{-t\lambda_i}-1)a_i^\dagger a_i)
. .
\label{fock3} \label{fock3}
\end{equation} \end{equation}
Furthermore, taking the $\dagger$ of\-~(\ref{fock3}), we find Combining\-~(\ref{fock2}) and\-~(\ref{fock3}), we find\-~(\ref{fock4}).
\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
.
\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
\end{equation}
we have
\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
.
\label{fock3'}
\end{equation}
This implies the second of\-~(\ref{fock4}).
\qed \qed
\bigskip \bigskip

2
README
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@ -28,6 +28,8 @@ Some extra functionality is provided in custom style files, located in the
gnuplot gnuplot
meankondo v1.5 meankondo v1.5
meankondo is available from http://ian.jauslin.org/software/meankondo
* Files: * Files:

4
figs/hierarchical_graphene.fig/graphene_vector_field.gnuplot
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@ -1,5 +1,5 @@
set ylabel "$\\alpha_1$" norotate set ylabel "$\\ell_1$" norotate
set xlabel "$\\alpha_0$" set xlabel "$\\ell_0$"
# default output canvas size: 12.5cm x 8.75cm # default output canvas size: 12.5cm x 8.75cm
set term lua tikz size 8,6 standalone set term lua tikz size 8,6 standalone