Commit a62653a4 authored by Jefferson Stafusa E. Portela's avatar Jefferson Stafusa E. Portela
Browse files

Notation correction in notebook testing_against_ED.

parent 59b99bd3
......@@ -12,7 +12,7 @@ run ALF with the Mz choice of Hubbard Stratonovitch transformation on a
four site ring, at \(U/t=4\) and inverse temperature \(\beta t = 2\).
For this set of parameters, the exact internal energy reads:\\
\[
\left\langle -t \sum_{\langle i,j\rangle, \sigma} c_{i,\sigma}^{\dagger} c_{j,\sigma}^{\phantom\dagger} + U \sum_{i=1}^{N} n_{i,\uparrow}n_{j,\downarrow} \right\rangle = -1.47261997 t
\left\langle -t \sum_{\langle i,j\rangle, \sigma} \hat{c}_{i,\sigma}^{\dagger} \hat{c}_{j,\sigma}^{\phantom\dagger} + U \sum_{i=1}^{N} \hat{n}_{i,\uparrow}\hat{n}_{j,\downarrow} \right\rangle = -1.47261997 t
\]
To reproduce this result we will have to carry out a systematic
......@@ -20,14 +20,14 @@ To reproduce this result we will have to carry out a systematic
\(\Delta \tau t L_\text{Trotter} = 2\) constant.\\
Recall that the formulation of the auxiliary field QMC approach is based
on the Trotter decomposition \[
e^{-\Delta \tau \left( A + B \right ) } = e^{-\Delta \tau A } e^{-\Delta \tau B } + {\cal O} \left( \Delta \tau^2 \right)
e^{ -\Delta \tau \left( \hat{A} + \hat{B} \right) } = e^{ -\Delta \tau \hat{A}/2} e^{ -\Delta \tau \hat{B} } e^{ -\Delta \tau \hat{A}/2} + \mathcal{O} \left (\Delta \tau^3\right)
\] The overall error produced by this approximation is of the order
\(\Delta \tau^2\).
Bellow we go through the steps for performing this extrapolation:
setting the simulation parameters, running it and analyzing the data.
\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}
\begin{center}\rule{0.5\linewidth}{\linethickness}\end{center}
\textbf{1.} Import \texttt{Simulation} class from the \texttt{py\_alf}
python module, which provides the interface with ALF, as well as
......@@ -36,10 +36,10 @@ mathematics and plotting packages:
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{1}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{k+kn}{from} \PY{n+nn}{py\PYZus{}alf} \PY{k+kn}{import} \PY{n}{Simulation} \PY{c+c1}{\PYZsh{} Interface with ALF}
\PY{k+kn}{from} \PY{n+nn}{py\PYZus{}alf} \PY{k}{import} \PY{n}{Simulation} \PY{c+c1}{\PYZsh{} Interface with ALF}
\PY{c+c1}{\PYZsh{} }
\PY{k+kn}{import} \PY{n+nn}{numpy} \PY{k}{as} \PY{n+nn}{np} \PY{c+c1}{\PYZsh{} Numerical library}
\PY{k+kn}{from} \PY{n+nn}{scipy}\PY{n+nn}{.}\PY{n+nn}{optimize} \PY{k+kn}{import} \PY{n}{curve\PYZus{}fit} \PY{c+c1}{\PYZsh{} Numerical library}
\PY{k+kn}{from} \PY{n+nn}{scipy}\PY{n+nn}{.}\PY{n+nn}{optimize} \PY{k}{import} \PY{n}{curve\PYZus{}fit} \PY{c+c1}{\PYZsh{} Numerical library}
\PY{k+kn}{import} \PY{n+nn}{matplotlib}\PY{n+nn}{.}\PY{n+nn}{pyplot} \PY{k}{as} \PY{n+nn}{plt} \PY{c+c1}{\PYZsh{} Plotting library}
\end{Verbatim}
\end{tcolorbox}
......@@ -228,7 +228,7 @@ For Dtau values [0.05 0.1 0.15] the measured energies are:
\end{center}
{ \hspace*{\fill} \\}
\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}
\begin{center}\rule{0.5\linewidth}{\linethickness}\end{center}
\hypertarget{exercises}{%
\subsection{Exercises}\label{exercises}}
......
......@@ -380,7 +380,7 @@ run ALF with the Mz choice of Hubbard Stratonovitch transformation on a
four site ring, at \(U/t=4\) and inverse temperature \(\beta t = 2\).
For this set of parameters, the exact internal energy reads:\\
\[
\left\langle -t \sum_{\langle i,j\rangle, \sigma} c_{i,\sigma}^{\dagger} c_{j,\sigma}^{\phantom\dagger} + U \sum_{i=1}^{N} n_{i,\uparrow}n_{j,\downarrow} \right\rangle = -1.47261997 t
\left\langle -t \sum_{\langle i,j\rangle, \sigma} \hat{c}_{i,\sigma}^{\dagger} \hat{c}_{j,\sigma}^{\phantom\dagger} + U \sum_{i=1}^{N} \hat{n}_{i,\uparrow}\hat{n}_{j,\downarrow} \right\rangle = -1.47261997 t
\]
To reproduce this result we will have to carry out a systematic
......@@ -388,14 +388,14 @@ To reproduce this result we will have to carry out a systematic
\(\Delta \tau t L_\text{Trotter} = 2\) constant.\\
Recall that the formulation of the auxiliary field QMC approach is based
on the Trotter decomposition \[
e^{-\Delta \tau \left( A + B \right ) } = e^{-\Delta \tau A } e^{-\Delta \tau B } + {\cal O} \left( \Delta \tau^2 \right)
e^{ -\Delta \tau \left( \hat{A} + \hat{B} \right) } = e^{ -\Delta \tau \hat{A}/2} e^{ -\Delta \tau \hat{B} } e^{ -\Delta \tau \hat{A}/2} + \mathcal{O} \left (\Delta \tau^3\right)
\] The overall error produced by this approximation is of the order
\(\Delta \tau^2\).
Bellow we go through the steps for performing this extrapolation:
setting the simulation parameters, running it and analyzing the data.
\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}
\begin{center}\rule{0.5\linewidth}{\linethickness}\end{center}
\textbf{1.} Import \texttt{Simulation} class from the \texttt{py\_alf}
python module, which provides the interface with ALF, as well as
......@@ -404,10 +404,10 @@ mathematics and plotting packages:
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{1}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{k+kn}{from} \PY{n+nn}{py\PYZus{}alf} \PY{k+kn}{import} \PY{n}{Simulation} \PY{c+c1}{\PYZsh{} Interface with ALF}
\PY{k+kn}{from} \PY{n+nn}{py\PYZus{}alf} \PY{k}{import} \PY{n}{Simulation} \PY{c+c1}{\PYZsh{} Interface with ALF}
\PY{c+c1}{\PYZsh{} }
\PY{k+kn}{import} \PY{n+nn}{numpy} \PY{k}{as} \PY{n+nn}{np} \PY{c+c1}{\PYZsh{} Numerical library}
\PY{k+kn}{from} \PY{n+nn}{scipy}\PY{n+nn}{.}\PY{n+nn}{optimize} \PY{k+kn}{import} \PY{n}{curve\PYZus{}fit} \PY{c+c1}{\PYZsh{} Numerical library}
\PY{k+kn}{from} \PY{n+nn}{scipy}\PY{n+nn}{.}\PY{n+nn}{optimize} \PY{k}{import} \PY{n}{curve\PYZus{}fit} \PY{c+c1}{\PYZsh{} Numerical library}
\PY{k+kn}{import} \PY{n+nn}{matplotlib}\PY{n+nn}{.}\PY{n+nn}{pyplot} \PY{k}{as} \PY{n+nn}{plt} \PY{c+c1}{\PYZsh{} Plotting library}
\end{Verbatim}
\end{tcolorbox}
......@@ -596,7 +596,7 @@ For Dtau values [0.05 0.1 0.15] the measured energies are:
\end{center}
{ \hspace*{\fill} \\}
\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center}
\begin{center}\rule{0.5\linewidth}{\linethickness}\end{center}
\hypertarget{exercises}{%
\subsection{Exercises}\label{exercises}}
......
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