The ALF package provides a general code for auxiliary-field Quantum Monte Carlo simulations and default analysis. In this tutorial we show how users from beginners to specialists can profit from ALF.
In the first part of the tutorial we make use of ALF's python interface
The first part of the tutorial is based on ALF's python interface -- \texttt{pyALF} -- which greatly simplifies using the code, making it ideal for
\part{Just running it}
\section*{Downloading the code and tutorial}
To download the code, type \texttt{ git clone git@git.physik.uni-wuerzburg.de:ALF/ALF\_code.git} in a shell. \\
To download the tutorial including solutions type: \\\texttt{git clone git@git.physik.uni-wuerzburg.de:ALF/ALF\_Tutorial.git} again in a shell.
\section*{Exercise 1) Testing against ED}
\exercise{Testing against ED}
Run the code 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:
\begin{equation}
\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}\rangle = -1.47261997 t
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@@ -197,9 +221,19 @@ Here is the result you should obtain when choosing $ \Delta \tau t = 0.05, 0.1,
\end{figure}
\newpage
\section*{Exercise 2) Dimensional crosover. }
\subsection*{a) Modifying the hopping}
\part{Getting your hands dirty - changing the code}
\setcounter{exercise}{1}
\section*{Downloading the code and tutorial}
To download the code, type \texttt{ git clone git@git.physik.uni-wuerzburg.de:ALF/ALF\_code.git} in a shell. \\
To download the tutorial including solutions type: \\\texttt{git clone git@git.physik.uni-wuerzburg.de:ALF/ALF\_Tutorial.git} again in a shell.
\exercise{Dimensional crossover}
\red{[To be updated.]}
\exerciseitem{Modifying the hopping}
Here we will modify the code so as to allow for different hopping matrix elements along the x and y directions of a square lattice. To do so, one merely has to do the following
\begin{itemize}
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@@ -265,8 +299,6 @@ LOBS_EN = 100
CPU_MAX = 0.1
/
&VAR_errors
n_skip = 2
N_rebin = 1
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@@ -287,7 +319,7 @@ For the Mz Hubbard-Stratonovitch transformation it is hence better to consider
\end{equation}
to compute the spin-spin correlations.
\subsection*{ (b) Adding a new observable}
\exerciseitem{Adding a new observable}
Here the aim is to include the new observable equal time observable $\langle\vec{S}_{i}\cdot\vec{S}_{j}\rangle$ in the \texttt{Hubbard\_Mz} code.
To achieve this, you will have to carry out the following steps.
\begin{itemize}
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@@ -304,13 +336,16 @@ In the program \texttt{Hamiltonian\_Examples.f90 } to be found in the directory
\caption{\label{Ladder.fig} Spin correlation functions along one leg for the Hubbard ladder. As $t_y$ grows the spin gap becomes large enough so as to detect the exponential decal of the spin correlation function on this small lattice size. The underlying physics of odd-even ladder systems is introduced in the article: Elbio Dagotto and T. M. Rice, Surprises on the way from one- to two-dimensional quantum magnets: The ladder materials, Science 271 (1996), no. 5249, 618?623. }
\end{figure}
\subsection*{ (c) The SU(2) HubbardStratonovitch transformation}
The SU(2) HubbardStratonovitch decomposition, conserves spin rotational symmetry. Run the ladder code with the SU(2) flag in the parameter file switched on (i.e. \texttt{Model = Hubbard\_SU2}) and compare results.
The SU(2) Hubbard-Stratonovich decomposition, conserves spin rotational symmetry. Run the ladder code with the SU(2) flag in the parameter file switched on (i.e. \texttt{Model = Hubbard\_SU2}) and compare results.
\newpage
\section*{ Exercise 3) Defining a new model: The one-dimensional t-V model. }
\exercise{Defining a new model: The one-dimensional t-V model.}
\red{[To be updated.]}
\exerciseitem{Define new model}
\subsection*{(a)}
In this section, one we will show what modifications have to be carried out for computing the physics of the one dimensional t-V model of spinless fermions.
\begin{equation}
H = -t \sum_{i}\left( c^{\dagger}_{i} c^{\phantom\dagger}_{i+a} + c^{\dagger}_{i+a} c^{\phantom\dagger}_{i}\right)
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@@ -377,7 +412,7 @@ with $J_{zz} = V $ and $J_{xx} = 2t$. Hence when $V/t = 2$ we reproduce the Hei
\end{figure}
\subsection*{(b)} How would you use the code to carry out simulations at $V/t < 0$?
\exerciseitem{Challenge} How would you use the code to carry out simulations at $V/t < 0$?
