Commit 426c7c3a authored by Jefferson Stafusa E. Portela's avatar Jefferson Stafusa E. Portela
Browse files

Improved wording; changed questions order in part 2 (according to ALF meeting...

Improved wording; changed questions order in part 2 (according to ALF meeting of 21.09.2020); and other minor.
parent 6ba27a9d
...@@ -293,9 +293,10 @@ git clone git@git.physik.uni-wuerzburg.de:ALF/ALF_Tutorial.git ...@@ -293,9 +293,10 @@ git clone git@git.physik.uni-wuerzburg.de:ALF/ALF_Tutorial.git
\exercise{Dimensional crossover} \exercise{Dimensional crossover}
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.
\exerciseitem{Modifying the hopping} \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 we start from the module \texttt{Hamiltonian\_Hubbard\_Plain\_Vanilla\_mod.F90} found in \texttt{<ALF\_DIR>/Prog/Hamiltonians/}, which we will simply call ``\texttt{Vanilla}'', and then: To do so we start from the module \texttt{Hamiltonian\_Hubbard\_Plain\_Vanilla\_mod.F90} found in \texttt{<ALF\_DIR>/Prog/Hamiltonians/}, which we will simply call ``\texttt{Vanilla}'', and then:
\begin{itemize} \begin{itemize}
\item Add \texttt{Ham\_Ty} to the \texttt{VAR\_Hubbard\_Plain\_Vanilla} name space in the parameter file \texttt{parameters}. \item Add \texttt{Ham\_Ty} to the \texttt{VAR\_Hubbard\_Plain\_Vanilla} name space in the parameter file \texttt{parameters}.
\item Declare a new variable, \texttt{Ham\_Ty}, in the module's specification (just search for the declaration of \texttt{Ham\_T} in \texttt{Vanilla}). \item Declare a new variable, \texttt{Ham\_Ty}, in the module's specification (just search for the declaration of \texttt{Ham\_T} in \texttt{Vanilla}).
...@@ -325,7 +326,7 @@ Note: If you'd like to run the simulation using MPI, you should also add the bro ...@@ -325,7 +326,7 @@ Note: If you'd like to run the simulation using MPI, you should also add the bro
In the directory \texttt{Solutions/Exercise\_1} we have duplicated ALF's code and commented the changes that have to be carried out to the file \texttt{Hamiltonian\_Hubbard\_Plain\_Vanilla\_mod.F90} in the \texttt{Prog} directory. The solution directory also includes reference data and the necessary \texttt{Start} directory (remember to copy its contents to every new \texttt{Run} directory, and to have a different \texttt{Run} directory for each simulation). In the directory \texttt{Solutions/Exercise\_1} we have duplicated ALF's code and commented the changes that have to be carried out to the file \texttt{Hamiltonian\_Hubbard\_Plain\_Vanilla\_mod.F90} in the \texttt{Prog} directory. The solution directory also includes reference data and the necessary \texttt{Start} directory (remember to copy its contents to every new \texttt{Run} directory, and to have a different \texttt{Run} directory for each simulation).
\noindent \noindent
As an application of this code, we can consider a 2-leg ladder system defined, e.g., with \texttt{L1=14}, \texttt{L2=2}, \texttt{Lattice\_type="Square"}, \texttt{Model="Hubbard\_Plain\_Vanilla"} and different values of \texttt{Ham\_Ty}. The results you should obtain are summarized in Fig.~\ref{fig:ladder}. As an application of this code, we can once again consider a ladder system (e.g, a 2-leg ladder with \texttt{L1=14} and \texttt{L2=2}), for different values of \texttt{Ham\_Ty}. The results you should obtain are summarized in Fig.~\ref{fig:ladder}.
\begin{figure}[h] \begin{figure}[h]
\begin{center} \begin{center}
...@@ -339,53 +340,26 @@ As an application of this code, we can consider a 2-leg ladder system defined, e ...@@ -339,53 +340,26 @@ As an application of this code, we can consider a 2-leg ladder system defined, e
\exerciseitem{The SU(2) Hubbard-Stratonovich transformation} \exerciseitem{The SU(2) Hubbard-Stratonovich transformation}
The SU(2) Hubbard-Stratonovich decomposition conserves spin rotational symmetry. Introduce into the moduel \texttt{Hamiltonian\_Hubbard\_mod.F90} and into the name space \texttt{VAR\_Hubbard} the same changes done to the \texttt{Vanilla} module, described in the previous item, and enter \texttt{Mz=.F.} in the \texttt{parameters} file (to choose the $SU(N)$ Hubbard interaction) and compare results. The SU(2) Hubbard-Stratonovich decomposition couples to the density and conserves spin rotational symmetry. Introduce into the module \texttt{Hamiltonian\_Hubbard\_mod.F90} and into the name space \texttt{VAR\_Hubbard} the same changes done to the \texttt{Vanilla} module, described in the previous item, and enter \texttt{Mz=.F.} in the \texttt{parameters} file (in order to choose the $SU(N)$ Hubbard interaction) and compare results to those of the $M_z$ decomposition above -- especially with regard to numerical convergence.
\red{[Should we state the expected result? Should we also provide a solution directory as done for the item (a)?]}
\exercise{Adding a new observable}
\red{[Stub]}
Here the task if to define a new observable, the kinetic energy correlation, given by
\begin{align}
&\left\langle \hat{O}_{i,\delta} \hat{O}_{j,\delta'} \right\rangle - \left\langle \hat{O}_{i,\delta} \right\rangle \left\langle \hat{O}_{j,\delta'} \right\rangle = S_O\big(i-j,\delta,\delta'\big)
\end{align}
where
\begin{align}
&\hat{O}_{i,x} = \sum_{\sigma}\left( \hat{c}^\dagger_{i,\sigma}\hat{c}^{\phantom\dagger}_{i+ax,\sigma} +H.c. \right).
\end{align}
[...]\\
In the 1-D Hubbard we have emergent $SO(4)$ symmetry:
\begin{align}
\left\langle \bar{S}(r)S(0) \right\rangle &\sim \frac{(-1)^r}{r}\ln^d(r)\\
\left\langle \hat{O}_{r,x} \hat{O}_{0,x} \right\rangle - \left\langle \hat{O}_{r,x} \right\rangle \left\langle \hat{O}_{0,x} \right\rangle &\sim \frac{(-1)^r}{r}\ln^\beta(r)
\end{align}
where $d=??$ and $\beta=??$ \cite{references}.
[It should be added to the Predefined Structures.]\\
\exercise{Defining a new model: The one-dimensional t-V model} \exercise{Defining a new model: The one-dimensional t-V model}
\exerciseitem{Define new model}
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. 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} \begin{equation}
\hat{H} = -t \sum_{i} \left( \hat{c}^{\dagger}_{i} \hat{c}^{\phantom\dagger}_{i+a} + \hat{c}^{\dagger}_{i+a} \hat{c}^{\phantom\dagger}_{i} \right) - \frac{V}{2} \sum_{i} \left( \hat{c}^{\dagger}_{i} \hat{c}^{\phantom\dagger}_{i+a} + \hat{c}^{\dagger}_{i+a} \hat{c}^{\phantom\dagger}_{i} \right)^2 \hat{H} = -t \sum_{i} \left( \hat{c}^{\dagger}_{i} \hat{c}^{\phantom\dagger}_{i+a} + \hat{c}^{\dagger}_{i+a} \hat{c}^{\phantom\dagger}_{i} \right) - \frac{V}{2} \sum_{i} \left( \hat{c}^{\dagger}_{i} \hat{c}^{\phantom\dagger}_{i+a} + \hat{c}^{\dagger}_{i+a} \hat{c}^{\phantom\dagger}_{i} \right)^2 .
\end{equation} \end{equation}
The above form is readily included in the ALF since the interaction is written in terms of a perfect square. Expanding the square yields (up to a constant) the desired model: The above form is readily included in the ALF since the interaction is written in terms of a perfect square. Expanding the square yields (up to a constant) the desired model:
\begin{equation} \begin{equation}
\hat{H} = -t \sum_{i} \left( \hat{c}^{\dagger}_{i} \hat{c}^{\phantom\dagger}_{i+a} + \hat{c}^{\dagger}_{i+a} \hat{c}^{\phantom\dagger}_{i} \right) + V \sum_{i} \left( \hat{n}_{i} - 1/2 \right) \left( \hat{n}_{i+a} - 1/2 \right) \hat{H} = -t \sum_{i} \left( \hat{c}^{\dagger}_{i} \hat{c}^{\phantom\dagger}_{i+a} + \hat{c}^{\dagger}_{i+a} \hat{c}^{\phantom\dagger}_{i} \right) + V \sum_{i} \left( \hat{n}_{i} - 1/2 \right) \left( \hat{n}_{i+a} - 1/2 \right).
\end{equation} \end{equation}
\red{[Mention tV Jupyter notebook.]} \red{[Mention more general tV implementation in pred. strutct. (for checking correctness), as well as the tV Jupyter notebook.]}
\exerciseitem{Define new model}
\red{[To be updated:]}\\ \red{[To be updated:]}\\
...@@ -444,14 +418,50 @@ with $J_{zz} = V $ and $J_{xx} = 2t$. Hence when $V/t = 2$ we reproduce the Hei ...@@ -444,14 +418,50 @@ with $J_{zz} = V $ and $J_{xx} = 2t$. Hence when $V/t = 2$ we reproduce the Hei
\end{center} \end{center}
\end{figure} \end{figure}
%\exerciseitem{Challenge} 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 $?
\exercise{Adding a new observable}
\red{[Stub]}
Here the task if to define a new observable, the kinetic energy correlation, given by
\begin{align}
&\left\langle \hat{O}_{i,\delta} \hat{O}_{j,\delta'} \right\rangle - \left\langle \hat{O}_{i,\delta} \right\rangle \left\langle \hat{O}_{j,\delta'} \right\rangle = S_O\big(i-j,\delta,\delta'\big)
\end{align}
where
\begin{align}
&\hat{O}_{i,x} = \sum_{\sigma}\left( \hat{c}^\dagger_{i,\sigma}\hat{c}^{\phantom\dagger}_{i+ax,\sigma} +H.c. \right).
\end{align}
[...]\\
In the 1-D Hubbard we have emergent $SO(4)$ symmetry:
\begin{align}
\left\langle \bar{S}(r)S(0) \right\rangle &\sim \frac{(-1)^r}{r}\ln^d(r)\\
\left\langle \hat{O}_{r,x} \hat{O}_{0,x} \right\rangle - \left\langle \hat{O}_{r,x} \right\rangle \left\langle \hat{O}_{0,x} \right\rangle &\sim \frac{(-1)^r}{r}\ln^\beta(r)
\end{align}
where $d=??$ and $\beta=??$ \cite{references}.
[It should be added to the Predefined Structures.]\\
\end{document} \end{document}
Exercise item excluded from exercise "Dimensional crossover": Exercise item excluded from exercise "Dimensional crossover":
\hrule \hrule
......
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