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}
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}
\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}).
...
...
@@ -356,6 +356,8 @@ The above form is readily included in the ALF since the interaction is written
\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}
Note that the t-V model is already implemented in ALF in the module \texttt{Hamiltonian\_tV\_mod.F90} found in \texttt{Prog/Hamiltonians/}. While it can be used for checking your own results\footnote{A short simulation of the t-V model can be conveniently run using its Jupyter notebook available in pyALF.}, you are not supposed to reproduce that implementation -- which is more general and makes use of predefined structures -- but instead to write a simpler one, based on the module \texttt{Hamiltonian\_Hubbard\_Plain\_Vanilla\_mod.F90} as detailed below.
\red{[Mention more general tV implementation in pred. strutct. (for checking correctness), as well as the tV Jupyter notebook.]}
