Commit fda2b989 authored by Jefferson Stafusa E. Portela's avatar Jefferson Stafusa E. Portela
Browse files

Finished Ex. 2 of part 2; deleted Ex. 2's Run directory; updated part 2 plots; some minor.

parent 7491980e
......@@ -296,7 +296,7 @@ git clone git@git.physik.uni-wuerzburg.de:ALF/ALF_Tutorial.git
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 here shorten to ``\texttt{Vanilla}'', and then:
To do so we start from the module \texttt{Hamiltonian\_Hubbard\_Plain\_Vanilla\_mod.F90}, which we here shorten to ``\texttt{Vanilla}'', found in \texttt{\$ALF\_DIR/Prog/Hamiltonians/}, proceeding as follows:
\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}).
......@@ -321,9 +321,9 @@ Do I = 1,Latt%N
Enddo
\end{lstlisting}
\end{itemize}
Note: If you'd like to run the simulation using MPI, you should also add the broadcasting call for \texttt{Ham\_Ty} to \texttt{Ham\_Set}. It is a good idea as well to include the new variable to the simulation parameters written into the file \texttt{info}, also in \texttt{Ham\_Set}.
Note: If you'd like to run the simulation using MPI, you should also add the broadcasting call for \texttt{Ham\_Ty} to \texttt{Ham\_Set}. It is a good idea as well to get the new simulation parameter written into the file \texttt{info}, also a change in \texttt{Ham\_Set}.
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}, found in the \texttt{Prog/Hamiltonians} directory. The solution directory also includes the modified and original modules, as well as 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
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}.
......@@ -361,7 +361,7 @@ Note that the t-V model is already implemented in ALF in the module \texttt{Hami
\exerciseitem{Define new model}
In the directory \texttt{\$ALF\_DIR/Solutions/Exercise\_2} we have duplicated the ALF and commented the changes that have to be carried out to the file \texttt{Hamiltonian\_Hubbard\_Plain\_Vanilla\_mod.F90} found in \texttt{\$ALF\_DIR/Prog/Hamiltonians/}, which we here shorten to ``\texttt{Vanilla}''. The following are the essential steps to be carried out:
In the directory \texttt{\$ALF\_DIR/Solutions/Exercise\_2} we have duplicated the ALF and commented the changes that have to be carried out to the file \texttt{Hamiltonian\_Hubbard\_Plain\_Vanilla\_mod.F90}, which we here shorten to ``\texttt{Vanilla}'', found in \texttt{\$ALF\_DIR/Prog/Hamiltonians/}. The following are the essential steps to be carried out:
\begin{itemize}
\item Add the \texttt{VAR\_t\_V} name space in the file \texttt{parameters} and set the necessary variables -- or simply rename the \texttt{VAR\_Hubbard\_Plain\_Vanilla} name space to \texttt{VAR\_t\_V} and, within it, \texttt{Ham\_U} to \texttt{Ham\_Vint}. (Ignore the name space \texttt{VAR\_tV}, which is used by the general implementation mentioned above.)
\item Declare a new variable, \texttt{Ham\_Vint}, in \texttt{Vanilla}'s specification.
......@@ -373,33 +373,68 @@ e^{\Delta \tau \frac{V}{2} \left( \hat{c}^{\dagger}_{i} \hat{c}^{\phantom\dagge
\sum_{l= \pm1, \pm 2} \gamma_l e^{ \sqrt{\Delta \tau \frac{V}{2}} \eta_l \left( \hat{c}^{\dagger}_{i} \hat{c}^{\phantom\dagger}_{i+a} + \hat{c}^{\dagger}_{i+a} \hat{c}^{\phantom\dagger}_{i} \right) } = \sum_{l= \pm1, \pm 2} \gamma_l e^{ g \eta_l \left( \hat{c}^{\dagger}_{i}, \hat{c}^{\dagger}_{i+a} \right) O
\left(\hat{c}^{\phantom\dagger}_{i}, \hat{c}^{\phantom\dagger}_{i+a} \right)^{T} }.
\end{equation}
Here is how this translates in the code.
Here is how this translates in the code (the new integer variable, \texttt{i2}, should be declared):
\begin{lstlisting}[style=fortran]
Allocate(Op_V(Latt%N,N_FL))
Allocate(Op_V(Ndim,N_FL))
do nf = 1,N_FL
do i = 1, N_coord*Ndim
do i = 1, Ndim
call Op_make(Op_V(i,nf),2)
enddo
enddo
Do nc = 1, Latt%N ! Runs over bonds = # of lattice sites in one-dimension.
I1 = nc
I2 = Latt%nnlist(I1,1,0)
Op_V(nc,1)%P(1) = I1
Op_V(nc,1)%P(2) = I2
Op_V(nc,1)%O(1,2) = cmplx(1.d0 ,0.d0, kind(0.D0))
Op_V(nc,1)%O(2,1) = cmplx(1.d0 ,0.d0, kind(0.D0))
Op_V(nc,1)%g = SQRT(CMPLX( DTAU*Ham_Vint/2.d0, 0.D0, kind(0.D0)))
Op_V(nc,1)%alpha = cmplx(0d0,0.d0, kind(0.D0))
Op_V(nc,1)%type =2
Call Op_set( Op_V(nc,1) )
Do i = 1, Ndim ! Runs over bonds = # of lattice sites in one-dimension.
i2 = Latt%nnlist(i,1,0)
Op_V(i,nf)%P(1) = i
Op_V(i,nf)%P(2) = i2
Op_V(i,nf)%O(1,2) = cmplx(1.d0 ,0.d0, kind(0.d0))
Op_V(i,nf)%O(2,1) = cmplx(1.d0 ,0.d0, kind(0.d0))
Op_V(i,nf)%g = sqrt(cmplx(Dtau*Ham_Vint/2.d0, 0.d0, kind(0.d0)))
Op_V(i,nf)%alpha = cmplx(0d0 ,0.d0, kind(0.d0))
Op_V(i,nf)%type = 2
Call Op_set( Op_V(i,nf) )
enddo
\end{lstlisting}
\item Finally you will have to update the \texttt{Obser} and \texttt{ObserT} routines for the calculation of the equal and time displaced correlations. For the \texttt{t\_V} model you can essentially use the same observables as for the \texttt{Hubbard\_SU(2)} model.
\item Finally, you have to update the \texttt{Obser} and \texttt{ObserT} routines for the calculation of equal- and time-displaced correlations. For the \texttt{t\_V} model you can essentially use the same observables as for the \texttt{Hubbard\_SU(2)} model in 1D -- a step which requires a number of changes with respect to the \texttt{Vanilla} base, such as:
\begin{lstlisting}[style=fortran]
!!!!! Modifications for Exercise 2
!Zpot = Zpot*ham_U ! Vanilla
Zpot = Zpot*Ham_Vint ! t-V
!!!!!
\end{lstlisting}
and
\begin{lstlisting}[style=fortran]
!Zrho = Zrho + Grc(i,i,1) + Grc(i,i,2) ! Vanilla
Zrho = Zrho + Grc(i,i,1) ! t-V
\end{lstlisting}
with the observables being coded in the routine \texttt{Obser} as
\begin{lstlisting}[style=fortran]
Z = cmplx(dble(N_SUN), 0.d0, kind(0.D0))
Do I1 = 1,Ndim
I = I1
no_I = 1
Do J1 = 1,Ndim
J = J1
no_J = 1
imj = latt%imj(I,J)
Obs_eq(1)%Obs_Latt(imj,1,no_I,no_J) = Obs_eq(1)%Obs_Latt(imj,1,no_I,no_J) + &
& Z * GRC(I1,J1,1) * ZP*ZS ! Green
Obs_eq(2)%Obs_Latt(imj,1,no_I,no_J) = Obs_eq(2)%Obs_Latt(imj,1,no_I,no_J) + &
& Z * GRC(I1,J1,1) * GR(I1,J1,1) * ZP*ZS ! SpinZ
Obs_eq(3)%Obs_Latt(imj,1,no_I,no_J) = Obs_eq(3)%Obs_Latt(imj,1,no_I,no_J) + &
& ( GRC(I1,I1,1) * GRC(J1,J1,1) * Z + &
& GRC(I1,J1,1) * GR(I1,J1,1 ) ) * Z * ZP*ZS ! Den
enddo
Obs_eq(3)%Obs_Latt0(no_I) = Obs_eq(3)%Obs_Latt0(no_I) + Z * GRC(I1,I1,1) * ZP * ZS
enddo
\end{lstlisting}
among other changes - with similar ones in the \texttt{ObserT} routine.
All necessary changes are implemented and clearly indicated in the solution provided in \texttt{Solutions/Exercise\_2/Hamiltonian\_Hubbard\_Plain\_Vanilla\_mod-Exercise\_2.F90}.
\end{itemize}
In the directory \texttt{Solutions/Exercise\_2} 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/Hamiltonians} 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).
You can now run the code for various values of $V/t$. A Jordan-Wigner transformation will map the \texttt{t\_V} model onto the XXZ chain:
\begin{equation}
\hat{H} = J_{xx} \sum_{i} \hat{S}^{x}_i \hat{S}^{x}_{i+a} + \hat{S}^{y}_i \hat{S}^{y}_{i+a} + J_{zz} \sum_{i}\hat{S}^{z}_i \hat{S}^{z}_{i +a}
......@@ -407,9 +442,8 @@ You can now run the code for various values of $V/t$. A Jordan-Wigner transform
with $J_{zz} = V $ and $J_{xx} = 2t$. Hence when $V/t = 2$ we reproduce the Heisenberg model. For $V/t > 2$ the model is in the Ising regime with long-range charge density wave order and is an insulator. In the regime $ -2 < V/t < 2$ the model is metallic and corresponds to a Luttinger liquid. Finally, at $ V/t < - 2$ phase separation between hole rich and electron rich phases occur. Fig.~\ref{tV.fig} shows typical results.
\begin{figure}[htb]
\begin{center}
\includegraphics[scale=.8]{tV.pdf}
\caption{Density-Density correlation functions of the t-V model. In the Luttinger liquid phase, $-2 < V/t < 2$ it is known that the density -density correlations decay as
$ \langle n(r) n(0)\rangle \propto \cos(\pi r) r^{-\left(1+K_\rho \right) } $ with $\left(1+K_\rho \right)^{-1}= \frac{1}{2} + \frac{1}{\pi} \arcsin \left( \frac{V}{2 | t | }\right) $
\includegraphics[width=0.6\columnwidth]{tV.pdf}
\caption{Density-Density correlation functions of the t-V model. In the Luttinger liquid phase, $-2 < V/t < 2$, it is known that the density-density correlations decay as $ \langle n(r) n(0)\rangle \propto \cos(\pi r) r^{-\left(1+K_\rho \right) } $ with $\left(1+K_\rho \right)^{-1}= \frac{1}{2} + \frac{1}{\pi} \arcsin \left( \frac{V}{2 | t | }\right) $
(A. Luther and I. Peschel, Calculation of critical exponents in two dimensions from quantum field theory in one dimension, Phys. Rev. B 12 (1975), 3908.) The interested reader can try to reproduce this result.}
\label{tV.fig}
\end{center}
......
......@@ -3,20 +3,15 @@ set size 0.8,0.8
set title "{L=14 Hubbard Ladder, {/Symbol b}t=10, U/t=4 }"
set out 'ladder.eps'
set fit errorvariables
#tmp='Ener_fit.dat'
#set print tmp
set xlabel "r"
set ylabel "S(r,0)"
f(x) = a + b * x
#print "0.0 ", a, a_err, b, b_err, "\n\n"
plot "ladder.dat" i 0 u 1:2:3 w e lt 2 t "t_y=0" ,\
"ladder.dat" i 1 u 1:2:3 w e lt 3 t "t_y=1" ,\
"ladder.dat" i 2 u 1:2:3 w e lt 4 t "t_y=2" ,\
"ladder.dat" i 0 u 1:2 w l lt 2 t "" ,\
"ladder.dat" i 1 u 1:2 w l lt 3 t "" ,\
"ladder.dat" i 2 u 1:2 w l lt 4 t ""
plot "ladder.dat" i 0 u 1:2:3 w e lc rgb "black" lt 1 pt 7 lw 2 t "V/t=1" , \
'' i 0 u 1:2 w l lc rgb "black" lt 1 lw 2 t "" , \
'' i 1 u 1:2:3 w e lc rgb "red" lt 1 pt 7 lw 2 t "V/t=2" , \
'' i 1 u 1:2 w l lc rgb "red" lt 1 lw 2 t "" , \
'' i 2 u 1:2:3 w e lc rgb "royalblue" lt 1 pt 7 lw 2 t "V/t=2.5" ,\
'' i 2 u 1:2 w l lc rgb "royalblue" lt 1 lw 2 t ""
!epstopdf ladder.eps
!open ladder.pdf
No preview for this file type
This diff is collapsed.
......@@ -3,20 +3,15 @@ set size 0.8,0.8
set title "{L=14 Hubbard Ladder, {/Symbol b}t=10, U/t=4 }"
set out 'ladder.eps'
set fit errorvariables
#tmp='Ener_fit.dat'
#set print tmp
set xlabel "r"
set ylabel "S(r,0)"
f(x) = a + b * x
#print "0.0 ", a, a_err, b, b_err, "\n\n"
plot "ladder.dat" i 0 u 1:2:3 w e lt 2 t "t_y=0" ,\
"ladder.dat" i 1 u 1:2:3 w e lt 3 t "t_y=1" ,\
"ladder.dat" i 2 u 1:2:3 w e lt 4 t "t_y=2" ,\
"ladder.dat" i 0 u 1:2 w l lt 2 t "" ,\
"ladder.dat" i 1 u 1:2 w l lt 3 t "" ,\
"ladder.dat" i 2 u 1:2 w l lt 4 t ""
plot "ladder.dat" i 0 u 1:2:3 w e lc rgb "black" lt 1 pt 7 lw 2 t "V/t=1" , \
'' i 0 u 1:2 w l lc rgb "black" lt 1 lw 2 t "" , \
'' i 1 u 1:2:3 w e lc rgb "red" lt 1 pt 7 lw 2 t "V/t=2" , \
'' i 1 u 1:2 w l lc rgb "red" lt 1 lw 2 t "" , \
'' i 2 u 1:2:3 w e lc rgb "royalblue" lt 1 pt 7 lw 2 t "V/t=2.5" ,\
'' i 2 u 1:2 w l lc rgb "royalblue" lt 1 lw 2 t ""
!epstopdf ladder.eps
!open ladder.pdf
......@@ -683,7 +683,7 @@
!!!!! Modifications for Exercise 2
!Zkin = Zkin + GRC(I,Ix,1) + GRC(Ix,I,1) &
! & + GRC(I,Ix,2) + GRC(Ix,I,2)
!!!Zkin = Zkin + GRC(I,Ix,1) + GRC(Ix,I,1) ! 1st attempt, wrong?, two lines above: orig.
!!!Zkin = Zkin + GRC(I,Ix,1) + GRC(Ix,I,1)
Zkin = Zkin + sum(Op_T(1,1)%O(:, i)*Grc(:, i, 1))
!!!!!
Enddo
......@@ -703,7 +703,7 @@
!!!!! Modifications for Exercise 2
!ZPot = ZPot + Grc(i,i,1) * Grc(i,i, 2)
i1 = Latt%nnlist(i,1,0)
ZPot = ZPot + Grc(i,i,1) * Grc(i1,i1, 1) + Grc(i,i1,1)*Gr(i,i1,1)
ZPot = ZPot + Grc(i,i,1)*Grc(i1,i1, 1) + Grc(i,i1,1)*Gr(i,i1,1)
Enddo
!Zpot = Zpot*ham_U
Zpot = Zpot*Ham_Vint
......@@ -753,11 +753,10 @@
J = J1 !List(J1,1)
no_J = 1 !List(J1,2)
imj = latt%imj(I,J)
! Green
Obs_eq(1)%Obs_Latt(imj,1,no_I,no_J) = Obs_eq(1)%Obs_Latt(imj,1,no_I,no_J) + &
& Z * GRC(I1,J1,1) * ZP*ZS ! Green
Obs_eq(2)%Obs_Latt(imj,1,no_I,no_J) = Obs_eq(2)%Obs_Latt(imj,1,no_I,no_J) + &
& Z * GRC(I1,J1,1) * GR(I1,J1,1) * ZP*ZS! SpinZ
& Z * GRC(I1,J1,1) * GR(I1,J1,1) * ZP*ZS ! SpinZ
!Obs_eq(3)%Obs_Latt(imj,1,no_I,no_J) = Obs_eq(3)%Obs_Latt(imj,1,no_I,no_J) + &
! & Z * GRC(I1,J1,1) * GR(I1,J1,1) * ZP*ZS ! SpinXY
Obs_eq(3)%Obs_Latt(imj,1,no_I,no_J) = Obs_eq(3)%Obs_Latt(imj,1,no_I,no_J) + &
......
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