### Add to new version of Exercises 2 and 3 of part 2.

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 ... ... @@ -18,8 +18,9 @@ \usepackage{soul} \usepackage{xspace} \usepackage{bm} %\usepackage{subfigure} % outdated package \usepackage{subcaption} %%\usepackage{subfigure} % outdated package %\usepackage{subcaption} %\usepackage{caption} \usepackage[breakable]{tcolorbox} \usepackage{dsfont} \usepackage{footnote} ... ... @@ -363,17 +364,8 @@ N_Cov = 0 ! slash terminates namelist statement - DO NOT REMOVE \end{lstlisting} \noindent When running the code for the above ladder system and analyzing the spin correlation functions, \texttt{SpinZ\_eqJR} and \texttt{SpinXY\_eqJR} one will notice that it is hard restore the SU(2) spin symmetry and that very long runs are required to obtain the desired equality \begin{equation} \langle S^{z}_{i} S^{z}_{j} \rangle = \langle S^{y}_{i} S^{y}_{j} \rangle = \langle S^{x}_{i} S^{x}_{j} \rangle. \end{equation} The structure of the output files \texttt{SpinZ\_eqJR} and \texttt{SpinXY\_eqJR} is described in the doucmentation. For the Mz Hubbard-Stratonovitch transformation it is hence better to consider the improved estimator \begin{equation} \langle \vec{S}_{i} \cdot \vec{S}_{j} \rangle \end{equation} to compute the spin-spin correlations. % [Excluding the introduction to the exercise item: improved estimator \langle \vec{S}_{i} \cdot \vec{S}_{j} \rangle] ... ... @@ -384,13 +376,19 @@ The SU(2) Hubbard-Stratonovich decomposition, conserves spin rotational symmet \exercise{Adding a new observable} \red{[Update with:]} \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,r}\hat{c}_{i+ax,\sigma} +H.c. \right) \text{, such that}\\ & \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_0\big(i-j,\delta,\delta'\big) &\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)\\ ... ... @@ -399,58 +397,40 @@ In the 1-D Hubbard we have emergent $SO(4)$ symmetry: [It should be added to the Predefined Structures.]\\ \hrule \red{[Old part:]}\\ 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} \item In the subroutine \texttt{Alloc\_obs} in the \texttt{Hamiltonian\_example.f90} file you will have to add a new equal time observable with a call to \texttt{Call Obser\_Latt\_make(Obs\_eq(I),Ns,Nt,No,Filename)} with \texttt{Ns = Latt\%N; No = Norb; Filename ="SpinT", Nt=1, I=5} \item In the subroutine \texttt{Obser} you will have to add the Wick decomposition of this observable. \end{itemize} In the program \texttt{Hamiltonian\_Examples.f90 } to be found in the directory \texttt{Solutions/Exercise\_2/Prog/} we have commented the changes that have to be carried out to add this observable. The new variable takes the name SpinT and the results you should obtain are summarized in Fig.~\ref{Ladder.fig}. \begin{figure}[h] \begin{center} \includegraphics[scale=.8]{Ladder.pdf} \end{center} \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} \exercise{Defining a new model: The one-dimensional t-V model} \red{[To be updated.]} \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. \begin{equation} H = -t \sum_{i} \left( c^{\dagger}_{i} c^{\phantom\dagger}_{i+a} + c^{\dagger}_{i+a} c^{\phantom\dagger}_{i} \right) - \frac{V}{2} \sum_{i} \left( c^{\dagger}_{i} c^{\phantom\dagger}_{i+a} + c^{\dagger}_{i+a} 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} 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} H = -t \sum_{i} \left( c^{\dagger}_{i} c^{\phantom\dagger}_{i+a} + c^{\dagger}_{i+a} c^{\phantom\dagger}_{i} \right) + V \sum_{i} \left( n_{i} - 1/2 \right) \left( 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} \red{[Mention tV Jupyter notebook.]} \red{[To be updated:]}\\ In the directory \texttt{Solutions/Exercise\_3} we have duplicated the ALF and commented the changes that have to be carried out to the file \texttt{Hamiltonian\_Examples.f90 } in the \texttt{Prog} directory so as to include the t$\_$V model. \texttt{Hamiltonian\_Examples.f90 } in the \texttt{Prog} directory so as to include the \texttt{t\_V} model. Here are the steps to be carried out. \begin{itemize} \item Add the $t-V$ name space in the parameter file so as to read in the appropriate variables. \item Add the t-V name space in the parameter file so as to read in the appropriate variables. \item Declare new variables in the \texttt{Hamiltonian\_Examples.f90 } file. \item In the \texttt{Ham\_set} subroutine of the file \texttt{Hamiltonian\_Examples.f90} set and read in the parameters for the new model, \texttt{t\_V}. For this model \texttt{NF=1} and \texttt{N\_SUN=1} since we are working with spinless fermions \item In the \texttt{Ham\_V} subroutine you will have to add the new interaction. For a given bond at a given time-slice, we need to decouple the interaction: \begin{equation} e^{\Delta \tau \frac{V}{2} \left( c^{\dagger}_{i} c^{\phantom\dagger}_{i+a} + c^{\dagger}_{i+a} c^{\phantom\dagger}_{i} \right)^2} = \sum_{l= \pm1, \pm 2} \gamma_l e^{ \sqrt{\Delta \tau \frac{V}{2}} \eta_l \left( c^{\dagger}_{i} c^{\phantom\dagger}_{i+a} + c^{\dagger}_{i+a} c^{\phantom\dagger}_{i} \right) } = \sum_{l= \pm1, \pm 2} \gamma_l e^{ g \eta_l \left( c^{\dagger}_{i}, c^{\dagger}_{i+a} \right) O \left(c^{\phantom\dagger}_{i}, c^{\phantom\dagger}_{i+a} \right)^{T} } e^{\Delta \tau \frac{V}{2} \left( \hat{c}^{\dagger}_{i} \hat{c}^{\phantom\dagger}_{i+a} + \hat{c}^{\dagger}_{i+a} \hat{c}^{\phantom\dagger}_{i} \right)^2} = \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. \begin{lstlisting}[style=fortran] ... ... @@ -479,20 +459,60 @@ e^{\Delta \tau \frac{V}{2} \left( c^{\dagger}_{i} c^{\phantom\dagger}_{i+a} 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} H = J_{xx} \sum_{i} S^{x}_i S^{x}_{i+a} + S^{y}_i S^{y}_{i+a} + J_{zz} \sum_{i}S^{z}_i S^{z}_{i +a} \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} \end{equation} 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}[h] \begin{center} \includegraphics[scale=.8]{tV.pdf} \end{center} \caption{ \label{tV.fig} 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 \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 $\left(1+K_\rho \right)^{-1}= \frac{1}{2} + \frac{1}{\pi} \arcsin \left( \frac{V}{2 | t | }\right)$ (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. } (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} \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 $V/t < 0$? \end{document} Exercise item excluded from exercise "Dimensional crossover": \hrule \noindent When running the code for the above ladder system and analyzing the spin correlation functions, \texttt{SpinZ\_eqJR} and \texttt{SpinXY\_eqJR} one will notice that it is hard restore the SU(2) spin symmetry and that very long runs are required to obtain the desired equality \begin{equation} \langle S^{z}_{i} S^{z}_{j} \rangle = \langle S^{y}_{i} S^{y}_{j} \rangle = \langle S^{x}_{i} S^{x}_{j} \rangle. \end{equation} The structure of the output files \texttt{SpinZ\_eqJR} and \texttt{SpinXY\_eqJR} is described in the doucmentation. For the Mz Hubbard-Stratonovitch transformation it is hence better to consider the improved estimator \begin{equation} \langle \vec{S}_{i} \cdot \vec{S}_{j} \rangle \end{equation} to compute the spin-spin correlations. 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} \item In the subroutine \texttt{Alloc\_obs} in the \texttt{Hamiltonian\_example.f90} file you will have to add a new equal time observable with a call to \texttt{Call Obser\_Latt\_make(Obs\_eq(I),Ns,Nt,No,Filename)} with \texttt{Ns = Latt\%N; No = Norb; Filename ="SpinT", Nt=1, I=5} \item In the subroutine \texttt{Obser} you will have to add the Wick decomposition of this observable. \end{itemize} In the program \texttt{Hamiltonian\_Examples.f90 } to be found in the directory \texttt{Solutions/Exercise\_2/Prog/} we have commented the changes that have to be carried out to add this observable. The new variable takes the name SpinT and the results you should obtain are summarized in Fig.~\ref{Ladder.fig}. \begin{figure}[h] \begin{center} \includegraphics[scale=.8]{Ladder.pdf} \caption{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.} \label{Ladder.fig} \end{center} \end{figure}
 ... ... @@ -224,7 +224,7 @@ For Dtau values [0.05 0.1 0.15] the measured energies are: \end{tcolorbox} \begin{center} \adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{trotter_error-output_12_2.png} \adjustimage{max size={0.6\linewidth}{0.9\paperheight}}{trotter_error-output_12_2.png} \end{center} { \hspace*{\fill} \\} ... ...
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