loos/DQMC
1
0
Fork 1
Code Issues Pull Requests Projects Releases Wiki Activity

minor corrections

Browse Source
This commit is contained in:
Pierre-Francois Loos 2022-12-16 13:57:17 +01:00
parent 0d964aaae6
commit 8bcea1f32f
1 changed files with 4 additions and 4 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

8
g.tex
View File

@ -619,7 +619,7 @@ The time-dependent Green's matrix, Eq.~\eqref{eq:cn}, is then rewritten as
\prod_{k=0}^{N-1} \mel{ i_k }{ T }{ i_{k+1} } . \prod_{k=0}^{N-1} \mel{ i_k }{ T }{ i_{k+1} } .
\ee \ee
$\cC_p$ can be partitioned further by considering the subset of paths, denoted by $\cC_p^{I,n}$, $\cC_p$ can be partitioned further by considering the subset of paths, denoted by $\cC_p^{I,n}$,
having $(\ket{I_k}, n_k)$ for $k=0$ to $p$ as exit states and trapping times. having $(\ket{I_k}, n_k)$, for $0 \le k \le p$, as exit states and trapping times.
We recall that, by definition of the exit states, $\ket{I_k} \notin \cD_{I_{k-1}}$. The total time must be conserved, so the relation $\sum_{k=0}^p n_k= N+1$ We recall that, by definition of the exit states, $\ket{I_k} \notin \cD_{I_{k-1}}$. The total time must be conserved, so the relation $\sum_{k=0}^p n_k= N+1$
must be fulfilled. must be fulfilled.
Now, the contribution of $\cC_p^{I,n}$ to the path integral is obtained by summing over all elementary paths $(i_1,...,i_{N-1})$ Now, the contribution of $\cC_p^{I,n}$ to the path integral is obtained by summing over all elementary paths $(i_1,...,i_{N-1})$
@ -1009,7 +1009,7 @@ The two variational parameters of the trial vector have been optimized and fixed
variational energy $E_\text{v}=-0.495361\ldots$. variational energy $E_\text{v}=-0.495361\ldots$.
In what follows, $\ket{I_0}$ is systematically chosen as one of the two N\'eel states, \eg, $\ket{I_0} = \ket{\uparrow,\downarrow, \uparrow,\ldots}$. In what follows, $\ket{I_0}$ is systematically chosen as one of the two N\'eel states, \eg, $\ket{I_0} = \ket{\uparrow,\downarrow, \uparrow,\ldots}$.
Figure \ref{fig3} shows the convergence of $H^{DMC}_p$, Eq.~\eqref{eq:defHp}, as a function of $p$ for different values of the reference energy $E$. Figure \ref{fig3} shows the convergence of $H^\text{DMC}_p$, Eq.~\eqref{eq:defHp}, as a function of $p$ for different values of the reference energy $E$.
We consider the simplest case where a single-state domain is associated to each state. We consider the simplest case where a single-state domain is associated to each state.
Five different values of $E$ have been chosen, namely $E=-1.6$, $-1.2$, $-1.0$, $-0.9$, and $-0.8$. Five different values of $E$ have been chosen, namely $E=-1.6$, $-1.2$, $-1.0$, $-0.9$, and $-0.8$.
Only $H_0$ is computed analytically ($p_\text{ex}=0$). Only $H_0$ is computed analytically ($p_\text{ex}=0$).
@ -1129,7 +1129,7 @@ The extrapolated values obtained from the five values of the energy with three d
Fitting the data using a simple linear function leads to an energy of $-0.7680282(5)$ (to be compared with the exact value of $-0.768068\ldots$). Fitting the data using a simple linear function leads to an energy of $-0.7680282(5)$ (to be compared with the exact value of $-0.768068\ldots$).
A small bias of about $4 \times 10^{-5}$ is observed. A small bias of about $4 \times 10^{-5}$ is observed.
This bias vanishes within the statistical error when resorting to more flexible fitting functions, such as a quadratic function of $E$ or the This bias vanishes within the statistical error when resorting to more flexible fitting functions, such as a quadratic function of $E$ or the
two-component representation given by Eq.(\ref{eq:2comp}). two-component representation given by Eq.~\eqref{eq:2comp}.
Our final value is in full agreement with the exact value with about six decimal places. Our final value is in full agreement with the exact value with about six decimal places.
Table \ref{tab4} shows the evolution of the average trapping times and extrapolated energies as a function of $U$ when using $\cD(0,1) \cup \cD(1,0)$ as the main domain. Table \ref{tab4} shows the evolution of the average trapping times and extrapolated energies as a function of $U$ when using $\cD(0,1) \cup \cD(1,0)$ as the main domain.
@ -1154,7 +1154,7 @@ All these aspects will be considered in a forthcoming work.
%%% TABLE II %%% %%% TABLE II %%%
\begin{table} \begin{table}
\caption{One-dimensional Hubbard model with $N=4$, $U=12$, and $E=-1$. \caption{One-dimensional Hubbard model with $N=4$, $U=12$, and $E=-1$.
Dependence of the statistical error on the energy with the number of $p$-components $p_{ex}$ calculated analytically in the expression Dependence of the statistical error on the energy with the number of $p$-components $p_\text{ex}$ calculated analytically in the expression
of the energy, $\cE^\text{DMC}(E,p_\text{ex},p_\text{max})$, Eq.~\eqref{eq:E_pex}. of the energy, $\cE^\text{DMC}(E,p_\text{ex},p_\text{max})$, Eq.~\eqref{eq:E_pex}.
The simulation is performed the same way as in Table \ref{tab1}. The simulation is performed the same way as in Table \ref{tab1}.
Results are presented when a single-state domain is used for all states and when $\cD(0,1) \cup \cD(1,0)$ is chosen as the main domain.} Results are presented when a single-state domain is used for all states and when $\cD(0,1) \cup \cD(1,0)$ is chosen as the main domain.}