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Measurement and averaging

The algorithm described in the previous section generated a Markov chain of states, favoring the selection of configurations that contribute to the partition function with relatively large weights. Thsi process is simple a generalization of the importance sampling. If we denote the set of generated states $\{\sigma_\nu\}$ with $\nu = 1,2\cdots,M$, then the monte Carlos estimates for the mnean value of an observable $A$ in a classical system will be given by

\begin{displaymath}
A\approx \frac{1}{M}\sum_{i=1}^M A(\sigma_\nu).
\end{displaymath} (301)

In this equation $A(\sigma_\nu)$ is the value of a function $A$ in the state $\sigma_\nu$. However, we must remember that in a classical system obtained from the Suzuki-Trotter decomposition, the variable $A$ is dynamic, and depends on the temperature. therefore, we must average some adecuate function $B$ associated to $A$, such that in reality we obtain:
\begin{displaymath}
A=\frac{1}{M}\sum_{i=1}^M B(\sigma_\nu).
\end{displaymath} (302)

For instance, let us consider the classical energies associated to the Heisenber Hamiltonian
$\displaystyle E_1^{(m)} = E_2^{(m)}$ $\textstyle =$ $\displaystyle \frac{\Delta \tau}{\beta}J/4$  
$\displaystyle E_3^{(m)} = E_4^{(m)}$ $\textstyle =$ $\displaystyle \frac{1}{\beta}(-J\Delta \tau/4-\ln \cosh\left(\frac{\Delta \tau J}{2}\right)$  
$\displaystyle E_5^{(m)} = E_6^{(m)}$ $\textstyle =$ $\displaystyle \frac{1}{\beta}(-J\Delta \tau/4-\ln \sinh\left(\frac{\Delta \tau J}{2}\right).$ (303)

The partition function for a single plaquette can written as
\begin{displaymath}
Z_{\mathrm plq.}^{(m)}=\sum_j \exp \left(-\beta E_j^{(m)}\right),
\end{displaymath} (304)

and the thermal average of the energy is finaly obtained as
$\displaystyle E_{\mathrm plq.}^{(m)}$ $\textstyle =$ $\displaystyle -\frac{\partial}{\partial \beta}\ln \left(Z_{\mathrm plq.}^{(m)}\right)$  
  $\textstyle =$ $\displaystyle \frac{1}{Z_{\mathrm plq.}^{(m)}}\sum_j\left[\frac{\partial}{\partial \beta}\left(\beta E_j^{(m)}\right)\right]\exp\left(-\beta E_j^{(m)}\right)$  
  $\textstyle =$ $\displaystyle \frac{1}{Z_{\mathrm plq.}^{(m)}}\sum_j F_j^{(m)} \exp \left(-\beta E_j^{(m)}\right).$ (305)

In the last step we have defined $F_j^{(m)}$, the value of the "energy function'' for the state $j$, such that tthe energy is the the thermodynamic average of a function $F$. The mean value of any observable can calculated in a similar way.

The equivalence between the quantum system and the classical counterpart is exact only in the limit of $m$ going to infinity. In practice we work always with finite values of $m$ (or $\delta \tau \neq 0$), which is a source of systematic error of the order $(\Delta \tau)^2$, which is in general small and under control. The error is independent of the volume for sufficiently large systems, and results porportional to the norm of the commutator $[H_1,H_2]$. For a large quantity of observables one can use the extrapolation

\begin{displaymath}
A(\Delta\tau) = A(0)+a/(\delta \tau)^2 + b/(\Delta \tau)^4+\cdots,
\end{displaymath} (306)

where $A(0)$ is the correct value. usually, only the lower oder temrs on the extrapolation are considered, such that we extrapolate with $1/(\Delta \tau)^2$. A possible approximation is to fix the value of $\Delta \tau$ to a very small number for all temperatures, such that the systematic error can be neglected, compared to the statistical error.


next up previous
Next: Determinantal (or Auxiliary Field) Up: World Line Monte Carlo Previous: Monte Carlo simulation with
Adrian E. Feiguin 2009-11-04