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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
with
, then the monte Carlos estimates for the mnean value of an observable in a classical system will be given by
|
(301) |
In this equation is the value of a function in the state . However, we must remember that in a classical system obtained from the Suzuki-Trotter decomposition, the variable is dynamic, and depends on the temperature. therefore, we must average some adecuate function associated to , such that in reality we obtain:
|
(302) |
For instance, let us consider the classical energies associated to the Heisenber Hamiltonian
The partition function for a single plaquette can written as
|
(304) |
and the thermal average of the energy is finaly obtained as
In the last step we have defined , the value of the "energy function'' for the state , such that tthe energy is the the thermodynamic average of a function . 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 going to infinity. In practice we work always with finite values of (or
), which is a source of systematic error of the order
, 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 . For a large quantity of observables one can use the extrapolation
|
(306) |
where is the correct value. usually, only the lower oder temrs on the extrapolation are considered, such that we extrapolate with
. A possible approximation is to fix the value of to a very small number for all temperatures, such that the systematic error can be neglected, compared to the statistical error.
Next: Determinantal (or Auxiliary Field)
Up: World Line Monte Carlo
Previous: Monte Carlo simulation with
Adrian E. Feiguin
2009-11-04