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DMC Decay Curves


The Monte Carlo solution to the diffusion equation can be written as a function of position, tex2html_wrap_inline6125 , and imaginary time, tex2html_wrap_inline6435 , as follows


where the coefficients, tex2html_wrap_inline8343 , are the overlap integrals of tex2html_wrap_inline8345 with the eigenfunctions of the many-body Hamiltonian, tex2html_wrap_inline6225 . The DMC method relies on the fact that in the limit of large imaginary time tex2html_wrap_inline8349 is dominated by the lowest energy solution, tex2html_wrap_inline8351 .

However, in the initial short imaginary time regime, it is clear that the above equation contains information about the energy differences, tex2html_wrap_inline8353 . For example the time-dependence of the energy estimate is given by


where the tex2html_wrap_inline8355 are the overlap integrals of the guiding wavefunction, tex2html_wrap_inline8357 , with the eigenfunctions of the many-body Hamiltonian, tex2html_wrap_inline6225 . Therefore, if one was to compute the energy estimate as a function of time, then standard curve fitting methods could be used to extract the excited state energies, tex2html_wrap_inline5849 . In practice however, obtaining energies from Eq.(gif) would be extremely difficult due to the statistical noise of the Monte Carlo simulation.

More sophisticated methods have been devised[80, 81, 82] to specifically measure the time dependence. Instead of the energy in Eq.(gif), consider the expectation value of the Green's function,


which can be sampled from the random walk by evaluating


where W is the cumulative branching weight,


which is essentially the total population. If we insert a complete set of eigenstates into Eq.(gif), the time behaviour of tex2html_wrap_inline8365 is


This is a simpler expression to attempt to fit than Eq.(gif). However, in practice it is still dominated by the statistical noise[23].

Andrew Williamson
Tue Nov 19 17:11:34 GMT 1996