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## DMC Decay Curves

The Monte Carlo solution to the diffusion equation can be written as a function of position, , and imaginary time, , as follows where the coefficients, , are the overlap integrals of with the eigenfunctions of the many-body Hamiltonian, . The DMC method relies on the fact that in the limit of large imaginary time is dominated by the lowest energy solution, .

However, in the initial short imaginary time regime, it is clear that the above equation contains information about the energy differences, . For example the time-dependence of the energy estimate is given by where the are the overlap integrals of the guiding wavefunction, , with the eigenfunctions of the many-body Hamiltonian, . 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, . In practice however, obtaining energies from Eq.( ) 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.( ), 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.( ), the time behaviour of is This is a simpler expression to attempt to fit than Eq.( ). However, in practice it is still dominated by the statistical noise.

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