If the wave function of the system
is real and positive, as it
happens in case of ground state of a bose system, it can be treated as population
density distribution2.1(the algorithm for fermions, where the wave function has nodes will be discussed in
Sec. 2.6)
Let us now interpret the action of the each of the three terms of the Hamiltonian
(2.20) on the population distribution or, being the same, the action of
the corresponding Green's functions (2.26, 2.28, 2.30).
In terms of Markov chains the Green's function is the
is the transition matrix which determines the evolution of
the distribution (see Eq. 2.18).
The first term means the diffusion of each of the walkers in the configuration
space
The second term describes the action of the drift force, which guides the walkers to
places in the configuration space, where the trial wave function is maximal. This is
the way how importance sampling acts in the algorithm
The Green's functions of steps (2.26),(2.28) are normalized to unity
. The normalization of wave function
is then conserved,
which means that the number of walkers remains constant.
The third term is the branching
The corresponding Green's function
(2.30) is no longer
normalized. That means that the weight of a walker
changes on this step, thus
walkers with lower local energy have larger weights and walkers with larger local
energy have smaller weights. On this step each walker is to be duplicated
times. In general the number
is not an
integer. A possible solution is to throw a random number
and
duplicate the walker
times, where the brackets
stand for the
integer part of a number.
Now it is clear that by adjusting the value of one can control the size of the
population and keep it within the desired range. If the value of
is taken to be
equal the estimator
(2.14) averaged over the population and the
population size does not change, it means that
is equal to the ground state
energy (see, also, Eq. 2.9).
The branching is an essential part of the DMC algorithm as it ``corrects'' the trial
wave function. Indeed, the first two steps (2.26), (2.28) alone without
(2.30) are equivalent to sampling the trial wave function and provide the same
result as the variational calculation (Sec. 2.2). If the trial wave function
is an exact eigenfunction of the Hamiltonian, the local energy equals to the
corresponding eigennumber and is independent of , thus the branching becomes
irrelevant as it acts in the same way on all walkers.