My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 3.
QUANTUM MONTE CARLO METHODS
which can be verified by substitution; the operator ˆT guides the walkers so that
their distribution follows the square of the trial wave function. This is importance
sampling: walkers are concentrated in important regions (where the magnitude of
the wave function is large). Allowing walkers to drift and diffuse in this manner is
therefore a sampling technique in its own right, and can be used as an alternative
to the Metropolis method in VMC.
However, the aim here is not simply to sample Ψ 2 T , but to sample Ψ T Φ 0 . This is
achieved by introducing weights. Initially, the walker weights are set to unity; then,
when a walker moves from R to R ′ during the time step τ, its weight is multiplied by
W (R, R ′ ; ∆τ). To see that this works, consider the density function for the walker
weights at R after a single move:
∫
f sim (R, ∆τ) = W (R, R ′ ; ∆τ)G D (R, R ′ ; ∆τ)f(R, 0) dR ′
∫
≈ ˜G(R, R ′ ; ∆τ)f(R, 0) dR ′
(3.56)
≈ f(R, ∆τ).
The initial distribution of walkers is f(R, 0); the probability that a walker moves
from R ′ to R during the time step ∆τ is G D (R, R ′ ; ∆τ), and the weight associated
with such a move is W (R, R ′ , ∆τ). The combination of weights and drift-diffusion
therefore accurately simulates the evolution of f, as long as the time step ∆τ is
small.
In order to reach the desired limit of large τ, many small steps of duration ∆τ
must be carried out. If, during this time, a walker spends a lot of time in a region
of space where E L < E T , its weight continues to increase, and may become very
large. In contrast, if a walker spends a lot of time in a region where E L > E T , its
weight may become very small, and it contributes little to the ultimate sampling of
f, which is dominated by walkers with large weights. This is one of the problems
alluded to in section 3.3.3, and makes the sampling inefficient.
The solution to this problem is to allow walkers to multiply or die out according
to their weight. Either when the weight of a walker becomes too large or too small,
or at regular intervals, the walker is replaced by a certain number of ‘descendants’
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