My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 3.
QUANTUM MONTE CARLO METHODS
eigenfunction of Ĥ, the local energy is subject to fluctuations; as a result, the walker
population also fluctuates. Using equation (3.58) allows the population fluctuations
to be kept under control; simply setting E T
= E 0 would not achieve this, even
though the average population would remain constant. In addition, of course, the
precise value of E 0 is not known — calculating E 0 is usually one of the aims of the
simulation.
Modifying the reference energy introduces a bias into the sampling, because the
equation being simulated is no longer (3.38), and therefore has different eigenfunctions.
The bias can be reduced by making τ g as large as possible, although this leads
to greater population fluctuations.
Even when τ g is large and the population-control bias is negligible, there remain
sampling errors, caused by the use of an approximate Green’s function. The true
Green’s function satisfies a form of detailed balance; using equation (3.44), and
noting that G(R, R ′ ; ∆τ) is symmetric in R and R ′ gives
Ψ 2 T (R ′ ) ˜G(R, R ′ ; ∆τ) = Ψ 2 T (R) ˜G(R ′ , R; ∆τ). (3.60)
This is not the usual version of detailed balance, because ˜G does not have the form
of a probability; specifically, the total density is not conserved between moves, and
may increase or decrease.
The approximate Green’s function does not satisfy equation (3.60). This can
be traced back to the expression for G D , equation (3.52), which is only correct to
O [∆τ]. The sampling error can be reduced by making the time step ∆τ smaller;
however, this is inefficient, because many more steps are required between uncorrelated
configurations. A better way of reducing the time-step error is to enforce the
detailed balance condition by introducing a Metropolis rejection step, as described
in section 3.2.2, with an acceptance probability now given by
(
A(R ′ ← R) = min 1, ˜G(R,
)
R ′ ; ∆τ)Ψ 2 T (R′ )
˜G(R ′ , R; ∆τ)Ψ 2 T (R) . (3.61)
For small time steps, the rejection probability tends to zero, since the approximate
drift-diffusion Green’s function becomes more exact. However, for non-zero time
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