12.4 Path-integral Monte Carlo 409within a cube (or a sphere). If we update the coordinateRm, keepingRm− 1 and
Rm+ 1 fixed, then in the heat-bath algorithm, the new valueR′mmust be generated
with distribution
ρ(R′m)=exp[
−
τ
(R′m−Rm)^2
2
τ^2−
τV(R′m)]
(12.88)
whereRm=(Rm+ 1 +Rm− 1 )/2. We may sample the new position directly from this
distribution by first generating a new position using a Gaussian random generator
with width 1/( 2
τ )and centred aroundRm, and then accepting or rejecting the
new position with a probability proportional to exp[−
τV(R′m)]. This procedure
guarantees 100% acceptance for zero potential. If there are hard-core interac-
tions between the particles, the Gaussian distribution might be replaced by a more
complicated form to take this into account [4].
A major drawback of the algorithm presented so far is that only one atom is
displaced at a time. To obtain a decent acceptance rate the maximal distance over
which the atom can be displaced is restricted by the harmonic interaction between
successive ‘beads’ on the imaginary time-chain to∼
√
τ. The presence of the
potentialVcan force us to decrease this step size even further. It will be clear that
our local update algorithm will cause the correlation time to be long, as this time
is determined by the long-wavelength modes of the chain. As it is estimated that
equilibration of the slowest modes takes roughlyO(M^2 )Monte Carlo sweeps (see
the next chapter), the relaxation time will scale asM^3 single-update steps. This
unfavourable time scaling behaviour is well known in computational field theory,
and a large part of the next chapter will be dedicated to methods for enhancing the
efficiency of Monte Carlo simulations on lattices. An important example of such
methods isnormal mode samplingin which, instead of single particle moves, one
changes the configuration via its Fourier modes [24, 25]. If one changes for example
thek=0 mode, all particles are shifted over the same distance. The transition
probability is calculated either through the Fourier-transformed kinetic (harmonic
interaction) term, followed by an acceptance/rejection based on the change in poten-
tial, or by using the Fourier transform of the full action. We shall not treat these
methods in detail here; in the next chapter, we shall discuss similar methods for
field theory.
A method introduced by Ceperley and Pollock divides the time slices up in a
hierarchical fashion and alters the values of groups of points in various stages
[ 3 , 4 ]. At each stage the step can be discontinued or continued according to some
acceptance criterion. It turns out[4]that with this method it is possible to reduce the
relaxation time fromM^3 toM1.4. The method seems close in spirit to the multigrid
Monte Carlo method of Goodman and Sokal, which we shall describe in the next
chapter.