484 Computational methods for lattice field theories
Table 15.1. Values of the renormalised mass.Lm g mRa mRb
8 0.2 0.04 0.374(5) 0.363(4)
12 0.1333 0.01778 0.265(5) 0.265(7)
16 0.1 0.01 0.205(7) 0.204(8)
24 0.06667 0.004444 0.138(4) 0.138(4)
aValues obtained from matching the measured cor-
relation function to the analytic form (15.59), for
different grid sizes.
bValues obtained from matching to formula(15.53).parameterZ) is then the renormalised mass. For each of the correlation functions
represented inFigure 15.1, parametersZandmRcan be found such that the analytic
form lies within the (rather small) error bars of the curves obtained from the simu-
lation. In Table 15.1, the values of the renormalised mass as determined using this
procedure are compared with those obtained using (15.53). Excellent agreement
is found. It is seen that for the larger lattices, the renormalised mass is more or
less inversely proportional to the linear lattice size. The physical mass, however,
should be independent of the lattice size. This is because masses are expressed in
units of the inverse lattice constant, and the lattice constant is obviously inversely
proportional to the linear lattice sizeLif the lattice represents the same physical
volume for different sizes.
The determination of the renormalised coupling constant is difficult. We use
Eq. (15.54), but this is subject to large statistical errors. The reason is that the result
is the difference of two nearly equal quantities, and this difference is subject to the
(absolute) error of these two quantities – hence therelativeerror of the difference
becomes very large. The renormalised coupling constant should not depend on
the lattice size for large sizes, as it is dimensionless. Table 15.2 shows the results.
The errors are large and it is difficult to check whether the renormalised coup-
ling constant remains the same, although the data are compatible with a coupling
constant settling at a size-independent value ofg≈0.11 for large lattices.
15.4.3 Molecular dynamicsHow can we use molecular dynamics for a field theory formulated on a lattice, which
has no intrinsic dynamics?^6 The point is that we assign afictitiousmomentum degree
(^6) The dynamics are here defined in terms of the evolution of the field configuration and not in terms of the
time axis of the lattice.