482 Computational methods for lattice field theories
is equal to that of the corresponding Poisson solver algorithm, that is, the relaxation
time will now scale asL. We should obviously check that the SOR method still
satisfies detailed balance. This is left as an exercise (Problem 15.5).
The SOR method works well for models with quadratic interactions. Including
aφ^4 termrendersthemethodlesssuitable(seehoweverRef.[12]). Fortunately,
the physically more interesting gauge theories which will be discussed later in this
chapter are quadratic. A problem with this method is that the optimal value of the
over-relaxation parameterω, which is 2 in the case of the scalar free field theory,
is not known in general and has to be determined empirically.
We have encountered the most straightforward methods for simulating the scalar
field theory. Most of these methods can easily be generalised to more complicated
field theories. Before discussing different methods, we shall analyse the behaviour
of the methods presented so far.
15.4.2 The MC algorithms: implementation and resultsThe implementation of the algorithms presented in the previous sections is straight-
forward. The reader is encouraged to try coding a few and to check the results given
below.
To obtain the renormalised mass and coupling constant,Eqs. (15.53)and(15.54)
can be used. However, it is nice to measure the full two-point correlation function.
This can be found by sampling this function for pairs of points which lie in the
same column or in the same row. To obtain better statistics, nonhorizontal and
nonvertical pairs can be taken into account as well. To this end we construct a
histogram, corresponding to equidistant intervals of the pair separation. We keep
two arrays in the program, one for the value of the correlation function, and the
other for the average distancercorresponding to each histogram column. At regular
time intervals we perform a loop over all pairs of lattice sites. For each pair we
calculate the closest distance within the periodic boundary conditions according to
the minimum image convention. Suppose this distance isrij. We calculate to which
column this value corresponds, and add the product of the field values at the two
sitesφiφjto the correlation function array. Furthermore we addrijto the average
distance array. After completing the loop over the pairs, we divide the values in the
correlation function array and in the average distance array by the number of pairs
that contributed to these values. The final histogram must contain the time averages
of the correlation function values thus evaluated, and this should be written to a file.
We can now check whether the scalarφ^4 theory is renormalisable. This means that
if we discretise the continuum field theory using finer and finer grids, the resulting
physics should remain unchanged. Equation (15.43) tells us how we should change
the various parameters of the theory when changing the grid constant. We now