480 Computational methods for lattice field theories
straightforwardly in order to sample field configurations with Boltzmann weight
exp(−S[φ]). Starting point is the action (15.44). An obvious method is the Metro-
polis MC algorithm, in which lattice sites are visited at random or in lexicographic
order, and at the selected site a change in the field is attempted by some random
amount. The change in the field is taken from a random number generator either
uniformly within some interval or according to a Gaussian distribution (with a suit-
able width). Then we calculate the change in the action due to this change in the
field. The trial value of the field is then accepted as the field value in the next step
according to the probability
PAccept=e−S[φnew]+S[φold] (15.55)where the exponent on the right hand side is the difference between the action of
the new and old field at the selected site, keeping the field at the remaining sites
fixed. IfPAccept>1, then the new configuration is accepted.
In Chapter 10 we have already encountered another method which is more effi-
cient as it reduces correlations between subsequent configurations: the heat-bath
algorithm. In this algorithm, the trial value of the field is chosen independently of
the previous value. Let us callWφ[φn]the Boltzmann factor e−S[φ]for a field which
is fixed everywhere except at the siten. We generate a new field value at siten
according to the probability distributionWφ[φn]. This is equivalent to performing
infinitely many Metropolis steps at the same sitensuccessively. The new value of
φncan be chosen in two ways: we can generate a trial value according to some distri-
butionρ(φn)and accept this value with probability proportional toWφ[φn]/ρ(φn),
or we can directly generate the new value with the required probabilityWφ[φn].
The Gaussian free field model will serve to illustrate the last method.
Consider the action (15.44). If we varyφn, and keep all the remaining field values
fixed, we see that the minimum of the action occurs forφ ̄n=
∑
μφn+μ/(^2 d+m(^2) ),
where the sum is overallneighbouring points, i.e. for both positive and negative
directions. The Boltzmann factorWφ[φn]as a function ofφnfor all remaining field
values fixed is then a Gaussian centred aroundφ ̄nand with a width 1/
√
2 d+m^2.
Therefore, we generate a Gaussian random numberrwith a variance 1, and then
we set the new field value according to
φn=φ ̄n+r/√
2 d+m^2. (15.56)An advantage of this method is that no trial steps have to be rejected, which
obviously improves the efficiency.
Unfortunately, this method is not feasible when aφ^4 interaction is present as
we cannot generate random numbers with an exp(−x^4 )distribution. Therefore we
treat this term with an acceptance/rejection step as described above. This is done
as follows. First we generate a ‘provisional’ value of the fieldφnaccording with