476 Computational methods for lattice field theories
values much larger than the lattice constant. This means that our parametersmand
gshould be chosen close to a critical point. Theφ^4 theory ind=4 dimensions has
one critical line [7], passing through the pointm=g=0, the massless free field
case. Therefore,mandgshould be chosen very close to this critical line in order
for the lattice representation to be justifiable.
As the experimental mass of a particle is a fixed number, varying the lattice
constantaforces us to varygandmin such a way that the correlation length
remains finite. Unfortunately this renders the use of finite size scaling techniques
impossible: the system sizeLmust always be larger than the correlation length:
a ξ<L.^4
The fact that the lattice field theory is always close to a critical point implies that
we will suffer fromcritical slowing down. Consider a Monte Carlo (MC) simulation
of the field theory. We change the field at one lattice point at a time. At very high
temperature, the field values at neighbouring sites are more or less independent, so
after having performed as many MC attempts as there are lattice sites (one MCS), we
have obtained a configuration which is more or less statistically independent from
the previous one. If the temperature is close to the critical temperature, however,
fields at neighbouring sites are strongly correlated, and if we attempt to change
the field at a particular site, the coupling to its neighbours will hardly allow a
significant change with respect to its previous value at that site. However, in order
to arrive at a statistically independent configuration, we need to change the field
over a volume of linear size equal to the correlation length. If that length is large,
it will obviously take a very long time to change the whole region, so this problem
gets worse when approaching the critical point. Critical slowing down is described
by a dynamic critical exponentzwhich describes the divergence of the decay time
τof the dynamic correlation function (see Chapter 7, Eq. (7.73)):
τ=ξz, (15.40)whereξis the correlation length of the system.
In recent years, much research has aimed at finding simulation methods for
reducing the critical time relaxation exponent. In the following section we shall
describe a few straightforward methods developed for simulating quantum field
theories, using theφ^4 scalar field theory in two dimensions as a testing model.
In Section 15.5 we shall focus on methods aiming at reducing critical slowing
down. We shall then also discuss methods devised for the Ising model and for a
two-dimensional model with continuous degrees of freedom.
(^4) In the case where physical particles are massless, so that the correlation length diverges, finite size scaling
can be applied. Finite size scaling applications in massive particle field theories have, however, been proposed;
seeRef.[ 8 ].