524 Computational methods for lattice field theories
It is now possible to calculate this average by updating the fieldφin a heat-bath
algorithm. As the matrixM(A)is local (it couples only nearest neighbours),W(A)
is local as well (it couples up to next nearest neighbours). Therefore the heat-bath
algorithm can be carried out efficiently (it should be possible to apply the SOR
method to this method). Each time we change the fieldA, the matrixW(A)changes
and a few heat-bath sweeps for the fieldφhave to be carried out. The value of the
fraction of the determinants is determined as the geometrical average of the two
estimators given inEq. (15.153).
The most efficient algorithms for dynamical fermions combine a molecular
dynamics method for the boson fields with a Monte Carlo approach for the fer-
mionic part of the action. We describe two of these here. The first one is a Langevin
approach, proposed by Batrouniet al.[17], and suitable for Fourier acceleration.
It is based on two observations: first, det(M)can be written as exp[Tr ln(M)], and
second, ifξnis a complex Gaussian random field on the lattice, so that
〈ξl†ξn〉=δnl (15.154)(the brackets〈〉denote an average over the realisations of the Gaussian random
generator), then the trace of any matrixKcan be written as
Tr(K)=∑
nl〈ξn†Knlξl〉. (15.155)In the Langevin approach, the force is given by the derivative of the action
with respect to the boson field. In the presence of fermions, the action reads (see
Eq. (15.148)):
S=SBoson−Tr ln[M(A)]. (15.156)
Therefore the derivative has the form
∂S(A)
∂An=
∂SBoson(A)
∂An−Tr[
M−^1 (A)
∂M(A)
∂An]
. (15.157)
To evaluate the trace, we make use of the auxiliary fieldξ:
Tr[
M−^1 (A)
∂M(A)
∂An]
=
〈
ξ†M−^1 (A)∂M(A)
∂Anξ〉
(15.158)
=
∑
ijl〈
ξi∗Mij−^1 (A)[
∂M(A)
∂An]
jlξl〉
. (15.159)
In the Langevin equation we do not calculate the average over theξby generating
many random fields for each step, but instead we generate a single random Gaussian
vectorξat every MD step, and evaluate the terms in angular brackets in (15.158)
only for this configuration. Below we justify this simplification. The MD step reads