526 Computational methods for lattice field theories

a factor det[M(A)]^2 which can be written as

det[M(A)]^2 =

∫

[Dφ][Dφ∗]e−φ

†[M(A)†M(A)]− (^1) φ

∫

[Dφ][Dφ∗]e−φ

†[W(A)]− (^1) φ
(15.164)
withW(A)defined in (15.149). Note that we need an even number of fermion
flavours here, because we cannot simply replace the matrixW(A)by its square root
in the following algorithm (see also the beginning of this subsection). This partition
function is much more convenient than(15.163)for generating MC configurations
of the fieldφ. This is done by an exact heat-bath algorithm, in which a Gaussian
random fieldξnis generated, and the fieldφis then found as
M(A)ξ=φ. (15.165)
For staggered fermions (see previous subsection) it turns out thatW(A) =
M(A)†M(A)couples only even sites with even sites, or odd sites with odd sites
[53, 54]. Therefore the matrixW(A)factorises into an even–even (ee) and an
odd–odd one (oo), so that we can write
det[W(A)]=det[W(A)ee]det[W(A)oo]. (15.166)
The matricesW(A)eeandW(A)ooare identical – therefore we have:
det[M(A)]=

∫

[Dφ][Dφ∗]e−φ

†[Wee(A)]− (^1) φ

. (15.167)

The matrixWeeis Hermitian and positive; it can be written asWee=Meo†(A)Moe(A),
where the two partial matricesMeoandMoeare again identical, so we can use the
heat-bath algorithm as described, withM(A)replaced byMeo(A).
The full path integral contains only integrations over boson fields:

Z=

∫

[DA][Dφ][Dφ∗]e−SBoson(A)−φ

†W− (^1) (A)φ
(15.168)
where subscripts ee forWshould be read in the case of staggered fermions. We want
to formulate a molecular dynamics algorithm for the boson fieldA, but generate the
auxiliary field configurationsφwith an MC technique. This procedure is justified
because the MD trajectory between an acceptance/rejection decision is reversible,
and the acceptance/rejection step ensures detailed balance.
We assign momenta to the boson fieldAonly:

Z=

∫

[DA][Dφ][Dφ∗][DP]e−^1 /^2

∑

nP^2 n(x)−SBoson(A)−φ†W−^1 (A)φ. (15.169)