15.7 Gauge field theories 523
Integrating out the fermion part of the path integral using(15.119)leads to a path
integral defined entirely in terms of bosons:
∫
[DA][Dψ ̄][Dψ]e−[SBoson(A)+ψ ̄M(A)ψ]=
∫
[DA]det[M(A)]e−LBoson(A)=
∫
[DA]e−LBoson(A)+ln[det(M(A))]
(15.148)(the inverse temperatureβis included in the action). Although the determinant
ofM(A)is real and usually positive,M(A)is not necessarily a positive definite
Hermitian matrix (a positive definite matrix has real and positive eigenvalues). It is
therefore sometimes useful to consider the matrix
W(A)=M†(A)M(A), (15.149)in terms of which the path integral can be written as
∫
[DA]e−LBoson(A)+
1
2 ln[det(W(A))]. (15.150)Now suppose that we want to perform a Metropolis update of theA-field. The
acceptance probability for a trial changeA→A′is
PAccept(A→A′)=e−SBoson(A
′)+SBoson(A)det[M(A′)]
det[M(A)]=e−SBoson(A′)+SBoson(A)√
det[W(A′)]
det[W(A)]. (15.151)
This is very expensive to evaluate since we must calculate the determinant of the
very large matrixM(A)[orW(A)] at every step. A clever alternative follows from
the observation that if the fieldAchanges on one site only (as is usually the case),
very few elements of the matrixM(A)change, which allows us to perform the
calculation more efficiently [50]. Another interesting suggestion is that of Bhanot
et al.who propose to evaluate the fraction of the determinants as follows [51]:
det[W(A′)]
det[W(A)]=
∫
[Dφ][Dφ∗]e−φ
†W(A)φ
∫
[Dφ][Dφ∗]e−φ†W(A′)φ, (15.152)
whereφis a boson field for which we can use the algorithms given earlier in this
chapter. DefiningW=W(A′)−W(A), we can express the ratio in terms of an
expectation value:
det[W(A′)]
det[W(A)]
=〈exp(φ†Wφ)〉A′= 1 /〈exp(−φ†Wφ)〉A. (15.153)