Computational Physics

15.4 Algorithms for lattice field theories 479

function〈φnφl〉. The Fourier transform of this correlation function can be found
quite easily:

〈φnφl〉=

1

L^2 d

∑

k,k′

〈φkφk′〉eik·neik

′·l

; (15.52a)

〈φkφk′〉=

L^2 d

Kk,−k

δk,−k′. (15.52b)

Taking the small-klimit in(15.49)and(15.52)leads to the form(15.31), as it should
be. Takingk=0, we find

〈φk^2 = 0 〉=

〈(

∑

n

φn

) 2 〉

=Ldζ/m^2 R, (15.53)

where the factorζon the right hand side represents the square of the scaling factor of
the field – from(15.43),ζ=ad−^2. This equation enables us therefore to determine
ζ/mRin a simulation simply by calculating the average value of〈^2 〉,=

∑

nφn.
We have seen that according to Wick’s theorem, the correlation functions to
arbitrary order for free fields can always be written as sums of products of two-point
correlation functions. If we switch on theφ^4 interaction, we will note deviations
from this Gaussian behaviour to all higher orders. Renormalisation ideas suggest
that it should be possible to express all higher order correlation functions in terms
of second and fourth order correlation functions, if the arguments of the Green’s
function are not too close (that is, much farther apart than the cut-offa). The second
order Green’s functions are still described by the free field form (15.52), but withm
in the kernelKk,k′being replaced by arenormalised mass,mR. The deviations from
the Gaussian behaviour manifest themselves in fourth and higher order correlation
functions. Therefore a natural definition of the renormalised coupling constantgRis

gR=

〈^4 〉− 3 〈^2 〉^2

〈^2 〉^2

(15.54)

where=

∑

nφn.

(^5) Equations(15.53)and(15.54)are used below to measure the
renormalised mass and coupling constant in a simulation.
15.4.1 Monte Carlo methods
The problem of calculating expectation values for the interacting scalar field theory
is exactly equivalent to the problem of finding expectation values of a statistical field
theory. Therefore we can apply the standard Monte Carlo algorithms ofChapter 10
(^5) This renormalisation scheme corresponds to defining the renormalised coupling constant as the four-point
one-particle irreducible (OPI) Green’s function in tree approximation at momentum zero [ 2 , 4 , 5 ].