Computational Physics

15.4 Algorithms for lattice field theories 483

g(

r)

0

1

2

3

4

5

6

7

8

0 1 2 3 4 5 6 7 8

8 × 8

12 × 12

16 × 16

24 × 24

r

Figure 15.1. The correlation function of the interacting scalar field theory for

various lattice sizes. The mass and coupling parameters for the different lattice

sizes have been scaled such as to keep the physical lattice size constant. Thex-axis

has been scaled accordingly. The values have been determined using the histogram

method described in the text.

present results for a field theory which on an 8×8 lattice is fixed by the parameter
valuesm=0.2 andg=0.04. Note that bothmandgshould be close to the
critical line (which passes throughm=0,g=0) to obtain long correlation lengths
justifying the discretisation. According to (15.43) we usem=0.1 andg=0.01
ona16×16 lattice, etc. The results are obtained using a heat-bath algorithm using
30 000 steps (8×8) to 100 000 steps (24×24).Figure 15.1shows the correlation
functions for various lattice sizes, obtained using the heat-bath algorithm. The
horizontal axis is scaled proportional to the lattice constant (which is obviously
twice as large for an 8×8 lattice as for a 16×16 lattice). The vertical axis is scaled
for each lattice size in order to obtain the best collapse of the various curves. It is
seen that for length scales beyond the lattice constant, scaling is satisfied very well.
Only on very small length scales do differences show up, as is to be expected.
The correlation functions obtained from the simulations can be compared with
the analytic form, which can be obtained by Fourier transforming

Gk,−k=

Z

4

∑

μsin(kμ/^2 )+m

2

R

(15.59)

(seeSection 15.3andEq. (15.53)). The parametermRthat gives the best match to
the correlation function obtained in the simulation (with an optimal value of the