Computational Physics

520 Computational methods for lattice field theories

0.01

0.1

1

0 5 10 15 20 25 30 35 40 45 50

W(C)

Area

16 × 16

32 × 32

Figure 15.8. The Wilson loop correlation function as a function of the enclosed

area for (1+1)-dimensional lattice QED. Note that the vertical scale is logarithmic,

so that the straight line is compatible with the area law. The values were determined

in a heat-bath simulation using 40 000 updates (first 2000 rejected).

Accept this trial value with probability

exp{−β ̃[cosφ+ 1 −(φ−π)^2 /( 2 σ^2 )]};

UNTIL Accepted;

θ=φ+

θ 1 +θ 2

2

.

(ii) Ifβ< ̃ 0 then the distribution is centred aroundφ=0. In that case, we do not
shift the Gaussian random variable overπ. The reader is invited to work out the
analogue of the algorithm for case (i).
In the simulation we calculate the averages of square Wilson loops, given in
Eq. (15.131)(it should be emphasised that for the area law it is not required to
haveT R). This is done by performing a loop over all lattice sites and cal-
culating the sum of theθμover the square loop with lower left corner at the
current site. The expectation values for different square sizes can be calculated
in a single simulation.Figure 15.8shows the average value of the Wilson loop
correlation functions as a function of the area enclosed by the loop for a 16× 16
anda32×32 lattice. The straight line is the theoretical curve with slopeαas in
(15.138). From this figure it is seen that the area law is satisfied well for loops which
are small with respect to half the lattice size. By implementing free boundary con-
ditions, the theoretical curve can be matched exactly, but this requires a little more
bookkeeping.