Computational Physics

516 Computational methods for lattice field theories

C

t

x

t

x

(a) (b)

Figure 15.7. The Wilson loop on a two-dimensional square lattice. (a) A general

Wilson loop. (b) A two-fermion loop in a gauge field theory with infinite-mass

fermions.

so-calledWilson loopcorrelation function:

W(C)=

〈

∏

n,μC

eiθμ(n)

〉

, (15.131)

where the product is over all linksn,μbetween sitenand its neighbourn+μlying
on the closed loopC(seeFigure 15.7)[ 41 ].
The Wilson loop correlation function has a physical interpretation. Suppose we
create at some timetia fermion–antifermion pair, which remains in existence at
fixed positions up to some timetf, at which the pair is annihilated again. Without
derivation we identify the partition function of the gauge field in the presence of the
fermion–antifermion pair with the Wilson loop correlation function in Figure 15.7b
timesthevacuumpartitionfunction–foradetailedderivationseeRefs.[ 6 , 42 , 43 ].
Now let us stretch the loop in the time direction,T=tf−ti→∞. The effective
interaction between two electrons at a distanceRis given by the difference between
the ground state energy in the presence of the fermion-antifermion pair (which we
denote by 2f) and the ground state energy of the vacuum:

V(R)=

〈

ψG(2f)|H|ψG(2f)

〉

2f

−

〈

ψG(vac)|H|ψG(vac)

〉

vac

. (15.132)

This expression can, however, be evaluated straightforwardly in the Lagrangian
picture. We have

e−TV(R)=

Z(C)

Z

=W(C) (15.133)

whereCis the rectangular contour of sizeT(time direction) andR(space direction);
Z(C)is the partition function evaluated in the presence of the Wilson loop; and