Computational Physics

536 Computational methods for lattice field theories

rectangular Wilson loop. The Wilson loop correlation function is defined as

W(C)=

∫ 2 π

0

∏

n,μdθμ(n)e

βcos

[∑

n;μνθμν(n)

]

ei

∑

(n,μ)Cθμ(n)

∫ 2 π

0

∏

n,μdθμ(n)e

βcos

[∑

n;μνθμν(n)

].

A plaquette sum over theθangles for a plaquette with lower-left corner atnreduces

in the temporal gauge to:

◦

∑

n;μν

θμν(n)=θ 1 (n 0 ,n 1 )−θ 1 (n 0 +1,n 1 ).

(a) Show that in the temporal gauge the Wilson loop sum can be written as

∑

(n,μ)C

θμ(n)=

∑

(n;μν)A

◦

∑

n;μν

θμν(n)

whereAis the area covered by the plaquettes enclosed by the Wilson loop.

(b) Use this to show that the Wilson loop correlation function factorises into a

product of plaquette-terms. Defining

θP(n)=◦

∑

n;μν

θμν(n),

wherePdenotes the plaquettes, we can write:

W(C)=

∫∏

∫P∏dθPexp[βcosθP+iθP]

PdθPexp[βcosθP]

(c) Show that this leads to the final result:

W(C)=

[

I 1 (β)

I 0 (β)

]A

whereIn(x)is the modified Bessel function andAis the area enclosed by the

Wilson loop.

References

[1] R. Balian and J. Zinn-Justin, eds.,Méthodes en théorie des champs / Methods in Field Theory,

Les Houches Summer School Proceedings, vol. XXVIII. Amsterdam, North-Holland, 1975.

[2] C. Itzykson and J.-B. Zuber,Quantum Field Theory. New York, McGraw-Hill, 1980.

[3] S. Weinberg,The Quantum Theory of Fields, vols. 1 and 2. Cambridge, Cambridge University

Press, 1995.

[4] D. Bailin and A. Love,Introduction to Gauge Field Theory. Bristol, Adam Hilger, 1986.

[5] J. Zinn-Justin,Quantum Field Theory and Critical Phenomena, 3rd edn. New York, Oxford

University Press, 1996.

[6] M. Creutz,Quarks, Gluons and Lattices. Cambridge, Cambridge University Press, 1983.

[7] M. Lüscher and P. Weisz, ‘Scaling laws and triviality bounds in the lattice-φ^4 theory. 1. One-

component model in the symmetric phase,’Nucl. Phys. B, 290 (1987), 25–60.

[8] M. Lüscher, ‘Volume dependence of the energy spectrum in massive quantum field theories (I).

Stable particle states,’Commun. Math. Phys., 104 (1986), 177–206.