Computational Physics

518 Computational methods for lattice field theories

of electrodynamics in which the interaction energy increases linearly with distance.
The high-temperature phase of the gauge theory considered so far is always confin-
ing, so this does not include this short-distance decay of the interaction, commonly
called ‘asymptotic freedom’ (G. ’t Hooft, unpublished remarks at the Marseille
conferenceongaugetheories, 1972 ;seealsoRefs.[ 6 , 42 , 43 , 45 ]).Weshallstudy
the more complex gauge theory which is believed to describe quarks later; this
theory is called ‘quantum chromodynamics’ (QCD).
The lattice version of quantum electrodynamics using variablesθμranging from
0to2πis often calledU( 1 )lattice gauge theory because the angleθμparametrises
the unit circle, which in group theory is calledU( 1 ). Another name for this field
theory is ‘compact QED’ because the values assumed by the variableθμform a
compact set, as opposed to the noncompactAμfield of continuum QED. Compact
QED can be formulated in any dimension, and in the next section we discuss an
example in 1 space+1 time dimension.

15.7.3 A lattice QED simulation

We describe a QED lattice simulation for determining the inter-fermion potential.
We do this by determining the Wilson loop correlation function described in the
previous section. Only the gauge field is included in the theory – the fermions
have a fixed position, and the photons exchanged between the two cannot generate
fermion–antifermion pairs (vacuum polarisation). This is equivalent to assigning
an infinite mass to the fermions. We use a square lattice with periodic boundary
conditions.
We consider the (1+1)-dimensional case. This is not a very interesting theory
by itself – it describes confined fermions, as the Coulomb potential in one spatial
dimension is confining:
V(x)∝|x|, (15.136)

but we treat it here because it is simple and useful for illustrating the method. The
theory can be solved exactly (see Problem 15.6)[42]: the result is that the Wilson
loop correlation function satisfies the area law

W=exp(−αA) (15.137)

(Ais the area enclosed within the loop) with the proportionality constantαgiven
in terms of the modified Bessel functionsIn:

α=−ln

[

I 1 (β)

I 0 (β)

]

; (15.138)

βis the coupling constant (inverse temperature). In fact the area law holds only for
loops much smaller than the system size; deviations from this law occur when the
linear size of the loop approaches half the system size.