15.7 Gauge field theories 519
The system can be simulated straightforwardly using the Metropolis algorithm,
but we shall use the heat-bath algorithm because of its greater efficiency. We want
the coefficientαto be not too large, as large values ofαcauseW(C)to decay very
rapidly with size, so that it cannot be distinguished from the simulation noise for
loops of a few lattice constants. From (15.138) we see thatβmust be large in that
case – we shall useβ=10. This causes the probability distributionP(θμ)for some
θμ, embedded in a particular, fixed configuration ofθμon neighbouring links, to be
sharply peaked. Therefore it is not recommended to takeθμrandom between 0 and
2 πand then accept with probabilityP(θμ)and retry otherwise, as in this approach
most trial values would end up being rejected. We shall therefore first generate a
trial value forθμaccording to a Gaussian probability distribution.
The distributionP(θ )has the form
P(θ )=exp{−β[cos(θ−θ 1 )+cos(θ−θ 2 )]} (15.139)whereθ 1 andθ 2 are fixed; they depend on theθ-values on the remaining links of
the plaquettes containingθ. The sum of the two cosines can be rewritten as
cos(θ−θ 1 )+cos(θ−θ 2 )=2 cos(
θ 1 −θ 2
2)
cos(
θ−θ 1 +θ 2
2)
. (15.140)
We define
β ̃= 2 βcos(
θ 1 −θ 2
2)
(15.141a)and
φ=θ−
θ 1 +θ 2
2(15.141b)so that our task is now to generate an angleφwith a distribution exp(−β ̃cosφ).
We distinguish between two cases: (i)β> ̃ 0. In that case the maximum of the
distribution occurs atφ=π. A Gaussian distribution centred atπand with width
σ=π/( 2
√
β) ̃ and amplitude exp(β) ̃ is always close to the desired distribution.
The Gaussian random numbers must be restricted to the interval[−π,π]. Therefore
the algorithm becomes:
REPEAT
REPEAT
Generate a Gaussian random variable−rwith width 1;
φ=σr;
UNTIL−π≤φ≤π;
φ→φ+π;