Computational Physics

Exercises 535

The relation between the coarse coupling constantsJNN′,KNN′and the fine ones is

JNN′=

∑

nn′|NN′

[Jnn′cos(φn−φn′)+Knn′sin(φn−φn′)];

KNN′=

∑

nn′|NN′

[Knn′cos(φn−φn′)−Jnn′sin(φn−φn′)];

seeFigure 15.5.

(a) Verify this.

(b) [C] Write a multigrid Monte Carlo program for theXYmodel. Calculate the

helicity modulus using(15.97)and and check the results by comparison with

Figure 15.4.

15.5 In this problem we verify that the SOR method for the free field theory satisfies
detailed balance.
(a) Consider a siten, chosen at random in the SOR method. The probability
distribution according to which we select a new value for the fieldφnin the heat
bath method is
ρ(φn)=exp[−a(φn−φ ̄n)^2 / 2 ],
whereφ ̄nis the average value of the field at the neighbouring sites. In the SOR
method we choose for the new valueφ′nat siten:

φ′n=φ ̃n+r

√

ω( 2 −ω)/a,

where

φ ̃n=ωφ ̄n+( 1 −ω)φn

and whereris a Gaussian random number with standard deviation 1. Show that

this algorithm corresponds to a transition probability

T(φn→φn′)∝exp

[

−

a

ω( 2 −ω)

(φn′−φ ̃n)^2

]

.

(b) Show that this transition probability satisfies the detailed balance condition:

T(φn→φn′)

T(φ′n→φn)

=

exp[−a(φ′n−φ ̄n)^2 / 2 ]

exp[−a(φn−φ ̄n)^2 / 2 ]

.

15.6 The Wilson loop correlation function for compact QED in (1+1) dimensions can be
solved exactly. Links in the time direction have indexμ=0, and the spatial links
haveμ=1. We must fix the gauge in order to keep the integrals finite. The so-called
temporal gaugeturns out convenient: in this gauge, the anglesθ 0 living on the
time-like bonds are zero, so that the partition sum is a sum over anglesθ 1 on spatial
links only. Therefore there is only a contribution from the two space-like sides of the