3469 Assignment 4 — Solution
Mike Peardon (mjp@maths.tcd.ie)
Michaelmas 2015-16
Normalisations were a bit of a problem with the homework sheet! Submissions with any sensible convention were
given full credit. I posted code which updates with the action not defined with the 1/2 factor in front of it. Also, the
magnetisation is more sensibly defined as
q
M (σ) =
X
2
1
qfk (q) − 1
q(q − 1)
(1)
k=1
so that as β → ∞, hM i → 1. The autocorrelation function can be modelled using
1
0.8
<f>
0.6
q=2
q=3
q=4
q=5
0.4
0.2
0
0.6
0.8
1
1.2
1.4
1.6
β
Figure 1: The magnetisation (normalisation fixed) as a function of inverse temperature (normalisation fixed).
C(τ ) = A exp(−τ /τA )
1
(2)
Autocorrelation function, C(τ)
1
0.8
0.6
0.4
0.2
0
0
5
10
Markov time, τ
15
20
Figure 2: The autocorrelation function of the magnetisation on a 10 × 10 grid for q = 5, β = 0.6. The red line is a fit
to Eqn. 1, giving τA = 2.1.
2