3469 - Assignment 4
Mike Peardon (mjp@maths.tcd.ie)
Instructions
Complete both parts of the assignment by Monday 25th January. You should submit the
C++ source files and a short (about 2 page) pdf file containing your solutions and plots via
tcd.blackboard.com. Do not submit executables. Marks will be deducted for late submissions and any submission more than a week late will not be marked.
1
The 2d q-state Potts model
The Potts model introduces an integer-valued spin, σx,y ∈ [1, 2, ...q] on every site of a regular
L × L grid. The Ising model is the case q = 2. The grid has periodic boundary conditions such
that σx+L,y = σx,y and σx,y+L = σx,y . Define the susceptibility M using
q
X
1
(qfk (σ) − 1)2 ,
M (σ) =
2
(q − 1) k=1
where fk is the total fraction of spins on the grid that take value k;
1 X
fk (σ) = 2
δk,σx,y .
L x,y
Thermodynamic expectation values of an observable O are given by
X
X
1X
e−βS(σ)
with S(σ) =
hOi =
O(σ)
(1 − δσx,y σx0 ,y0 ),
Z
2
0
0
x,y
σ∈Σ
(x ,y )∈n(x,y)
and where Z is the constant that ensures h1i = 1. For this homework, these expectation values
should be evaluated with importance sampling Monte Carlo using a Markov Chain method
such as the Metropolis algorithm to generate the sampling ensemble needed.
The problem with Markov Chain Monte Carlo is that the configurations near each other
on the chain are not statistically independent. These autocorrelations should be monitored to
establish how many Metropolis updates are needed to give effectively independent samples.
The autocorrelation function of observable X after update τ is
P
2
t Xt Xt+τ − X̄
CX (τ ) = P
2
2
t Xt − X̄
1. On a grid of size 16 × 16, measure the thermodynamic expectation value of this magnetisation for a range of values of β ∈ [0.2, 1.0] for q = 2, 3, 4 and 5. Make a plot of
your data, including a measure of the statistical uncertainty.
2. For a grid of size 10×10, compute the autocorrelation function for fk for β = 0.6, q = 5.
Approximately how many updates of the whole grid are required to give an independent
sample (where independent means the autocorrelation is below 10%).