Computational Physics - Department of Physics

13.7 Correlation Functions and Further Analysis of the Ising Model 447

kBT/J= 2. 4

kBT/J= 1. 5

t

|M

|

0 100 200 300 400 500 600 700 800

1.2

1

0.8

0.6

0.4

0.2

0

-0.2

Fig. 13.10Absolute value of the mean magnetisation as function of timet. Time is represented by the number
of Monte Carlo cycles. The calculations were performed witha 100 × 100 lattice for temperatureskBT/J=

1. 5 andkBT/J= 2. 4. For the lowest temperature, an ordered start configurationwas chosen, while for the
   temperature close toTC, a disordered configuration was used.

the temperature close tokBTC/J≈ 2. 269 it takes more time to reach the actual steady state.
It seems thus that the time needed to reach a steady state is longer for temperatures close
to the critical temperature than for temperatures away. In the next subsection we will define
more rigorously the equilibration timeteqin terms of the so-called correlation timeτ. The
correlation time represents the typical time by which the correlation function discussed in
the next subsection falls off. There are a number of ways to estimate the correlation timeτ. It
is normal to set the equilibration timeτ=teq. The correlation time is a measure of how long it
takes the system to get from one state to another one that is significantly different from the
first. Normally the equilibration time is longer than the correlation time, mainly because two
states close to the steady state are more similar in structure than a state far from the steady
state.
Here we mention also that one can show, using scaling relations [79], that at the critical
temperature the correlation timeτrelates to the lattice sizeLas

τ∼Ld+z,

withdthe dimensionality of the system. For the Metropolis algorithm based on a single spin-
flip process, Nightingale and Blöte obtainedz= 2. 1665 ± 0. 0012 [86]. This is a rather high
value, meaning that our algorithm is not the best choice whenstudying properties of the
Ising model nearTC.
We can understand this behavior by studying the developmentof the two-dimensional Ising
model as function of temperature. The first figure to the left shows the start of a simulation
of a 40 × 40 lattice at a high temperature. Black dots stand for spin downor− 1 while white
dots represent spin up (+ 1 ). As the system cools down, we see in the picture to the right that
it starts developing domains with several spins pointing inone particular direction.
figure=figures/pict4.ps,width=height=6cm figure=figures/pict2.ps,width=height=6cm