13.7 Correlation Functions and Further Analysis of the Ising Model 445
" 100 × 100 "" 80 × 80 "" 40 × 40 "" 10 × 10 "kBTχ2 2.1 2.2 2.3 2.4 2.5 2.6250200150100500Fig. 13.8Susceptibility per spin as function of the lattice size for the two-dimensional Ising model. Note that
we have computed the susceptibility asξ= (〈M^2 〉−〈|M|〉^2 )/kbT.
The Metropolis algorithm is not very efficient close to the critical temperature. Other algo-
rihms such as the heat bath algorithm, the Wolff algorithm and other clustering algorithms,
the Swendsen-Wang algorithm, or the multi-histogram method [84,85] are much more effi-
cient in simulating properties near the critical temperature. For spin models like the class
of higher-order Potts models discussed in section 13.8, theefficiency of the Metropolis algo-
rithm is simply inadequate. These topics are discussed in depth in the textbooks of Newman
and Barkema [79] and Landau and Binder [80].
13.7 Correlation Functions and Further Analysis of the Ising Model
13.7.1Thermalization.
In the code discussed above we have assumed that one performsa calculation starting with
low temperatures, typically well belowTC. For the Ising model this means to start with an
ordered configuration. The final set of configurations that define the established equilibrium
at a givenT, will then be dominated by those configurations where most spins are aligned
in one specific direction. For a calculation starting at lowT, it makes sense to start with an
initial configuration where all spins have the same value, whereas if we were to perform a
calculation at highT, for example well aboveTC, it would most likely be more meaningful
to have a randomly assigned value for the spins. In our code example we use the final spin
configuration from a lower temperature to define the initial spin configuration for the next
temperature.
In many other cases we may have a limited knowledge on the suitable initial configurations
at a givenT. This means in turn that if we guess wrongly, we may need a certain number of
Monte Carlo cycles before we reach the most likely equilibrium configurations. When equilib-
rium is established, various observable such as the mean energy and magnetization oscillate
around their mean values. A parallel is the particle in the box example discussed in chap-