11 Transfer matrix and diagonalisation of spin chains
11.1 Introduction
In Chapters 8 and 10 we studied methods for simulating classical systems con-
sisting of many interacting degrees of freedom. In these methods, a sequence of
system configurations is generated, and from this sequence averages of physical
quantities, given as functions of the degrees of freedom (positions and momenta, or
spins), can be determined. These quantities are called mechanical quantities, and
the expressions of their expectation values are calledmechanical averages.
There exist, however, quantities that cannot be determined straightforwardly
using these methods. These quantities include free energies and chemical potentials.
The point is that these quantities are not given as anormalisedaverage, which for
mechanical quantities is replaced by an average over the configurations generated in
the simulation. In the previous chapters we have seen that it is not straightforward
to find free energies and chemical potentials using MC and MD methods (see
Section 10.5).
In this chapter we discuss a method which enables us to find free energies for
lattice spin models with very high accuracy; this is thetransfer matrix method.
This method calculates the free energy of a model defined on a strip of finite width
and infinite length directly in terms of the largest eigenvalue of a large matrix, the
transfer matrix. This matrix contains essentially the Boltzmann weights for adding
an extra row of spins to the strip. InChapter 12we shall see that the transfer matrix
is the analogue of the time evolution operator in quantum mechanics. The fact that
we can apply matrix methods to both quantum mechanics problems and statistical
problems results from the fact that statistical and quantum mechanics are intimately
related, as will be shown in the next chapter.
The transfer matrix is an operator acting in a dimension which is one lower
than the dimension of the original, classical system. In this lower dimensional
space, we must solve a quantum problem. This can be an interesting application by
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