Computational Physics

11.2 The one-dimensional Ising model and the transfer matrix 339

itself, even without reference to a transfer matrix relating the quantum problem to
a classical one. We shall discuss one-dimensional quantum problems extensively
in this chapter. Not too long ago, a new method was developed for treating this
problem very efficiently. It is based on renormalisation and on the density matrix –
hence its name,density matrix renormalisation group. We shall consider a few
applications of this method in the last sections of this chapters.
In the next section we solve the one-dimensional Ising model exactly using the
transfer matrix. We shall derive the free energy and the pair correlation function
from this matrix. In the following section we shall describe the transfer matrix
for two-dimensional spin models and see how transfer matrix techniques allow
for numerically exact solutions of two-dimensional models on an infinite strip
of finite width. This, in combination with finite-size scaling methods, makes the
transfer matrix method very useful for obtaining accurate data for two-dimensional
statistical models. For details concerning the numerical transfer matrix method, see
Ref.[ 1 ].
Later sections are then dedicated to numerically diagonalising quantum chains
(Section 11.5), to quantum renormalisation methods (11.6), and a particular form
of this, the density matrix renormalisation group method (11.7).

11.2 The one-dimensional Ising model and the transfer matrix

In this section we present the exact solution of the one-dimensional Ising model.
To arrive at the solution we must introduce a new concept, thetransfer matrix,
whose largest eigenvalue determines the partition function. The one-dimensional
Ising model (see also Section 7.2.2) consists of a chain of spins numbered 1 through
L; the spins assume valuessi=±, and the Hamiltonian is given by

H[{si}] =

∑L

i= 1

(−Jsisi+ 1 −Hsi). (11.1)

Jis the coupling strength between the spins, and the external fieldHfavours spins
of one particular sign. Periodic boundary conditions are imposed by identifying the
spins at sites 1 andL+1. The partition function is

Z=

∑

{si}

exp

[

β

∑L

i= 1

(Jsisi+ 1 +Hsi)

]

. (11.2)

The sum runs over all possible spin configurations{si}.
The exponent can be written as a product over all nearest neighbour pairs:

Z=

∑

{si}

∏L

i= 1

exp[βJsisi+ 1 +βHsi]. (11.3)