Computational Physics

11.3 Two-dimensional spin models 343

11.3 Two-dimensional spin models

We shall now describe the transfer matrix analysis for the two-dimensional Ising
model; generalisation to models with more spin degrees of freedom is straightfor-
ward. Using the transfer matrix method we can calculate the free energy of an Ising
model on an infinite strip of widthMwith periodic boundary conditions (PBC)
connecting the two sides of the strip – it is therefore convenient to imagine the
strip to be wrapped around a cylinder. The Hamiltonian of the Ising model in this
geometry is given by

H=

∑M

i= 1

∑∞

j=−∞

(−Jsi,jsi,j+ 1 −Jsi,jsi+1,j−Hsi,j), (11.20)

wheres1,j≡sM+1,jas a result of the PBC.
For this model, the transfer matrix is the contribution to the Boltzmann factor
of two adjacent lattice rows (the transfer matrix acts along the axis of the cylinder;
the rows are perpendicular to this direction). The possible states of the rows are the
indices of the transfer matrix. If the rows containMsites, the size of the transfer
matrix is 2M× 2 M. We represent the configuration of rowjby the state|Sj〉:

|Sj〉=|s1,js2,j...sM,j〉=|s1,j〉⊗|s2,j〉⊗...⊗|sM,j〉. (11.21)

The transfer matrix for rowsjandj+1 is found as

〈Sj|T|Sj+ 1 〉=exp

[

β

∑M

i= 1

J

(

1

2

si,jsi+1,j+

1

2

si,j+ 1 si+1,j+ 1 +si,jsi,j+ 1

)

+

β

2

H

∑M

i= 1

(si,j+si,j+ 1 )

]

, (11.22)

The eigenvalues can now be found from a numerical diagonalisation of this matrix:
its largest eigenvalues (in absolute value) determine the free energy and correlation
functions of the model on a semi-infinite strip of widthM.
Diagonalisation can be performed straightforwardly up to matrices of size
10 000×10 000 (and beyond if one uses powerful machines). However, we
need only the largest few eigenvalues and there exist special numerical meth-
ods for calculating these which, for sparse matrices, are much more efficient than
standard methods. Most convenient is Lanczos’ method, which is described in
Appendix A8.2. In this method, the matrix enters only via the multiplication with a
vector, and this multiplication can be carried out efficiently, in particular for a sparse
matrix, provided we take only the nonzero entries into account. Unfortunately, the
transfer matrix of the Ising model is not sparse at all – it follows from Eq. (11.22)
that it has a nonzero value for each pair of possible row configurationsSjandSj+ 1.