Computational Physics

11.4 More complicated models 347

f

0.93

0.931

0.932

0.933

0.934

0.935

0.936

0 0.006 0.012 0.018 0.024

0.879

0.88

0.881

0.882

0.883

1/M^2

Figure 11.2. Free energy per site of the Ising model on a strip as a function of

1 /M^2. Values forMrun from 7 to 20. The pluses correspond to the critical point

J=0.4407; the crosses (right axis) are the data for an off-critical point:J=0.4.

The straight line through the data has slopeπ/12.

Therefore, if we plot our results for the Ising model in the form(lnλ 0 )/Mvs.
1 /M^2 we should obtain a straight line with slopeπ/12. Figure 11.2 shows that this
is indeed the case. As this is a finite size correction to the bulk free energy, the
eigenvectorsλ 0 must be determined with a precision of about six significant digits.
The magnetic exponent can be obtained by comparing the free energies for sys-
temswithperiodicandantiperiodicboundaryconditions,seeRef.[4]. Antiperiodic
means the bonds across a seam along the cylinder are antiferromagnetic. The free
energy difference then scales with the widthMas

FAP−FP=

2 π

M

x, (11.29)

wherexis the magnetic exponent – indeed,x= 1 /8 can be found in this way for
the Ising model.

11.4 More complicated models

The linear size of the transfer matrix increases exponentially with the strip width.
This puts severe limits on the system sizes that can be treated with this method. In
particular, this prohibits calculations with reasonable system size for models with
larger numbers of degrees of freedom. There exist models, such as the clock or the