March 13, 2008

As my previous report showed, the assumption of scale invariance in the critical Ising model provides enormous simplifications. In particular, we showed that it makes calculation of the two point spin-spin correlation function quite trivial. In report I, however, we merely assumed the scale invariance property, motivated by the observation that the statistical properties of critical-point spin configurations seemed to be invariant under spin-block transformations. Here we will attempt to put this hypothesis on a firmer footing.

How? We need to find some testable predictions of scale invariance. Fortunately, in the previous report, we already derived a powerful prediction. Specifically, we found that scale invariance forces the spin-spin correlation function to be pure power-law. The most obvious way to test scale invariance, then, is to check the power-law dependence of the correlation function. Here we will consider three ways in which this can be tested: numerically, experimentally and analytically. We will see that the analytical results due to Onsager more or less steal the show, but the numerical and experimental results are nevertheless interesting in their own right.

Fig. 1: Comparison of the exact spin-spin correlation
function (red) with numerical approximations computed using
Monte-Carlo simulations. The code used to produce these
correlation functions can be found here. |

Given that the Ising Hamiltonian only involves nearest-neighbor interactions, computing the energy of a spin configuration is an easy numerical task. Thus the Ising model lends itself easily to computer simulations. This suggests an easy way to test the scale invariance hypothesis: simply perform a simulation to check that the correlation functions are power-law.

We applied the Monte Carlo algorithm to a 5000 by 500 spin lattice (with toroidal boundary conditions) to produce typical spin configurations. We then used the resulting spin array to compute average correlation functions in the longitudinal direction. The results are shown in the figure at left. The fat red line marks the theoretical prediction for "strip" geometry, treating the array as infinite in the longitudinal direction and periodic with circumference 500 in the vertical direction. (This functional form was derived in the previous report.) The thinner lines denote the results for different numbers of Monte Carlo iterations. As the number of iterations increases, the correlation function becomes more and more like the theoretical prediction, until the two are virtually indistinguishable (50,000 iterations).

Clearly, the agreement with numerical results is excellent. Not only is the essential behavior the same, but the agreement is close enough to confirm the exponent 1/4 (i.e. the right conformal field theory correspondence was identified).

Unfortunately, pure 2D Ising systems are difficult to find. Most real ferromagnets are of the Heisenberg form, where the spin is allowed to point along any axis, rather than being restricted to "up" or "down". Moreover, and most annoyingly, we live in a world with 3 spatial dimensions.

Both of these obstacles can be overcome somewhat with creative use of special materials. Specifically, by choosing a material for which a large degree of uniaxial anisotropy, (i.e. the spins strongly prefer align either up or down with respect to a single axis), we can produce pseudo-Ising behavior. Moreover if we choose a material formed from "sandwiches" of 2 dimensional layers, and ensure that the "filling" is sufficiently thick, the "slices of bread" will be too far apart to interact, and thus each slice will behave like an isolated 2D system.

A small number of materials with these properties
have been discovered. For example, in 1969 and 1971, Birgeneau *et
al.* [3-5] performed neutron scattering experiments to probe the
magnetic correlations in the compound K_{2}NiF_{4}. This
material consists of layers of NiF_{2} ("bread") separated by
double layers of KF, with a geometry that suitably insulates the layers
magnetically. Moreover, the compound NiF_{2} exhibits strong
uniaxial anisotropy, making it a good candidate for Ising-like
behavior.

The results of Birgeneau *et al.* were mixed.
Most crucially for our purposes, the neutron scattering results did
indicate that the correlation length diverges as the critical
temperature is approached, presumably leaving power-law correlators
precisely at T_{c}. However, the critical exponents that
describe the approach to the critical temperature did not agree well
with the Ising model predictions, therefore it is unlikely that the
correlation function would reduce to the 1/4 power dependence we have
come to expect.

What can we conclude from this experiment? Essentially, the experiment was a confirmation that spin magnetic systems can exhibit scale invariance. Unfortunately, however, it does not constitute a real, physical manifestation of an Ising system. While it does lend support to the scaling hypothesis, it does not constitute any form of proof of scale invariance in the critical Ising model.

The Ising Hamiltonian is simple. Each spin interacts with its nearest neighbors in more or less the simplest imaginable way. Evaluating partition functions analytically, even for simple systems, however, is a challenging task. Formally, it involves summing over all possible configurations of spins. For certain cases, however, this sum can be carried out explicitly. For the 1 dimensional Ising model, (i.e. a chain of spins), the partition function for a chain of N spins can be written as the limit of the trace of N powers of a 2 by 2 matrix, known as the transfer matrix. This matrix can be easily diagonalized, so that the partition function is given by the sum of the Nth powers of the two eigenvalues. Unfortunately, although the 1D model is easily solvable, it is not particularly interesting. Specifically, it doesn't exhibit scale invariance above zero temperature.

In 2 dimensions, where the interesting critical
behavior is, evaluating the partition function is vastly more difficult.
The basic transfer matrix approach applies, but the matrix is no longer 2
by 2. Instead, for a n by n configuration of spins, the matrix is
2^{n} by 2^{n}. Thus finding the partition function is
equivalent to finding the large n behavior of the trace of a matrix whose
dimensionality grows exponentially with n. Diagonalizing this matrix, and
finding the leading eigenvalues, is a formidable mathematical challenge.
Indeed, the problem went unsolved until 1944, when Lars Onsager presented
his famous exact solution.[1,2].

The results of Onsager's calculation demonstrated that
critical behavior does emerge at finite temperature. Most importantly for
our purposes, the solution demonstrates that, precisely at the critical
temperature, the correlation function is pure power law, falling off as
the 1/4 power of separation. Onsager showed us that the Ising model is one
of the beautiful problems in physics where the behavior is difficult to
probe experimentally or numerically, but possible to find *exactly*
using analytical techniques.

© 2007 D. Stanford. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

[1] L. Onsager, Phys. Rev. **65**, 117 (1944).

[2] K. Huang. *Statistical Mechanics.* (Wiley,
1963), p.349.

[3] R.J. Birgeneau *et al.*, Phys. Rev. Lett.
**22**, 720 (1969).

[4] R.J. Birgeneau *et al.*, Phys. Rev. B
**3**, 1736 (1971).

[5] C. Domb and M. S. Green. *Phase Transitions and
Critical Phenomena*, (Academic Press, 1976). pp.154-62.