February 26, 2008

Fig. 1: An example of a block-spin transformation in
the critical Ising model. Coordinates of blocks for
statistically equivalent distributions is produced by the
code here. |

The Ising Model is a simple description for the magnetic interaction of an array of spins. It has been known for some time that, at the critical temperature, the model exhibits scale-invariance. This property renders the system exactly solvable. As a demonstration of the power of conformal invariance, we will calculate and display the 2-point correlation function for a toriodal geometry.

The Ising Model describes a collection of spins,
*s*, interacting with a nearest neighbor coupling *J*. The
Hamiltonian, or energy function, is given by

where the sum is taken over nearest-neighbor lattice
sites, and *J* is the reduced coupling constant, inversely
proportional to the temperature. Because of the overall minus sign,
spins tend to line up parallel to each other. Indeed, the
zero-temperature ground state simply consists of all spins pointing in
the same direction. If we raise the temperature slightly, however,
thermal fluctuations will force the configuration of spins to depart
from uniformity. For very low temperatures, the majority of the spins
will still be aligned. In experimental terms, the sample will have a net
magnetization. If the temperature is high, however, thermal fluctuations
will completely destroy the system's order, and the spins will be evenly
distributed up and down, with zero net magnetizaion. At some
intermediate temperature, the "critical temperature", or
*T _{c}*, the Ising system undergoes a phase transition.

Precisely at the critical temperature, the Ising model exhibits several interesting behaviors, including the property of scale invariance. This concept can be described in many ways: stated simply, scale invariance means that a configuration of spins looks the same at all levels of magnification. More formally, it means that if we trace out the short length degrees of freedom in the partition function, the hamiltonian for the remaining degrees of freedom will be precisely the same as the original H, with an identical coupling. Equivalently, the critical Ising model defines a fixed point of the Renormalization Group.

This phenomenon can be seen at work in the figures
at left. The top figure represents a typical configuration of spins in
the critical Ising model. Solid squares represent lattice sites with
spin up, and blank squares represent sites with spin down. If we "put on
blurry glasses," and group each block of 9 spins together as one unit,
either up or down, according to the majority of the individuals within
the block, we can form a new configuration of spins. The bottom figure
is the result of applying this process to the top. You can see the
original configuration, smaller and slightly "blurred," in the lower
left corner of the second image. The distribution of cluster sizes in
the two systems looks strikingly similar. The exact scale invariance of
the critical Ising model ensures that they are *statistically*
identical, in the sense that they each have equal probabilities.

Scale invariance of the critical Ising model can be
made rigorous (see the Appendix report for a partial review of the
evidence). For simplicity, however, we will instead *assume* scale
invariance (motivated by the statisticaly similarity of the two
figures), and see what we can derive. Specifically, let's see what
constraints this invariance imposes on correlation functions. The
correlation function is defined as the expectation value of the product
of two spins, separated by distance r:

Suppose that we perform a Renormalization Group
transformation with scale factor *b*. This is equivalent to
grouping together a block of spins, with * b* spins on each side,
into a single unit. Let's consider a correlation function between two of
these units, *r* blocks apart. In the blocked system, this is a
simple two-point correlation function,* G*(*r*). But we
can also write the correlator of the blocks in terms of correlations of
individual sites in the original system that make up the blocks.
Specifically, this correlator will consist of correlation functions
between the sites in block A and the sites in block B, multiplied by a
complicated factor accounting for the number of spins in each block, and
the interactions between them. Crucially, this factor is insensitive to
the distance between the blocks, it can only depend upon the scaling
parameter *b*. We write this statement as * G*(*r*)* =
f*(*b*)*G*(*br*). Now,consider compounding two
spin-blocking transformations, one with scale factor *b*, and the
other with factor *a*. We can relate the correlation function in
the doubly transformed system to the one in the original system, simply
by iterating our transformation: *G*(*r*)* =
f*(*b*)*f*(*a*)*G*(*r/ab*). But we
can also perform exactly the same shift in a single transformation, with
factor ab. Applying our rule to this case, we get *G*(*r*)*
= f*(*ab*)*G*(*r/ab*). We conclude that
*f*(*ab*)* = f*(*a*)*f*(*b*), which
implies that *f*(*b*)*=b ^{y}* for some constant

Motivated by the scale invariance of the critical
Ising model, we have heuristically derived that the correlation function
must be a simple power law. If we augment our simply analysis with the
mathematical tools of conformal field theory, we can pin down the
scaling dimension, *x*. The details of this calculation lie outside
the scope of this report, but the result (obtained by identifying the
continuum Ising model with a CFT sharing the same symmetry properties
and operator content) turns out to be * x = 1/8 *. Plugging this
value in our expression above, we easily obtain the exact spin-spin
correlation function for the 2D critical Ising model:

We can do better, though. The fact that the Ising model is scale
invariant, together with the fact that the interactions are short
ranged, actually implies invariance under a more general group of
transformations, known as the conformal transformations. Locally, such
maps are combinations of dilations, rotations and translations. For a
rotationallly invariant, translationally invariant theory like the Ising
model, only the dilation factor is important. Moreover, for a
sufficiently short-ranged interaction, only the local properties of the
transformation affect the correlations. With these considerations in
mind, it is very natural to generalize the transformation rule derived
above to the conformal case simply by replacing *b* with
*f'*(*r*), the jacobian of the transformation, i.e. the local
dilation factor:

A rigorous proof of this relation (for primary
operators) is possible using basic Conformal Field Theory [1]. Using
this result, we can find the correlation function in a variety of new
geometries by conformally transforming our coordinates, and then
applying the above equation to find G. Thus, by solving the exact
correlation function in the infinite 2D case, we have effectively found
*G* for all geometries conformally equivalent to the plane. In 2
dimensions, this result is particularly powerful, since any analytic
function generates a conformal transformation. Specifically, any
transformation of the form *z = x+iy → w*(*z*) can be
shown to be conformal.

Let's apply this to a concrete example. Consider the transformation w(z)=L/2π ln z. This maps the infinite 2D plane to a strip of width L with periodic boundary conditions in the finite dimension. Using our transformation formula, and the expression for the exact correlation function in the original geometry, we easily find the correlation function in the strip geometry:

Fig. 2: Ising correlators for strip vs. plane
geometry, function of longitudinal distance. |

If the distance between the points is small compared to L, the correlation function reduces to the form for the infinite 2D geometry. On the other hand, if the distance is large compared to L, the correlator takes the form of a negative exponential, with length scale L/2π. Conformal invariance would forbid the existence of such a length scale in the fully infinite 2D case, but for the strip geometry, the length in the finite direction sets a natural length scale for the theory. The figure at right displays a plot of the correlation function in the strip geometry (L=2), in red, and the correlation function in the plane, in green. For small values of r, the two are equivalent, since the spins don't "know" what type of geometry they reside in. But for distances comparable to L, the correlator in the strip is exponentially suppressed with respect to the correlator in the plane. This can be understood as follows: for large separations along the strip, the geometry begins to "look" one dimensional. In 1D, there is no finite temperature phase transition in the Ising model, thus we expect an exponential decay in the correlations.

© 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] J. Cardy, *Scaling and Renormalization in
Statistical Physics* (Cambridge, 1996).