The Kadanoff procedure, often called the block-spin method, is a lattice procedure. It is also the primary method that we use since we make specific calculations involving various Ising models. The procedure is as follows. Start with a large lattice of spins,σr, whereris a vector index labeling the spin’s position in the lattice. The action isS[σ;K], whereKare the coupling constants. The partition function is then,
Z=X
σ
exp(−S[σ;K]). (3.1)
Next, group neighboring spins together. These spins form a block. The block can be formed in all sorts of ways. For example, if we grouped nearest neighbor spins together in a block around a spin at site
r, then the grouped spins would consist ofσrand the spins located one lattice unit away from it; since the nearest neighbors are in the block, the block will look like a diamond.
Next, replace the spins in the block with a single spin at site r that summarizes the spins in the block,σ0r. There are a variety of ways to do this. One way is to take the average value of all the spins
in the block,σr0 =average of all the spins in a block centered atσrand consisting ofσrand all spins
located one lattice unit away fromσr. Another possibility is to take the majority spin in the block, aka the majority rule (e.g., if most of the spins were up, then the spin that replaces the block will also be up). So for example,σr0 =the spin that occurred the most frequently in the block centered atσr. If a tie-breaker is needed, the new spin takes a random value of the allowed spin values. See Figure3.1for an example of blocking spins in a 2D lattice.
Figure 3.1: An example of grouping spins into blocks in a 2D lattice. In this particular case of blocking, the blocks are squares and contain nine spins; the center spin is chosen for the block, resulting in a lattice with fewer spins and a greater spacing between spins. Other blocking methods could be chosen, e.g., triangles, and different amounts of spins could be included in the blocks.
The replacing of the spins in the block by one spin can be described as a transformationT[σ0|σ], which is normalized as,
X
σ0
T[σ0|σ] = 1. (3.2)
The transformationT is known as a block-spin transformation. Notice the similarity to a conditional probability distribution. This similarity will be exploited in later sections.
The new effective action is then defined as,
exp(−Sef f[σ0;K0]) =X
σ
T[σ0|σ] exp(−S[σ;K]), (3.3)
where we have taken a further step and changed the coupling constants toK0, re-writing the action in terms of local spin operators (to be explained momentarily). Note that the new partition function is the
same as the old partition function, Znew =X σ0 exp(−Sef f[σ0;K0]) =X σ0 X σ T[σ0|σ] exp(−S[σ;K]) =X σ exp(−S[σ;K]) =Z. (3.4)
Because each block of spins was replaced with a single spin (known as a block spin), we now have a lattice of spins that have fewer spins than before. Because the block spins are located at the center of each block, the resulting spins are separated by a larger distance (from center to center), thereby indicating a change of scale to lower energy. Once the spins are replaced with new spin variables, the form of the action can be re-written in terms of the interactions between sitesσ0. Hence, we have new couplingsK0
for these interactions. We now have a new actionS[σ0;K0]that has the same form of interactions as the earlier action but with effective coupling constants and defined on a lattice with fewer sites. This effective action in general will also include higher order interactions than the original action, which can be understood to have had zero coupling constant in the original action.
Continuing our example of using a block centered atr, if the lattice originally had a spacinga, it now has a spacing√2a. The new lattice will be rotated relative to the original lattice. Since the new spins are separated by a larger distance, our description of the spin lattice can be understood as a lower energy effective description in terms of the new spin variablesσ0r.
Finally, the new lattice spacing is rescaled to be equivalent to the old spacing. This rescaling will also involve rescaling the spin operators so that the action remains invariant; however, the coupling constants will be unaffected by the rescaling operation. The rescaling is done so that the effective action looks like the original action but with the different coupling constants generated by the renormalization transformation, i.e., the new spins with new couplings are now described by the old lattice. The rescaling is necessary so that fixed points can be defined after iterating the Kadanoff procedure a number of times. However, it might not be possible to iterate the Kadanoff procedure without an approximation if there are new higher order interactions in the effective action; this is a mere practical problem with the procedure arising from the inability to exactly solve models with the higher order interactions.
Aside from a constant (i.e., spin-independent) shift in the action, the only thing that will change in the action is the coefficients in front of the spin operators in the action. The partition function is held fixed, so the physics is the same (alternatively, the constant shift in the action can be absorbed into the partition function, in which case the new partition function will differ from the old by an exponential factor of the constant shift).