Ising model renormalization

Keywords: ising model renormalization
Description: Example: The one-dimensional Ising model

The figure shows the one-dimension spin lattice numbered in two different ways - one a straight numbering and one using blocks of three spins, with spins in each block numbered 1-3. The block spin transformation to be employed here is that the spin of a block will be determined by the value of the spin in the center of the block. Thus, for block 1, it is the value of spin 2, for block 2, it is the value of spin 5, etc. This rather undemocratic choice should be reasonable at low temperature, where local ordering is expected, and spins close to the center spin would be expected to be aligned with it, anyway. The transformation function, T for this case is

The idea is then to find a K ' such that when the sum over and are performed, the new interaction between and is of the form . which preserves the functional form of the old Hamiltonian. The sum over and is

When summed over and . most terms in the above expression will cancel, yielding the following expression:

where the last expression puts the interaction into the original form with a new coupling constant K '. One of the possible choices for the new coupling constant is

This, then, is the RG equation for this particular block spin transformation. With this identification of K ', the new Hamiltonian can be shown to be

Thus, apart from the additional term, the new Hamiltonian is exactly the same functional form as the original Hamiltonian but with a different set of spin variables and a different coupling constant.

The transformation can now be applied to the new Hamiltonian, yielding the same relation between the new and old coupling constants. This is equivalent to iterating the RG equation. Since the coupling constant K depends on temperature through K =J /kT. the purpose of the iteration would be to find out if, for some value of K. there is an ordered phase. In an ordered phase, the transformed lattice would be exactly the same as the old lattice, and hence the same coupling constant and Hamiltonian would result. Such points are called fixed points of the RG equations and are generally given by the condition

The fixed points correspond to critical points. For the one-dimensional Ising model, the fixed point condition is

Since K is restricted to . the only solutions to this equation are x =0 and x =1, which are the fixed points of the RG equation.

Since K =J /kT. at high T. and . At low temperature, and . Viewing the RG equation as an iteration or recursion of the form

if we start at . each successive iteration will yield 1. However, for any value of x less than 1, the iteration eventually goes to 0 in some finite (though perhaps large) number of iterations of the RG equation. This can be illustrated pictorially as shown below:

The iteration of the RG equation produces an RG flow through coupling constant space. The fixed point at x =1 is called an unstable fixed point because any perturbation away from it, if iterated through the RG equation, flows away from this point to the other fixed point, which is called a stable fixed point. As the stable fixed point is approached, the coupling constant gets smaller and smaller until, at the fixed point, it is 0. The absence of a fixed point for any finite, nonzero value of temperature tells us that there can be no ordered phase in one dimension, hence no critical point in one dimension. If there were a critical point at a temperature . then, at that point, long range order would set it, and there would be a fixed point of the RG equation at . Note, however, that at T =0, . there is perfect ordering in one dimension. Although this is physically meaningless, it suggests that ordered phases and critical points will be associated with the unstable fixed points of the RG equations.

Another way to see what the T =0 unstable fixed point means is to study the correlation length. The correlation is a quantity that has units of length. However, if we choose to measure it in units of the lattice spacing, then the correlation length will be a number that can only depend on the coupling constant K or .

Under an RG transformation, the lattice spacing increases by a factor of 3 as a result of coarse graining. Thus, the correlation length, in units of the lattice spacing, must decrease by a factor of 3 in order for the same physical distance to be maintained:

Note, also, that at very low T. where K is large, motion toward the stable fixed point is initially extremely slow. To see this, rewrite the RG equation as

Notice that the term in brackets is extremely close to 1 if K is large. To leading order, we can say

Since the interactions between blocks are predominantly mediated by interactions between boundary spins, which, for one dimension, involves a single spin pair between blocks, we expect that a block spin transformation in 1 dimension yields a coupling constant of the same order as the original coupling constant when T is low enough that there is alignment between the blocks and the new lattice is similar to the original lattice. This is reflected in the above statement.

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