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Block-spin transformation in one dimension. The block spins only have nearest-neighbor interactions.
Fig. 18.2
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The partition function for an N-spin chain is the trace of the Nth power of the transfer matrix P, as shown in (14.81). If we want to describe the system in terms of 2-spin blocks, the same partition function should be re-expressed as the trace of a new transfer matrix P' raised to the power N /2. This is achieved through a trivial rewriting of the partition function: (18.11) where
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The limits on u and v come from the conditions J > 0 and h > o. The former restricts us to the ferromagnetic case. The second condition occasions no loss in generality because the model is invariant under h ~ - h. The transfer matrix for block spins is
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We demand that P' have the same form as P:
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u'v' u'
u' v' u'
which defines the parameters u' and v' in the block-spin system. A new parameter C must be introduced, because to match (18.13) with (18.14) requires matching three matrix elements, which is generally impossible with only two variables u' and v'. With C, we have three unknowns to satisfy the three
Cu' = v + v
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The block-spin transformation can be regarded as a mapping (u, v) ~ (u', v') in parameter space. By carrying out the block-spin transformation repeatedly, an initial point (u, v) in parameter space generates a sequence of points, which may be connected to form a trajectory. By doing this for different initial points, we obtain the "flow diagram" of Fig. 18.3. A salient feature of the mapping are the "fixed points"-values (u, v) that remain unchanged under the transformation:
u = 0, u = 1,
v= 1
all v
( 00
(0 interaction, any field)
The fixed point (0,1) is unstable in the sense that any point in its neighborhood will tend to go away from it under a block-spin transformation. The fixed points on the line u = 1 are stable. At the fixed points the correlation length ~ is invariant under a scale change, and therefore can only be or 00. The line u = corresponds to T = 0, with ~ = 00 at (0,1). The line u = corresponds to T = 00, with ~ = all along this line. Unfortunately the fixed point (0,1) is inaccessible. You are either already there or you go away from it. This expresses the fact that the system has no critical point.
Line of stable fixed points
Flow diagram of one-dimensional Ising model, showing how the coupling constant { and the external field H change under successive block-spin transformations.
Fig. 18.3
The operation of coarse-graining followed by rescaling is called a "renormalization-group" (RG) transformation, of which the block-spin transformation in an Ising model is an example. We now give a formal and general definition of the latter. This is possible owing to the simplicity of the model. * But the results are very instructive, and the concepts illustrated in this case are useful in more general models. Consider an Ising model with N spin variables Si = 1 defined on sites of a d-dimensional cubic lattice. We will have to include the most general types of interactions, in order that a block-spin transformation, which may generate arbitrarily complicated interactions, can be described as a mapping in parameter space. To this end, let I a denote an arbitrary set of site labels. Let Sa denote the product of all spins on the sites in I a :
We follow Th. Niemeijer and J. M. J. van Leeuwen, in Phase Transitions and Critical Phenomena, Vol. 6, C. Domb and M. S. Green, eds. (Academic Press, New York, 1976), pp. 425-505.