650 Critical phenomena
d. What is the probability, for largeN, that the angles will alternate trans,
gauche, trans, gauche, ...?16.6. Consider the Ising Hamiltonian in eqn. (16.3.3) for which each spinhasz
neighbors on the lattice. Each spin variableσican take on three values,
− 1 , 0 ,1. Using mean-field theory, find the transcendental equation forthe
magnetization, and determine the critical temperature of this model. What
are the critical exponents?16.7. In 1978, H. J. Maris and L. J. Kadanoff introduced an RG procedure for the
two-dimensional zero-field Ising model (Maris and Kadanoff, 1978;Chandler,
1987). This problem explores this procedure step by step, leading to the RG
equation.
a. Consider the following labeling of spins on the two-dimensional periodic
lattice shown in Fig. 16.21(a).... ...
......1
2 5 4
7 6 3
8
... ...
......1
2 4
7 3
8
(a) (b)Fig. 16.21(a) Labeling of spins on a two-dimensional lattice.
(b) Spin lattice after decimation.The first step is to sum over half of the spins on the lattice by partitioning
the summand of the partition function in such a way that each spin to
be summed over appears in only one Boltzmann factor. Show that the
resulting partition function, for one choice of the spins summed over,
corresponds to the spin lattice shown below and takes the formQ=
∑
remaining spins···
[
eK(σ^1 +σ^2 +σ^3 +σ^4 )+ e−K(σ^1 +σ^2 +σ^3 +σ^4 )]
×
[
eK(σ^2 +σ^3 +σ^7 +σ^8 )+ e−K(σ^2 +σ^3 +σ^7 +σ^8 )]
···,
whereK=βJ(see Fig. 16.21(b)).