1549380323-Statistical Mechanics Theory and Molecular Simulation

Renormalization Group 637

Requiring that the Hamiltonian itself remain unchanged under an RG transforma-
tion is stronger than simply requiring that the functional form of the Hamiltonian be
preserved. When the Hamiltonian is unchanged by the RG transformation, then the
parametersK′obtained via eqn. (16.9.6) are unaltered, implying that

K=R(K). (16.9.7)

A pointKin parameter space that satisfies eqn. (16.9.7) is called afixed pointof the
RG transformation. Eqn. (16.9.7) indicates that the Hamiltonian of an ordered phase
emerges from a fixed point of the RG equations.

16.9.1 RG example: The one-dimensional Ising model

In the zero-field limit, the Hamiltonian for the one-dimensional Ising model is

H 0 ({σ}) =−J

∑N

i=1

σiσi+1. (16.9.8)

Let us define a dimensionless Hamiltonian Θ 0 =βH 0 and a dimensionless coupling
constantK=βJso that Θ 0 ({σ}) =−K

∑N

i=1σiσi+1. With these definitions, the
partition function becomes

Q(N,T) = Trσe−Θ^0 ({σ}). (16.9.9)

Consider the simple block spin transformation illustrated in Fig. 16.14.The figure

σ 1 σ 2 σ 3 σ 4 σ 5 σ 6 σ 7 σ 8 σ 9

σ 1 σ 2 σ 3 σ 1 σ 2 σ 3 σ 1 σ 2 σ 3

σ’ 1 σ’ 2 σ’ 3

Fig. 16.14Example of the block spin transformation applied to the one-dimensional Ising
model. The three spins that result are shown below.

shows the one-dimensional spin lattice with two different indexing schemes: The upper
scheme is a straight numbering of the nine spins in the figure, while thelower scheme
numbers the spins in each block. As Fig. 16.14 indicates, the block spintransforma-
tion employed in this example replaces each block of three spins with a single spin
determined solely by the spin at the center of the block. Thus, for the left block, the
new spinσ′ 1 =σ 2 , for the middle block,σ′ 2 =σ 5 ,σ′ 3 =σ 8 , and so forth. Though not
particularly democratic, this block spin transformation should be reasonable at low