1549380323-Statistical Mechanics Theory and Molecular Simulation

636 Critical phenomena

T(σ′;σ 1 ,...,σ 9 ) =







1 σ′

∑ 9

i=1σi>^0

0 otherwise

. (16.9.2)

This function ensures that when the spin sum over the original lattice is performed,
only those terms that conform to the rule of the block spin transformation are nonzero.
That is, the only nonzero terms are those for whichσ′and

∑ 9

i=1σihave the same sign.
When eqn. (16.9.2) is inserted into eqn. (16.9.1), the functionT(σ′;σ 1 ,...,σ 9 ) projects
out those configurations that are consistent with the block spin transformation rule,
while the sum over the old spin variablesσ 1 ,...,σNleaves a function of only the new
spin variables{σ′ 1 ,...,σN′′}. Note thatT(σ′;σ 1 ,...,σ 9 ) satisfies the property
∑

σ′=± 1

T(σ′;σ 1 ,...,σ 9 ) = 1, (16.9.3)

which means simply that only one of the two values ofσ′can satisfy the block spin
transformation rule. The new spin variables{σ′}can now be used to define a new
partition function. To see how this is done, let the Hamiltonian of the new lattice be
defined according to

e−βH

′ 0 ({σ′})

= Trσ

[

∏

blocks

T(σ′;σ 1 ,...,σ 9 )

]

e−βH^0 ({σ}), (16.9.4)

which follows from eqn. (16.9.3). Summing both sides of eqn. (16.9.4) over the relevant
spin variables yields
Trσ′e−βH

′ 0 ({σ′})

= Trσe−βH^0 ({σ}). (16.9.5)

Eqn. (16.9.5) states that the partition function is preserved by the block spin trans-
formation and, consequently, so are the equilibrium properties.
If the block spin transformation is devised in such a way that the functional form
of the Hamiltonian is preserved, then the transformation can be iterated repeatedly on
each new lattice generated by the transformation: each iterationwill generate a system
that is statistically equivalent to the original. Importantly, in a truly ordered state,
each iteration will produce precisely the same lattice in the thermodynamic limit, thus
signifying the existence of a critical point. If the functionalformof the Hamiltonian is
maintained, then only its parameters (e.g., the strength of the spin-spin coupling) are
affected by the transformation, and thus, we can regard the transformation as one that
acts on these parameters. If the original Hamiltonian contains parametersK 1 ,K 2 ,...,≡
K(for example, the couplingJin the Ising model), then the transformation yields a
Hamiltonian with a new set of parametersK′= (K′ 1 ,K 2 ′,...) that are functions of the
old parameters
K′=R(K). (16.9.6)

The vector functionRdefines the transformation. These equations are called the
renormalization group equationsorrenormalization group transformations. By iterat-
ing the RG equations, it is possible to determine if a system has an ordered phase and
for what parameter values the ordered phase occurs. In an ordered phase, each itera-
tion of the RG equations yields the same lattice with exactly the same Hamiltonian.