176
16. Lattice Field Theory, and the Renormalization Group Revisited
function, defined on a lattice of size 2a, has the same form as the old one, but
′
with a new coupling K related to the old one K by (16.112).
Equation (16.112) is an example of a renormalization transformation: the
number of degrees of freedom has been halved, the lattice spacing has doubled,
′
and the coupling K has been renormalized to K .
It is clear that we could apply the same procedure to the new Hamiltonian,
′′
′
introducing a coupling K which is related to K , and thence to K by
′′
′ )
2
tanh K = (tanh K = (tanh K)
4 .
(16.114)
This is equivalent to iterating the renormalization transformation; after n
iterations, the effective lattice constant is 2
n a, and the effective coupling is
given by
tanh K
(n) = (tanh K)
n .
(16.115)
′′
The successive values K
′ , K , . . . of the coupling under these iterations can
be regarded as a ‘flow’ in the (one-dimensional) space of K-values: a renormalization flow.
Of particular interest is a point (or points) K
∗ such that
tanh K
∗ = tanh
2 K
∗ .
(16.116)
This is called a fixed point of the renormalization tranformation. At such a
point in K-space, changing the scale by a factor of 2 (or 2
n for that matter)
will make no difference, which means that the system must be in some sense
ordered. Remembering that K = J/(k B T ), we see that K = K
∗ when the
temperature is ‘tuned’ to the value T = T
∗ = J/(k B K
∗ ). Such a T
∗ would
be the temperature of a critical point for the thermodynamics of the system,
corresponding to the onset of ordering. In the present case, the only fixed
points are K
∗ = ∞ and K
∗ = 0. Thus there is no critical point at a non-zero
T
∗ , and hence no transition to an ordered phase. However, we may describe
the behaviour as T → 0 as ‘quasi-critical’. For large K, we may use
tanh K ⋍ 1 − 2e
−2K
(16.117)
to write (16.115) as
1
K
(n) = K − ln n,
(16.118)
2
which shows that K
n changes only very slowly (logarithmically) under iterations when in the vicinity of a very large value of K, so that this is ‘almost’
a fixed point.
We may represent the flow of K, under the renormalization transformation
(16.115), as in figure 16.6. Note that the flow is away from the quasi-fixed
point at K
∗ = ∞ (T = 0) and towards the (non-interacting) fixed point at
K
∗ = 0.
A renormalization transformation which has a fixed point at a finite (neither zero nor infinite) value of the coupling is clearly of greater interest, since
16. Lattice Field Theory, and the Renormalization Group Revisited
function, defined on a lattice of size 2a, has the same form as the old one, but
′
with a new coupling K related to the old one K by (16.112).
Equation (16.112) is an example of a renormalization transformation: the
number of degrees of freedom has been halved, the lattice spacing has doubled,
′
and the coupling K has been renormalized to K .
It is clear that we could apply the same procedure to the new Hamiltonian,
′′
′
introducing a coupling K which is related to K , and thence to K by
′′
′ )
2
tanh K = (tanh K = (tanh K)
4 .
(16.114)
This is equivalent to iterating the renormalization transformation; after n
iterations, the effective lattice constant is 2
n a, and the effective coupling is
given by
tanh K
(n) = (tanh K)
n .
(16.115)
′′
The successive values K
′ , K , . . . of the coupling under these iterations can
be regarded as a ‘flow’ in the (one-dimensional) space of K-values: a renormalization flow.
Of particular interest is a point (or points) K
∗ such that
tanh K
∗ = tanh
2 K
∗ .
(16.116)
This is called a fixed point of the renormalization tranformation. At such a
point in K-space, changing the scale by a factor of 2 (or 2
n for that matter)
will make no difference, which means that the system must be in some sense
ordered. Remembering that K = J/(k B T ), we see that K = K
∗ when the
temperature is ‘tuned’ to the value T = T
∗ = J/(k B K
∗ ). Such a T
∗ would
be the temperature of a critical point for the thermodynamics of the system,
corresponding to the onset of ordering. In the present case, the only fixed
points are K
∗ = ∞ and K
∗ = 0. Thus there is no critical point at a non-zero
T
∗ , and hence no transition to an ordered phase. However, we may describe
the behaviour as T → 0 as ‘quasi-critical’. For large K, we may use
tanh K ⋍ 1 − 2e
−2K
(16.117)
to write (16.115) as
1
K
(n) = K − ln n,
(16.118)
2
which shows that K
n changes only very slowly (logarithmically) under iterations when in the vicinity of a very large value of K, so that this is ‘almost’
a fixed point.
We may represent the flow of K, under the renormalization transformation
(16.115), as in figure 16.6. Note that the flow is away from the quasi-fixed
point at K
∗ = ∞ (T = 0) and towards the (non-interacting) fixed point at
K
∗ = 0.
A renormalization transformation which has a fixed point at a finite (neither zero nor infinite) value of the coupling is clearly of greater interest, since