16.4. Renormalization, and the renormalization group, on the lattice
179
We have emphasized that, at a critical point, and in the continuum limit,
the correlation length ξ → ∞, or equivalently the mass parameter (cf (16.102))
m = ξ
−1
→ 0. In this case, the Fourier transform of the spin-spin correlation
function should behave as
G ˜ (k
2 ) ∝
1 .
(16.125)
k 2
This is indeed the k
2 -dependence of the propagator of a free, massless scalar
particle, but – as we learned for the fermion propagator in section 15.5 – it
is no longer true in an interacting theory. In the interacting case, (16.125)
generally becomes modified to
G ˜ (k
2 ) ∝
1
,
(16.126)
(k 2 ) 1−
η
2
or equivalently
1
G(x) ∝
(16.127)
1+η
|x|
in three spatial dimensions, and in the continuum limit. Thus, at a critical
point, the spin-spin correlation function exhibits scaling under the transforma′
tion x = f x, but it is not free-field scaling. Comparing (16.126) with (15.75),
we see that η/2 is precisely the anomalous dimension of the field s(x), so –
just as in section 15.5 – we have an example of scaling with anomalous dimension. In the statistical mechanics case, η is a critical exponent, one of a
number of such quantities characterizing the critical behaviour of a system.
In general, η will depend on the coupling constant η(K): at a non-trivial
fixed point, η will be evaluated at the fixed point value K
∗ , η(K
∗ ). Enormous
progress was made in the theory of critical phenomena when the powerful
methods of quantum field theory were applied to calculate critical exponents
(see for example Peskin & Schroeder 1995, chapter 13, and Binney et al.
1992).
In our discussion so far, we have only considered simple models with just
one ‘coupling constant’, so that diagrams of renormalization flow were onedimensional. Generally, of course, Hamiltonians will consist of several terms,
and the behaviour of all their coefficients will need to be considered under a
renormalization transformation. The general analysis of renormalization flow
in multi-dimensional coupling space was given by Wegner (1972). In simple
terms, the coefficients show one of three types of behaviour under renormalization transformations such that a → f a, characterized by their behaviour in
the vicinity of a fixed point: (i) the difference from the fixed point value grows
as f increases, so that the system moves away from the fixed point (as in the
single-coupling examples considered earlier); (ii) the difference decreases as f
increases, so the system moves towards the fixed point; (iii) there is no change
in the value of the coupling as f changes. The corresponding coefficients are
called, respectively, (i) relevant, (ii) irrelevant and (iii) marginal couplings; the
terminology is also frequently applied to the operators in the Hamiltonians
179
We have emphasized that, at a critical point, and in the continuum limit,
the correlation length ξ → ∞, or equivalently the mass parameter (cf (16.102))
m = ξ
−1
→ 0. In this case, the Fourier transform of the spin-spin correlation
function should behave as
G ˜ (k
2 ) ∝
1 .
(16.125)
k 2
This is indeed the k
2 -dependence of the propagator of a free, massless scalar
particle, but – as we learned for the fermion propagator in section 15.5 – it
is no longer true in an interacting theory. In the interacting case, (16.125)
generally becomes modified to
G ˜ (k
2 ) ∝
1
,
(16.126)
(k 2 ) 1−
η
2
or equivalently
1
G(x) ∝
(16.127)
1+η
|x|
in three spatial dimensions, and in the continuum limit. Thus, at a critical
point, the spin-spin correlation function exhibits scaling under the transforma′
tion x = f x, but it is not free-field scaling. Comparing (16.126) with (15.75),
we see that η/2 is precisely the anomalous dimension of the field s(x), so –
just as in section 15.5 – we have an example of scaling with anomalous dimension. In the statistical mechanics case, η is a critical exponent, one of a
number of such quantities characterizing the critical behaviour of a system.
In general, η will depend on the coupling constant η(K): at a non-trivial
fixed point, η will be evaluated at the fixed point value K
∗ , η(K
∗ ). Enormous
progress was made in the theory of critical phenomena when the powerful
methods of quantum field theory were applied to calculate critical exponents
(see for example Peskin & Schroeder 1995, chapter 13, and Binney et al.
1992).
In our discussion so far, we have only considered simple models with just
one ‘coupling constant’, so that diagrams of renormalization flow were onedimensional. Generally, of course, Hamiltonians will consist of several terms,
and the behaviour of all their coefficients will need to be considered under a
renormalization transformation. The general analysis of renormalization flow
in multi-dimensional coupling space was given by Wegner (1972). In simple
terms, the coefficients show one of three types of behaviour under renormalization transformations such that a → f a, characterized by their behaviour in
the vicinity of a fixed point: (i) the difference from the fixed point value grows
as f increases, so that the system moves away from the fixed point (as in the
single-coupling examples considered earlier); (ii) the difference decreases as f
increases, so the system moves towards the fixed point; (iii) there is no change
in the value of the coupling as f changes. The corresponding coefficients are
called, respectively, (i) relevant, (ii) irrelevant and (iii) marginal couplings; the
terminology is also frequently applied to the operators in the Hamiltonians