Feynman Path Integral Formulation

3.3 Non-linear Sigma Model in the Large-N Limit 79
importantly, with high precision experiments on systems belonging to the same
universality class of the O(N) model.
3.3 Non-linear Sigma Model in the Large-N Limit
A rather fortunate circumstance is represented by the fact that in the large N limit the
non-linear σ-model can be solved exactly. This allows an independent verification
of the correctness of the general ideas developed in the previous section, as well as
a direct comparison of explicit results for universal quantities. The starting point is
the functional integral of Eq. (3.1),
∫
Z =
[dφ(x)]∏ δ [ φ 2 (x) − 1 ] exp[−S(φ)] (3.46)
x
with
S(φ) = 1 ∫
d d x ∂ μ φ(x) · ∂ μ φ(x) . (3.47)
2T
The constraint on the φ field can be implemented via an auxiliary Lagrange multiplier
field α(x). One writes
∫
Z = [dφ(x)][dα(x)] exp[−S(φ,α)] (3.48)
with
S(φ,α) = 1 ∫
d d x [ [∂ μ φ(x)] 2 + α(x)(φ 2 (x) − 1) ] . (3.49)
2T
Since the action is now quadratic in φ(x) one can integrate over N − 1 φ-fields
(denoted previously by π). The resulting determinant is then re-exponentiated, and
one is left with a functional integral over the remaining first field φ 1 (x) ≡ σ(x), as
well as the Lagrange multiplier field α(x),
∫
Z = [dσ(x)dα(x)] exp[−S N (φ,α)] (3.50)
with now
S N (φ,α) = 1 ∫
d d x [ (∂ μ σ) 2 + α(σ 2 − 1) ]
2T
+ 1 2 (N − 1)trln[−∂ 2 + α] . (3.51)
In the large N limit one can neglect, to leading order, fluctuations in the α and σ
fields. For a constant α field, = m 2 , the last (trace) term can be written in
momentum space as