Feynman Path Integral Formulation

3.3 Non-linear Sigma Model in the Large-N Limit 79importantly, with high precision experiments on systems belonging to the sameuniversality class of the O(N) model.3.3 Non-linear Sigma Model in the Large-N LimitA rather fortunate circumstance is represented by the fact that in the large N limit thenon-linear σ-model can be solved exactly. This allows an independent verificationof the correctness of the general ideas developed in the previous section, as well asa direct comparison of explicit results for universal quantities. The starting point isthe functional integral of Eq. (3.1),∫Z =[dφ(x)]∏ δ [ φ 2 (x) − 1 ] exp[−S(φ)] (3.46)xwithS(φ) = 1 ∫d d x ∂ μ φ(x) · ∂ μ φ(x) . (3.47)2TThe constraint on the φ field can be implemented via an auxiliary Lagrange multiplierfield α(x). One writes∫Z = [dφ(x)][dα(x)] exp[−S(φ,α)] (3.48)withS(φ,α) = 1 ∫d d x [ [∂ μ φ(x)] 2 + α(x)(φ 2 (x) − 1) ] . (3.49)2TSince 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, andone is left with a functional integral over the remaining first field φ 1 (x) ≡ σ(x), aswell as the Lagrange multiplier field α(x),∫Z = [dσ(x)dα(x)] exp[−S N (φ,α)] (3.50)with nowS 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 σfields. For a constant α field, = m 2 , the last (trace) term can be written inmomentum space as