CriticalZ N spin systems andφ 4 theory

The continuum limit ofZ _N (N≧4) spin systems at the critical point is identified with a two-component massless scalar field with quartic self-interactionf _αβγδ φ _α φ _β φ _γ φ _δ , withf _αβγδ = 1/3 (δ _αβ δ _γδ +δ _αγ δ _βδ +δ _αδ δ _βγ ) ≡S _αβγδ forN≧5 andf _αβγδ =S _αβγδ ? 2/3δ _αβ δ _αγ δ _αδ forN=4. The result is based on the study of the two and four-point irreducible correlation functions in the one-loop approximation.