Generalized simplicial chiral models

Abstract

Using the auxiliary field representation of the simplicial chiral models on a ( d−1)-dimensional simplex, the simplicial chiral models are generalized through replacing the term Tr (AA †) in the Lagrangian of these models by an arbitrary class function of AA † ; V(AA †) . This is the same method used in defining the generalized two-dimensional Yang–Mills theories (gYM 2) from ordinary YM 2. We call these models the “generalized simplicial chiral models”. Using the results of the one-link integral over a U( N) matrix, the large- N saddle-point equations for eigenvalue density function ρ( z) in the weak ( β> β c) and strong ( β< β c) regions are computed. In d=2, where the model is in some sense related to the gYM 2 theory, the saddle-point equations are solved for ρ( z) in the two regions, and the explicit value of critical point β c is calculated for V(B)= Tr B n (B=AA †) . For V(B)= Tr B 2, Tr B 3 , and Tr B 4, the critical behaviour of the model at d=2 is studied, and by calculating the internal energy, it is shown that these models have a third order phase transition.