Critical Behaviour of a Fermionic Random Matrix Model at Large-N

We study the large-N limit of adjoint fermion one-matrix models. We find one-cut solutions of the loop equations for the correlators of these models and show that they exhibit third order phase transitions associated with m-th order multi-critical points with string susceptibility exponents γstr = −1/m. We also find critical points which can be interpreted as points of first order phase transitions, and we discuss the implications of this critical behaviour for the topological expansion of these matrix models. Work supported in part by the Natural Sciences and Engineering Research Council of Canada. Work supported in part by a University of British Columbia Graduate Fellowship. Hermitian matrix models are the classic example of a D = 0 quantum field theory where ’tHooft’s topological large-N expansion [1] can be solved explicitly [2, 3]. They have recently been of interest in the study of the statistical mechanics of random surfaces [4] particularly for non-perturbative approaches to lower dimensional string theory [5]. There, the large-N expansion coincides with the genus expansion and the large-N limit exhibits phase transitions which correspond to the continuum limits of the discretized random surface theories [5]. Unitary matrix models also play a role in 2-dimensional QCD [6], mean-field computations in lattice gauge theory [7] and various other approaches to higher dimensional gauge theories such as induced QCD [8]. In this Letter we study a matrix model where the degrees of freedom are matrices whose elements are anticommuting Grassmann numbers. The partition function is Z = ∫ dψ dψ̄ e 2 tr V (ψ̄ψ) (1) where V is a polynomial potential, V (ψ̄ψ) = K ∑ k=1 gk k (ψ̄ψ) , (2) ψ and ψ̄ are independent N × N matrices with anticommuting nilpotent elements and the integration measure, dψ dψ̄ ≡ ∏i,j dψij dψ̄ij , is defined using the usual rules for integrating Grassmann variables, ∫ dψij ψij = 1, ∫ dψij 1 = 0. We normalize all traces here and in the following as tr V ≡ 1 N ∑ i Vii. Matrix models of this kind have been studied recently by Makeenko and Zarembo [9], and Ambjørn, Kristjansen and Makeenko [10]. They are motivated by models of induced gauge theories using adjoint matter where the Yang-Mills interactions of gluons are induced by loops with heavy adjoint scalar fields [8] or other kinds of matter such as heavy adjoint fermions [9, 11]. Makeenko and Zarembo [9] have shown that the adjoint fermion matrix model (1) has many of the features of the more familiar Hermitian one-matrix model in the large-N limit [5, 12], including multi-critical behaviour with a third order phase transition and string susceptibility with critical exponent γstr = −1/m, m ∈ ZZ. They also showed that the loop equations for the model (1) are identical to those for the Hermitian one-matrix model with generalized Penner potential [10, 13] ZP = ∫