Second quantization approach to characteristic polynomials in RMT

Abstract

The distribution of the characteristic polynomial $Z(U,\theta)$ of $N\times N$ matrices $U$ in the Circular Unitary Ensemble is studied by the method of second quantization for one-dimensional fermions. For infinite $N$ the Gaussian distribution of $Z(U,\theta)$ is established straightforwardly by bosonization. A general expression for the $n$-point correlation function of the characteristic polynomial at different points is given by this method. The case of finite $N$ is discussed.