Schrodinger equation approach to non-linear $\sigma$-models in the large N-limit

Abstract

Non-linear d-dimensional vector $\sigma$-models are studied in the large N-limit. It is found that a two-point correlation function obeys a standard Schrodinger equation for a free quantum particle moving in the $\delta$-function quantum well. The threshold problem for bound states in this equation is shown to be equivalent to a critical behavior of these models above and below the Curie point. The SU(N)- symmetric Ginzburg-Landau (GL) $\sigma$-model subject to a uniform magnetic field H is considered in the large-N limit within the Schrodinger equation approach. A upper critical magnetic field line $H_{c2}(T)$ of type-II superconductors for an arbitrary external H is obtained without exploiting the lowest Landau level (LLL) approximation. Both low-H perturbation expansion terms and exponentially small corrections to the LLL approximation are calculated. Correspondences between the one-particle quantum mechanics and critical phenomena as well as some applications of the above method to other models of statistical mechanics are also discussed.