Dyson–Maleev representation of nonlinear sigma models

Abstract

For nonlinear sigma models in the unitary symmetry class, the nonlinear target space can be parameterized with cubic polynomials. This choice of coordinates has been known previously as the Dyson–Maleev parameterization for spin systems, and we show that it can be applied to a wide range of sigma models. The practical use of this parameterization includes simplification of diagrammatic calculations (in perturbative methods) and of algebraic manipulations (in non-perturbative approaches). We illustrate the use and specific issues of the Dyson–Maleev parameterization with three examples: the Keldysh sigma model for time-dependent random Hamiltonians, the supersymmetric sigma model for random matrices and the supersymmetric transfer-matrix technique for quasi-one-dimensional disordered wires. We demonstrate that nonlinear sigma models of unitary-like symmetry classes C and B/D also admit the Dyson–Maleev parameterization.