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Localization transition on the Bethe lattice.

The critical behavior of disordered single-particle systems without time-reversal invariance is described by a nonlinear \ensuremath{\sigma} model formulated in terms of graded pseudounitary matrices. It is shown that the symmetry-breaking pattern of this model predicts the correct analytical structure of the two-point Green's functions of the disordered system, which are expected to be singular (finite) for localized (extended) states in the limit of vanishing frequency. The main purpose of the paper is to obtain exact solutions for the graded nonlinear \ensuremath{\sigma} model on a Bethe lattice. A combination of analytical and numerical methods is used to determine the critical behavior of all two-point Green's functions. In contrast with other work on the graded nonlinear \ensuremath{\sigma} model, no minimum metallic conductivity is found. Instead, the averaged inverse conductivity has an exponential singularity at the critical point.