Global features of nonlinear renormalization-group equations

Abstract

The analysis of selected nonlinear problems in the renormalization group is found to show striking contrasts between the usual local linearized fixed-point analysis and the properties of global solutions of nonlinear equations derived from an approximation of the Wegner-Houghton differential formulation. The competition between various fixed points that is incorporated in general global solutions can upset the asymptotically valid critical behavior deduced from the local analysis. In general, the critical-point exponents of such a solution will not satisfy equalities, but rather the corresponding inequalities. However, these nonscaling solutions have extraneous singularities that are not related to the thermodynamic singularities of the system. If singularities of this type are excluded, then the global solution has the same critical-point exponents as the local solution derived by linearizing around the stablest fixed point. It is shown that in this case the critical surface in the Hamiltonian space is closely related to the surface of order-2 critical points in a thermodynamic field space. The boundaries of this surface are correspondingly related to the critical points of higher order in this thermodynamic space. The nonlinear global solution predicts multiple power scaling behavior from a single scaling equation deduced from the renormalization group. Previously such behavior was obtained by postulating the simultaneous validity of two of more "linear" scaling hypotheses.