Symmetry, phase transition, and polymer limitn=0

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Abstract
We argue for the logical necessity of the existence of a phase transition in a polymer system which is identical to the n=0 limit of two different magnetic systems, one possessing a continuous symmetry, the other one possessing a discrete symmetry. The phase transition occurs across a curve AC passing through the critical point, as has been argued recently by Gujrati. We accomplish this by demonstrating explicitly a phase transition across AC in the mean-field calculation of the discrete model for nAC in this model has no polymer counterpart, and corresponds to a complete restoration of symmetry among the n components, even though the magnetic field couples to only one of the components. We discuss the nature of the magnetization and the susceptibilities on and near AC in the context of (i) the mean-field calculation with and without fluctuations and (ii) the ε expansion. Within the general context of renormalization and scaling, we argue that in the scaling regime near the critical point, the longitudinal susceptibility must be positive for small magnetic fields, and the magnetization must not be less than the spontaneous magnetization, provided there is no transition across AC. Even though this result seems to be in contradiction with the ε expansion result, we argue that there is no contradiction, and that our results are consistent with the ε expansion results provided the existence of the phase transition is acknowledged.