Potential Flow of Renormalization Group in Quasi-One-Dimensional Systems

We address the issue why the phase diagrams for quasi-one-dimensional systems are rather simple, while the renormalization group equations behind the scene are non-linear and messy looking. The puzzle is answered in two steps -- we first demonstrate that the complicated coupled flow equations are simply described by a potential $V(h_i)$, in an appropriate basis for the interaction couplings $h_i$. The renormalization-group potential is explicitly constructed by introducing the Majorana fermion representation. The existence of the potential prevents chaotic behaviors and other exotic possibilities such as limit cycles. Once the potential is obtained, the ultimate fate of the flows are described by a special set of fixed-ray solutions and the phase diagram is determined by Abelian bosonization. Extension to strong coupling regime and comparison with the Zamolodchikov c-theorem are discussed at the end.