ar X iv : h ep - t h / 00 12 20 8 v 1 2 1 D ec 2 00 0 Wave - function renormalization for the Coulomb - gas in Wegner - Houghton ’ s RG method

The well-known phase structure of the two-dimensional Coulomb-gas was investigated in several papers by using the differential RG approach with a smooth cut-off in the momentum space (or equivalently using a sharp cut-off in the coordinate space) (Kosterlitz 1973, Kosterlitz 1974, Jose 1977, Huang 1991, Gersdorff 2000). Our goal is in this work to determine the RG flow for the two-dimensional Coulomb-gas in the framework of the RG approach of Wegner and Houghton (Wegner 1973) taking the field-independent wave-function renormalization into account. Results of two different methods were compared: renormalization by means of the blocking construction in the coordinate space using the dilute gas approximation (real space RG) (Kosterlitz 1973, Kosterlitz 1974, Jose 1977, Huang 1991), and the renormalization of the equivalent sine-Gordon scalar field model, performing the blocking transformations in the momentum space in Wegner-Houghton’s framework with a sharp cut-off and usage of the derivative expansion (Wegner-Houghton RG method) (Wegner 1973, Hasenfratz 1986, Polonyi 2000). It is believed that several different models, like the sine-Gordon, Thirring, and the X-Y planar models belong to the same universality class, namely to that of the two-dimensional Coulomb-gas. The X-Y model with external field, which is a classical two-component spin-model described by the action,