The renormalization of the periodic potential is investigated in the framework of the Euclidean onecomponent scalar field theory by means of the differential RG approach. Some known results about the sine-Gordon model are recovered in an extremely simple manner. There are two phases: an ordered one with asymptotical freedom and a disordered one where the model is nonrenormalizable and trivial. The order parameter of the periodicity, the winding number, indicates spontaneous symmetry breaking in the ordered phase where the fundamental group symmetry is broken and the solitons acquire dynamical stability. It is argued that the periodicity and the convexity are such strong constraints on the effective potential that it always becomes flat. This flattening is reproduced by integrating out the RG equation.

The renormalization group flow is presented for the two-dimensional sine-Gordon model within the framework of the functional renormalization group method by including the wave-function renormalization constant. The Kosterlitz-Thouless-Berezinski type phase structure is recovered as the interpolating scaling law between two competing IR attractive area of the global renormalization group flow.

The effective action for the charge density and the photon field is proposed as a generalization of the density functional. A simple definition is given for the density functional, as the functional Legendre transform of the generator functional of connected Green functions for the density and the photon field, offering systematic approximation schemes. The leading order of the perturbation expansion reproduces the Hartree-Fock equation. A renormalization-group motivated method is introduced to turn on the Coulomb interaction gradually and to find corrections to the Hartree-Fock and the Kohn-Sham schemes.

The well-known phase structure of the two-dimensional sine-Gordon model is reconstructed by means of its renormalization group flow, the study of the sensitivity of the dynamics on microscopic parameters. Such an analysis resolves the apparent contradiction between the phase structure and the triviality of the effective potential in either phases, provides a case where usual classification of operators based on the linearization of the scaling relation around a fixed point is not available and shows that the Maxwell-cut generates an unusually strong universality at long distances. Possible analogies with four-dimensional Yang-Mills theories are mentioned, too.Comment: 12 pages, 7 figures. Revised form, to appear in Phys. Lett.

The scheme dependence of the renormalization group (RG) flow has been investigated in the local potential approximation for two-dimensional periodic, sine-Gordon type field-theoretic models discussing the applicability of various functional RG methods in detail. It was shown that scheme-independent determination of such physical parameters is possible as the critical frequency (temperature) at which Kosterlitz-Thouless-Berezinskii type phase transition takes place in the sine-Gordon and the layered sine-Gordon models, and the critical ratio characterizing the Ising-type phase transition of the massive sine-Gordon model. For the latter case, the Maxwell construction represents a strong constraint on the RG flow, which results in a scheme-independent infrared value for the critical ratio. For the massive sine-Gordon model also the shrinking of the domain of the phase with spontaneously broken periodicity is shown to take place due to the quantum fluctuations.