First-order phase transitions, the Maxwell construction, and the momentum-space renormalization group

The behavior of a nonperturbative momentum-space renormalization group (RG) is analyzed both above and below the critical temperature. The case of a scalar order parameter and of the Ising model is studied in detail by analytical and numerical means. It is shown that this RG transformation is always well defined even inside the coexistence curve. van der Waals loops are suppressed by long-wavelength fluctuations which enforce the convexity of the free energy. The RG description emerging from this study is then compared with exact results and other approximate theories.