  The functional renormalization group (FRG) has been used widely to
investigate phase diagrams, in particular the one of the two-dimensional
Hubbard model. So far, the study of one-dimensional models has not attracted as
much attention. We use the FRG to investigate the phases of a one-dimensional
spinless tight-binding chain with nearest and next-nearest neighbor
interactions at half filling. The phase diagram of this model has already been
established with other methods, and phase transitions from a metallic phase to
ordered phases take place at intermediate to strong interactions. The model is
thus well suited to analyze the potential and the limitations of the FRG in
this regime of interactions. We employ flow equations that are exact up to
second order in the interaction, which implies that we take into account the
frequency dependence of the two-particle vertex as well as the feedback of the
dynamic self-energy. For intermediate nearest neighbor interactions, our scheme
captures the phase transition from a metallic phase to a charge density wave
with alternating occupation. The critical interaction, at which this transition
occurs, is underestimated due to our approximations. Similarly, for
intermediate next-nearest neighbor interactions, we observe a transition to a
charge density wave with occupation pattern ..00110011... We show that taking
into account a feedback of the two-particle vertex in the flow equation is
essential for the detection of those phases.
