Functional renormalization-group calculation of the equation of state of one-dimensional uniform matter inspired by the Hohenberg-Kohn theorem

We present to our knowledge the first successful functional renormalization group (FRG)–aided density-functional theory (DFT) calculation of the equation of state (EOS) of infinite nuclear matter (NM) in (1+1) dimensions composed of spinless nucleons. We give a formulation to describe infinite matters in which the “flowing” chemical potential is introduced to control the expectation value of the particle number during the flow. The resultant saturation energy of the NM coincides with that obtained by the Monte Carlo method within a few percent. Our result demonstrates that the FRG-aided DFT can be as powerful as any other methods in quantum many-body theory.

Figure 1

The diagrammatic representations of ${\tilde{G}}_0^{(2,3,4)}$. The solid line is the free fermion propagator ${\tilde{G}}_{\mathrm{F},0}^{\left(2\right)}\left(P\right)$. In the diagrams for ${\tilde{G}}_0^{\left(3\right)}$ and ${\tilde{G}}_0^{\left(4\right)}$, $P_3=-P_1-P_2$ and $P_4=-P_1-P_2-P_3$, respectively.

Figure 2

Energy per particle ${\overline{E}}_{\mathrm{gs}}$ as a function of density $n$ in the case of FRG-DFT (solid red line). The energy of free fermions (green dotted line), the contribution from the internucleon potential (purple dashed line) and the result of the extrapolated energy given by a MC simulation [28] at a density close but not equal to saturation density (black cross) are also shown. The red point is the saturation point derived from the FRG-DFT calculation. The units for ${\overline{E}}_{\mathrm{gs}}$ and $n$ are such that the mass of a nucleon is 1.

Figure 3

Energy per particle ${\overline{E}}_{\mathrm{gs}}$ as a function of density $n$ near the saturation point. The results of FRG-DFT (solid red line), the first-order perturbation (blue dashed line), FRG-DFT with $c_{\lambda }=1$ (green dot-dashed line), and a MC simulation [28] (black cross) are shown. The density $n=1.16$ used in the MC simulation is an assumed one which is close but not equal to saturation density. The units for ${\overline{E}}_{\mathrm{gs}}$ and $n$ are such that the mass of a nucleon is 1.