Coupled self-consistent random-phase approximation equations for even and odd particle numbers: Tests with solvable models

Coupled equations for even and odd particle number correlation functions are set up via the equation of motion method. For the even particle number case this leads to self-consistent random-phase approximation equations already known from the literature. From the equations of the odd particle number case the single-particle occupation probabilities are obtained in a self-consistent way. This is the essential new procedure of this work. Both even and odd particle number cases are based on the same correlated vacuum and, thus, are coupled equations. Applications to the Lipkin model and to the one-dimensional Hubbard model give very good results.

Figure 1

Schematic representation of the three-body interaction $\mathcal{D}$. The wiggly line stands for the (collective) “ph” modes. The full dot represents the two-body interaction.

Figure 2

The difference between occupation number of the two levels in the Lipkin model, normalized by $N$ as a function of $\chi =V(N-1)$ for different values of $N$: (a) $N=4$, (b) $N=10$, (c) $N=20$, (d) $N=100$. This with standard RPA (red dots), SCRPA (violet dashed-dot), eo-SCRPA (blue dashed line) with the eom method for odd particle excitations and exact solution (full black line). Note that our approach gives the exact result for $N=4$.

Figure 3

Same as Fig. 2 but for the square of the difference between occupation number of the two levels in the Lipkin model, normalized by $N^2$.

Figure 4

Same as Fig. 2, but for the first-excited state for different values of $N$: (a) $N=4$, (b) $N=20$, (c) $N=100$, (d) $N=200$. Please note that one may make the hypothesis that the eo-SCRPA approach becomes exact in the $N\to \infty$ limit.

Figure 5

The correlation energy as a function of $\chi =V(N-1)$ for different values of $N$: (a) $N=4$, (b) $N=10$, (c) $N=20$, (d) $N=100$ with eo-SCRPA (blue dashed line) compared with the exact solution (full black line). Note again that, for $N=4$, the exact result is obtained with our approach.

Figure 6

Excitation energy between the system $N+1$ and $N$ particles as a function of $\chi =V(N-1)$ for different values of $N$: (a) $N=4$ and (b) $N=10$ with eo-SCRPA (blue dashed line) (A9) compared with the exact solution (full black line) ${\lambda }_{+}=E_{\alpha }^{N+1}-E_0^N$.

Figure 7

Same as Fig. 6 but for the excitation energy between the system $N-1$ and $N$ particles as a function of $\chi =V(N-1)$ for different values of $N$: (a) $N=4$ and (b) $N=10$. Note that, for $N=4$ the exact result ${\lambda }_{-}=E_{\alpha }^{N-1}-E_0^N$ is obtained with our approach, Eq. (A9).

Figure 8

Hartree-Fock states at $U=0$ for the chain with six sites at half filling and projection of spin $m_s=0$. The occupied states are represented by full arrows and those not occupied are represented by dashed arrows.

Figure 9

Occupation numbers as function of the interaction $U/t$ for various values of the momenta $k_6=-\pi$, $k_5=-2\pi /3$, $k_4=2\pi /3$ for states above the Fermi level. Notice that the modes $k_4=2\pi /3$ and $k_5=-2\pi /3$ are degenerate. For each approximation, sRPA (red dots) and eo-SCRPA (blue crosses) are compared with the exact solution (full black line). Also we have $n_{k_1}=1-n_{k_6}$ and $n_{k_2}=n_{k_3}=1-n_{k_4}=1-n_{k_5}$.

Figure 10

Same as Fig. 9 but for the energies of excited states for different channels: (a) $\left|q\right|=\pi /3$, (b) $\left|q\right|=2\pi /3$, and (c) $\left|q\right|=\pi$.

Figure 11

Same as Fig. 9 but for the ground-state energy.