Quasiparticle coupled cluster theory for pairing interactions

We present an extension of the pair coupled cluster doubles (p-CCD) method to quasiparticles and apply it to the attractive pairing Hamiltonian. Near the transition point where number symmetry gets spontaneously broken, the proposed BCS-based p-CCD method yields energies significantly better than those of existing methods when compared to the exact results obtained via solution of the Richardson equations. The quasiparticle p-CCD method has a low computational cost of $\mathcal{O}\left(N^3\right)$ as a function of system size. This together with the high quality of results here demonstrated points to considerable promise for the accurate description of strongly correlated systems with more realistic pairing interactions.

Figure 1

Fraction of correlation energy recovered in the half-filled pairing Hamiltonian with 100 levels. We show both the p-CCD and the number-projected BCS. A BCS solution emerges at the point we have labeled as $G_c$. Past the point at which the coupled cluster curve is cut off, the method predicts a complex energy.

Figure 2

Fraction of the correlation energy recovered in the half-filled pairing Hamiltonian with 100 levels. BCS p-CCD and BCS p-BCCD refer to p-CCD based on a BCS or on a BCS-Brueckner reference, respectively.

Figure 3

Occupation numbers from the BCS and BCS-Brueckner determinants as a function of $G$ for the half-filled pairing Hamiltonian with 100 levels. We show the two occupation numbers that bracket the Fermi level at $G=0$.

Figure 4

Deviations from the exact occupation number for the half-filled pairing Hamiltonian with 100 levels and $G/\Delta \epsilon$ = 1.

Figure 5

Fluctuations in particle number in the BCS and BCS p-CCD wave functions for the half-filled pairing Hamiltonian with 100 levels.

Figure 6

Pairing parameter ${\Delta }_c$ defined in Eq. (19) for the half-filled pairing Hamiltonian with 100 levels.

Figure 7

Fraction of the correlation energy recovered in half-filled pairing Hamiltonians as a function of the number of levels. We have scaled the interaction strength $G$ to keep a roughly constant BCS gap and have used $G=0.24$ and $G=0.40$ for 100 levels.

Figure 8

BCS gap $\Delta$ as a function of the number of levels with pairing strength $G$ scaled according to Eq. (21).

Figure 9

Fraction of correlation recovered in the pairing Hamiltonian with 100 levels. Solid lines denote half filling while dotted lines indicate 20% filling. The interaction strength $G$ is expressed in units of $G_c$, the value of $G$ for which the Hartree-Fock to BCS transition occurs.

Figure 10

Fraction of the correlation energy recovered in the pairing Hamiltonian with 100 levels plotted against the filling fraction.