Polynomial similarity transformation theory: A smooth interpolation between coupled cluster doubles and projected BCS applied to the reduced BCS Hamiltonian

We present a similarity transformation theory based on a polynomial form of a particle-hole pair excitation operator. In the weakly correlated limit, this polynomial becomes an exponential, leading to coupled cluster doubles. In the opposite strongly correlated limit, the polynomial becomes an extended Bessel expansion and yields the projected BCS wave function. In between, we interpolate using a single parameter. The effective Hamiltonian is non-Hermitian and this polynomial similarity transformation theory follows the philosophy of traditional coupled cluster, left projecting the transformed Hamiltonian onto subspaces of the Hilbert space in which the wave function variance is forced to be zero. Similarly, the interpolation parameter is obtained through minimizing the next residual in the projective hierarchy. We rationalize and demonstrate how and why coupled cluster doubles is ill suited to the strongly correlated limit, whereas the Bessel expansion remains well behaved. The model provides accurate wave functions with energy errors that in its best variant are smaller than 1% across all interaction strengths. The numerical cost is polynomial in system size and the theory can be straightforwardly applied to any realistic Hamiltonian.

Figure 1

Fraction of correlation energy recovered with respect to Hartree-Fock in the 16-site, half-filled pairing Hamiltonian with equally spaced levels. We have defined $G_c$ as the point at which number symmetry is spontaneously broken by the mean field. The pCCD curve is truncated; for larger $G$ the method predicts a complex total energy.

Figure 2

Root-mean-square values of amplitudes defining excitation operators in the 12-site, half-filled pairing Hamiltonian with equally spaced levels using the coupled cluster parametrization of the wave function [Eq. (9)]. Results for larger systems are broadly similar.

Figure 3

Root-mean-square values of amplitudes defining excitation operators in the 12-site, half-filled pairing Hamiltonian with equally spaced levels using the configuration interaction parametrization of the wave function [Eq. (12)]. Results for larger systems are broadly similar.

Figure 4

Root-mean-square values of amplitudes defining excitation operators in the 12-site, half-filled pairing Hamiltonian with equally spaced levels using the Bessel parametrization of the wave function [Eq. (22)]. Compare to the coupled cluster and configuration interaction parametrizations in Figs. 2 and 3. Results for larger systems are broadly similar, although higher excitation amplitudes become fairly large before decaying to zero for large enough $G/G_c$.

Figure 5

Fraction of correlation energy recovered in the half-filled 16-site pairing Hamiltonian as a function of $G$. We show pCCD and PBCS, while “Bessel” denotes the use of our PoST equations (see below) with $c_2=1/4\iff \alpha =2$. Put differently, “Bessel” indicates a wave function of PBCS form [Eq. (22)] where $T_2$ has been obtained projectively as in pCCD.

Figure 6

Fraction of correlation energy recovered in the half-filled 16-site pairing Hamiltonian as a function of $G$. We show PBCS and our PoST method choosing that parameter $\alpha$ which minimizes $T_4$ given the exact $T_2$.

Figure 7

The $\alpha$ parameter defining the PoST correlator $F\left(T\right)$ as a function of $G$ in the half-filled 16-site pairing Hamiltonian. We include the $\alpha$ that minimizes $T_4$ given the exact $T_2$ (labeled as ${\alpha }_{T_4})$ and the $\alpha$ which gives the exact energy (labeled as ${\alpha }_E)$. We also include the $\alpha$ that minimizes the $R_4$ residual (see below), labeled as ${\alpha }_{R_4}$. Note that in all cases $\alpha$ goes from 1 at small $G$ (as expected from pCCD) to 2 at large $G$ (as expected from PBCS).

Figure 8

Fraction of correlation energy recovered in the half-filled 16-site pairing Hamiltonian as a function of $G$.

Figure 9

Fraction of correlation energy recovered in the half-filled 16-site pairing Hamiltonian as a function of $G$. We show pCCD and PBCS, along with our new PoST model and the expectation value of the PoST wave function as defined in Eq. (34).

Figure 10

Singular values of the PoST $T_2$ amplitudes in the half-filled 16-site pairing Hamiltonian as a function of $G$. Note that as $G$ becomes large, all singular values but one tend to zero; in other words, the $T_2$ amplitudes factorize in the PBCS form as $G\to \infty$.

Figure 11

Overlap of the PoST wave function $F_{\alpha }\left(T\right)|0\rangle$ with the exact wave function for the half-filled 16-site pairing Hamiltonian as a function of $G$.

Figure 12

Fraction of correlation energy recovered in the half-filled 16-site pairing Hamiltonian as a function of $G$. We show PBCS, and our PoST model using the similarity transforms defined by $F_{\alpha }\left(T\right)$ given in Eq. (24b) and by $F_x\left(T\right)$ given in Eq. (45), which we label as ${\mathrm{PoST}}_{\alpha }$ and ${\mathrm{PoST}}_{\mathrm{x}}$, respectively.