Polynomial Similarity Transformation Theory: A smooth interpolation between coupled cluster doubles and projected BCS applied to the reduced BCS Hamiltonian

TL;DR

This paper introduces a polynomial similarity transformation theory that smoothly interpolates between coupled cluster doubles and projected BCS methods, providing accurate wave functions across different correlation regimes.

Contribution

It develops a novel polynomial similarity transformation framework that unifies coupled cluster and projected BCS approaches for the reduced BCS Hamiltonian.

Findings

Accurately models wave functions with less than 1% energy error.

Provides a polynomial cost method applicable to realistic Hamiltonians.

Demonstrates effective interpolation between weak and strong correlation limits.

Abstract

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 wavefunction. 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 stengths. The numerical cost is polynomial in system size and the theory can be straightforwardly applied to any realistic Hamiltonian.