Eigenstate Estimation for the Bardeen-Cooper-Schrieffer (BCS) Hamiltonian

TL;DR

This paper introduces the Shifted Harmonic Approximation (SHA), a novel method for accurately estimating eigenstates of finite BCS Hamiltonians, simplifying calculations and preserving symmetries without relying on traditional BCS approximations.

Contribution

The paper presents the SHA as a new approach to approximate BCS Hamiltonians with differential operators, enabling easier derivation of BCS results and efficient eigenstate computation.

Findings

SHA accurately approximates finite BCS Hamiltonians in strong pairing regimes.

SHA preserves symmetries of the original Hamiltonians.

SHA enables low-dimensional eigenstate calculations in large Hilbert spaces.

Abstract

We show how multi-level BCS Hamiltonians of finite systems in the strong pairing interaction regime can be accurately approximated using multi-dimensional shifted harmonic oscillator Hamiltonians. In the Shifted Harmonic Approximation (SHA), discrete quantum state variables are approximated as continuous ones and algebraic Hamiltonians are replaced by differential operators. Using the SHA, the results of the BCS theory, such as the gap equations, can be easily derived without the BCS approximation. In addition, the SHA preserves the symmetries associated with the BCS Hamiltonians. Lastly, for all interaction strengths, the SHA can be used to identify the most important basis states -- allowing accurate computation of low-lying eigenstates by diagonalizing BCS Hamiltonians in small subspaces of what may otherwise be vastly larger Hilbert spaces.