Generalized canonical purification for density matrix minimization

TL;DR

This paper introduces a new Lagrangian-based canonical purification method for density matrix minimization that converges systematically to the ground state without trace adjustment, offering a computationally efficient alternative to diagonalization.

Contribution

It presents a novel closed-form canonical purification approach derived from a Lagrangian formulation, generalizing existing methods via hole-particle duality.

Findings

Converges systematically to the ground state.

No need for a posteriori trace adjustment.

Potentially more efficient than diagonalization.

Abstract

A Lagrangian formulation for the constrained search for the $N$-representable one-particle density matrix based on the McWeeny idempotency error minimization is proposed, which converges systematically to the ground state. A closed form of the canonical purification is derived for which no a posteriori adjustement on the trace of the density matrix is needed. The relationship with comparable methods are discussed, showing their possible generalization through the hole-particle duality. The appealing simplicity of this self-consistent recursion relation along with its low computational complexity could prove useful as an alternative to diagonalization in solving dense and sparse matrix eigenvalue problems.