Projective purification of correlated reduced density matrices

TL;DR

This paper introduces an efficient algorithm for purifying reduced density matrices in many-body quantum systems, ensuring physicality, symmetry preservation, and consistency, thereby improving the accuracy of approximate solutions in quantum dynamics.

Contribution

The paper presents a novel purification algorithm that maintains all conserved quantities and symmetry considerations while restoring N-representability in reduced density matrices.

Findings

Outperforms previous purification methods in preserving symmetries.

Ensures contraction consistency across reduced density matrices.

Demonstrated effectiveness in Fermi-Hubbard model dynamics.

Abstract

In the search for accurate approximate solutions of the many-body Schr\"odinger equation, reduced density matrices play an important role, as they allow to formulate approximate methods with polynomial scaling in the number of particles. However, these methods frequently encounter the issue of $N$-representability, whereby in self-consistent applications of the methods, the reduced density matrices become unphysical. A number of algorithms have been proposed in the past to restore a given set of $N$-representability conditions once the reduced density matrices become defective. However, these purification algorithms have either ignored symmetries of the Hamiltonian related to conserved quantities, or have not incorporated them in an efficient way, thereby modifying the reduced density matrix to a greater extent than is necessary. In this paper, we present an algorithm capable of efficiently performing all of the following tasks in the least invasive manner: restoring a given set of $N$-representability conditions, maintaining contraction consistency between successive orders of reduced density matrices, and preserving all conserved quantities. We demonstrate the superiority of the present purification algorithm over previous ones in the context of the time-dependent two-particle reduced density matrix method applied to the quench dynamics of the Fermi-Hubbard model.