Projective purification of correlated reduced density matrices

In the search for accurate approximate solutions of the many-body Schrödinger equation, reduced density matrices play an important role, as they allow one to formulate approximate methods with polynomial scaling in the number of particles. However, these methods frequently encounter the issue of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>N</a:mi></a:math>-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 <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>N</b:mi></b:math>-representability conditions once the reduced density matrices become defective. However, these purification algorithms either have 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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>N</c:mi></c:math>-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.