Exact solutions to the quantum many-body problem using the geminal density matrix

TL;DR

This paper introduces a novel approach using the geminal density matrix to solve the quantum many-body problem, enabling the calculation of eigenstates in strongly correlated systems through a unitary evolution that preserves N-representability.

Contribution

It develops a new method based on the geminal density matrix to address the N-representability problem and compute eigenstates of many-electron systems efficiently.

Findings

Successfully diagonalized atomic Hamiltonians.

Reduced the problem to solving two-electron eigenstates.

Demonstrated the method on the Helium atom.

Abstract

It is virtually impossible to directly solve the Schr\"odinger equation for a many-electron wave function due to the exponential growth in degrees of freedom with increasing particle number. The two-body reduced density matrix (2-RDM) formalism reduces this coordinate dependence to that of four particles irrespective of the wave function's dimensionality, providing a promising path to solve the many-body problem. Unfortunately, errors arise in this approach because the 2-RDM cannot practically be constrained to guarantee that it corresponds to a valid wave function. Here we approach this so-called $N$-representability problem by expanding the 2-RDM in a complete basis of two-electron wave functions and studying the matrix formed by the expansion coefficients. This quantity, which we call the geminal density matrix (GDM), is found to evolve in time by a unitary transformation that preserves $N$-representability. This evolution law enables us to calculate eigenstates of strongly correlated systems by a fictitious adiabatic evolution in which the electron-electron interaction is slowly switched on. We show how this technique is used to diagonalize atomic Hamiltonians, finding that the problem reduces to the solution of $\sim N(N-1)/2$ two-electron eigenstates of the Helium atom on a grid of electron-electron interaction scaling factors.