Extensive v2DM study of the one-dimensional Hubbard model for large lattice sizes: Exploiting translational invariance and parity

TL;DR

This paper applies variational density matrix optimization with symmetry exploitation to study large one-dimensional Hubbard models, revealing the importance of three-index conditions for accurate physical properties.

Contribution

It demonstrates the effective use of translational and parity symmetries in large-scale variational density matrix studies of the Hubbard model, and compares semidefinite programming algorithms.

Findings

Three-index conditions are essential for accurate phase diagrams.

The boundary point method is most efficient for large lattice sizes.

Physical properties like correlation functions can be inaccurate even with close ground-state energies.

Abstract

Using variational density matrix optimization with two- and three-index conditions we study the one-dimensional Hubbard model with periodic boundary conditions at various filling factors. Special attention is directed to the full exploitation of the available symmetries, more specifically the combination of translational invariance and space-inversion parity, which allows for the study of large lattice sizes. We compare the computational scaling of three different semidefinite programming algorithms with increasing lattice size, and find the boundary point method to be the most suited for this type of problem. Several physical properties, such as the two-particle correlation functions, are extracted to check the physical content of the variationally determined density matrix. It is found that the three-index conditions are needed to correctly describe the full phase diagram of the Hubbard model. We also show that even in the case of half filling, where the ground-state energy is close to the exact value, other properties such as the spin-correlation function can be flawed.