---
abstract: 'The reduced density matrix is variationally optimized for the two-dimensional Hubbard model. Exploiting all symmetries present in the system, we have been able to study $6\times6$ lattices at various fillings and different values for the on-site repulsion, using the highly accurate but computationally expensive three-index conditions. To reduce the computational cost we study the performance of imposing the three-index constraints on local clusters of $2\times2$ and $3\times3$ sites. We subsequently derive new constraints which extend these cluster constraints to incorporate the open-system nature of a cluster on a larger lattice. The feasibility of implementing these new constraints is demonstrated by performing a proof-of-principle calculation on the $6\times6$ lattice. It is shown that a large portion of the three-index result can be recovered using these extended cluster constraints, at a fraction of the computational cost.'
author:
- Brecht Verstichel
- Ward Poelmans
- Stijn De Baerdemacker
- Sebastian Wouters
- Dimitri Van Neck
bibliography:
- 'cluster.bib'
title: 'Variational optimization of the 2DM: approaching three-index accuracy using extended cluster constraints.'
---

Introduction
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Although the underlying physical laws and mathematical formalism describing many-electron systems are fully understood, solving the associated quantum many-body problem remains a very challenging task. This is 