Quantum phase transitions in one-dimensional nanostructures: a comparison between DFT and DMRG methodologies

In the realm of quantum chemistry, the accurate prediction of electronic structure and properties of nanostructures remains a formidable challenge. Density functional theory (DFT) and density matrix renormalization group (DMRG) have emerged as two powerful computational methods for addressing electronic correlation effects in diverse molecular systems. We compare ground-state energies ( \(\varvec{e}_{\textbf{0}}\) ${\mathit{e}}_{\mathbf{0}}$ ), density profiles ( \(\varvec{n}\) $\mathit{n}$ ), and average entanglement entropies ( \(\bar{\varvec{S}}\) $\overline{\mathit{S}}$ ) in metals, insulators and at the transition from metal to insulator, in homogeneous, superlattices, and harmonically confined chains described by the fermionic one-dimensional Hubbard model. While for the homogeneous systems, there is a clear hierarchy between the deviations, \(\varvec{D\%(\bar{S})<D\%(e_0)< \bar{D}\%(n)}\) $\mathit{D}\mathbf{\%}\mathbf{(}\overline{\mathit{S}}\mathbf{)}\mathbf{<}\mathit{D}\mathbf{\%}\mathbf{(}{\mathit{e}}_{\mathbf{0}}\mathbf{)}\mathbf{<}\overline{\mathit{D}}\mathbf{\%}\mathbf{(}\mathit{n}\mathbf{)}$ , and all the deviations decrease with the chain size; for superlattices and harmonic confinement, the relation among the deviations is less trivial and strongly dependent on the superlattice structure and the confinement strength considered. For the superlattices, in general, increasing the number of impurities in the unit cell represents lower precision in the DFT calculations. For the confined chains, DFT performs better for metallic phases, while the highest deviations appear for the Mott and band-insulator phases. This work provides a comprehensive comparative analysis of these methodologies, shedding light on their respective strengths, limitations, and applications.

The DFT calculations were performed using the standard Kohn-Sham scheme within the BALDA approach. It integrated the numerical Bethe-Ansatz (BA) solution of the Hubbard model as the homogeneous density functional within a local-density approximation (LDA) for the exchange-correlation energy. The DMRG algorithms were implemented using the ITensor library, which is based on the matrix product states (MPS) ansatz. The calculations were performed until the energy reaches convergence of at least \(\textbf{10}^{\mathbf {-8}}\) ${\mathbf{10}}^{-\mathbf{8}}$ .