We have implemented the dynamical vertex approximation $\left(\mathrm{D}\Gamma \mathrm{A}\right)$ in its full parquet-based version to include spatial correlations on all length scales and in all scattering channels. The algorithm is applied to study the electronic self-energies and the spectral properties of finite-size one-dimensional Hubbard models with periodic boundary conditions (nanoscopic Hubbard rings). From a methodological point of view, our calculations and their comparison to the results obtained within dynamical mean-field theory, plain parquet approximation, and the exact numerical solution allow us to evaluate the performance of the $\mathrm{D}\Gamma \mathrm{A}$ algorithm in the most challenging situation of low dimensions. From a physical perspective, our results unveil how nonlocal correlations affect the spectral properties of nanoscopic systems of various sizes in different regimes of interaction strength.

• Received 13 October 2014
• Revised 4 February 2015

DOI:https://doi.org/10.1103/PhysRevB.91.115115