Dynamical vertex approximation in its parquet implementation: application to Hubbard nano-rings

We have implemented the dynamical vertex approximation (D$\mathrm{\Gamma }$A) in its full parquet-based version to include spatial correlations on all length scales and in {\sl 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 D$\mathrm{\Gamma }$A algorithm in the most challenging situation of low dimensions. From a physical perspective, our results unveil how non-local correlations affect the spectral properties of nanoscopic systems of various sizes in different regimes of interaction strength.