Dynamical vertex approximation in its parquet implementation:   application to Hubbard nano-rings
  We have implemented the dynamical vertex approximation (D$\Gamma$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$\Gamma$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. 
A. Valli, T. Sch\"afer, P. Thunstr\"om, G. Rohringer, S. Andergassen,   G. Sangiovanni, K. Held, and A. Toschi
cond-mat.str-el