Taste-split staggered actions: eigenvalues, chiralities and Symanzik improvement

The eigenvalue spectra of staggered fermions with an Adams and/or Hoelbling mass term are studied. The chiralities of the eigenmodes reflect whether the chirality linked to the unflavored approximate (\gamma_5 \times 1) or the flavored exact (\gamma_5 \times \xi_5) staggered symmetry is considered, and which one of the RR, LR, RL, LL eigenmode definitions is used. In either case a sensitivity to the topological charge of the gauge background is found. We discuss how to remove the leading cut-off effects of these actions by means of a properly tuned improvement term and/or the overlap procedure. The combination of Symanzik improvement and link smearing radically improves the properties of the physical branch.