Taste-splittings of staggered, Karsten-Wilczek and Borici-Creutz fermions under gradient flow in 2D

Karsten-Wilczek and Borici-Creutz fermions show a near-degeneracy of the $2$ species involved, similar to the $2^{d/2}$ species of staggered fermions. Hence in $d=2$ dimensions all three formulations happen to be minimally doubled (two species). This near-degeneracy shows up both in the eigenvalue spectrum of the respective Dirac operator and in spectroscopic quantities (e.g. the pion mass), but in the former case it is easier to quantify. We use the quenched Schwinger model to determine the low-lying eigenvalues of these fermion operators at a fixed gradient flow time $\tau$ (either in lattice units or in physical units, hence keeping either $\tau/a^2$ or $e^2 \tau$ fixed at all $\beta$).