Tunably realizing flat-bands and exceptional points in kinetically frustrated systems: An example on the non-Hermitian Creutz ladder

We study a non-Hermitian extension of the Creutz ladder with generic non-reciprocal hopping. By mapping the ladder onto two decoupled non-Hermitian Su--Schrieffer--Heeger (SSH) chains, we uncover a rich structure in parameter space under different boundary conditions. Under periodic boundary conditions, the spectrum admits a fine-tuned line in parameter space with entirely real eigenvalues, while deviations from this line induce a real--complex spectral transition without crossing exceptional points. In contrast, an exact analytical diagonalization under open boundary conditions reveals extended regions in parameter space with purely real or purely imaginary spectra, separated from complex spectral domains by exceptional lines. The intersections of these exceptional lines define triple-junction points where distinct spectral regimes meet, giving rise to a structured phase diagram that is absent under periodic boundary conditions. We further show that flat bands in this system can occur both as Hermitian diabolical points and as non-Hermitian exceptional points, known as exceptional flat bands, where the dynamics is more stringent than in the Hermitian case, leading to distinct spectral and dynamical signatures.