Non-Abelian geometry, topology, and dynamics of a nonreciprocal Su-Schrieffer-Heeger ladder

Non-Hermiticity naturally breaks down the adiabaticity and thus leads to non-Abelian behaviors in multiband systems. Here, we study how non-Abelian properties emerge in non-Hermitian systems by considering a multiband non-Hermitian model: the nonreciprocal Su-Schrieffer-Heeger (SSH) ladder that is formed by coupling two nonreciprocal SSH chains. Under periodic boundary conditions, we analytically obtain the exact phase diagrams of the geometry of band structure classified by its complex value and gap type, and of the non-Abelian topology based on a newly defined gauge-invariant winding number under the chiral symmetry. Under open boundary conditions, we find that the bulk-boundary correspondence survives in the thermodynamic limit but breaks down for finite sizes along with the emergence of critical non-Hermitian skin effects when the interleg coupling is weak, where the decaying length $\ensuremath{\xi}$ of the bulk skin modes varies with the system size $L$, satisfying the scale-free power law $\ensuremath{\xi}\ensuremath{\propto}L$. Finally, we demonstrate the non-Abelian dynamics of a Bloch state subject to an external constant force in the pseudo-Hermitian symmetric regime in comparison with the non-Hermitian Wilson lines. Our work may stimulate further interests in nontrivial non-Abelian behaviors in non-Hermitian and open quantum systems.