Acoustic Su–Schrieffer–Heeger chain with phase nonreciprocal couplings

The Su–Schrieffer–Heeger model becomes non-Hermitian in general when the reciprocity of wave propagation is broken in the couplings. Here, we introduce phase non-reciprocity in the coupling of an acoustic Su–Schrieffer–Heeger chain by adding direction-dependent imaginary parts in the coupling strengths while keeping the real parts independent on the direction. We theoretically and numerically investigate the effect of the phase non-reciprocity on the band structure and the topology of the chain. The hermiticity is broken in general, but there is still one special case where the model remains Hermitian. An acoustic analog of this case is build where fluid flow is introduced in the couplings. In the non-Hermitian cases, the skin effect is visible except for one specific transition case where the system remains non-Hermitian but where reciprocity is maintained.