The topological counterparts of non-Hermitian SSH models

Inspired by the relevance between the asymmetric coupling amplitude and the imaginary gauge field, we construct the counterpart of the non-Hermitian SSH model. The idea is the nonzero imaginary magnetic flux vanishing when the boundary condition changes from periodic to open. The zero imaginary magnetic flux of the counterpart leads to the eliminating of the non-Hermitian skin effect and the non-Hermitian Aharonov-Bohm effect which ensures the recovery of the conventional bulk-boundary correspondence from the non-Bloch bulk-boundary correspondence. We explain how some the non-Hermitian models can be transformed to the non-Hermitian SSH models and how the non-reciprocal hopping in the non-Hermitian SSH models can be transformed from one term to the other terms by the similarity transformations. We elaborate why the effective imaginary magnetic flux disappears due to the interplay of the non-reciprocal hoppings in the partner of the non-Hermitian SSH model. As the results, we obtain the topological invariants of the non-Hermitian SSH model in analytical form defined in conventional Brillouin zone. The non-Hermitian SSH model in domain configuration on a chain is discussed with this method. The technique gives an alternative way to study the topological properties of non-Hermitian systems.