The bulk-boundary correspondence in Hermitian systems is governed by wave-function topology, whereas in non-Hermitian systems exhibiting the skin effect, it is determined by complex energy topology. In this work, we establish the bulk-boundary correspondence in non-Hermitian class-${\mathrm{D}}^{†}$ topological systems within the framework of wave-function topology. Specifically, we demonstrate that bulk polarization serves as a universal topological invariant, effectively characterizing anisotropic in-gap edge states in Hermitian class-D systems and isotropic in-gap edge states in non-Hermitian class-${\mathrm{D}}^{†}$ systems. Additionally, the spectral winding number determines the localization direction of the isotropic in-gap edge states in class-${\mathrm{D}}^{†}$ systems. By employing the Wilson loop approach to compute the bulk polarization, our work not only restores the bulk-boundary correspondence in non-Hermitian class-${\mathrm{D}}^{†}$ systems, but it also provides valuable insights into the connection between wave-function topology and non-Hermitian band theory.

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• Topological insulators
• Topological phases of matter
• Topological materials