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We study the characterization and realization of higher-order topologicalAnderson insulator (HOTAI) in non-Hermitian systems, where the non-Hermitianmechanism ensures extra symmetries as well as gain and loss disorder.Weilluminate that the quadrupole moment $Q_{xy}$ can be used as the real spacetopological invariant of non-Hermitian higher-order topological insulator(HOTI). Based on the biorthogonal bases and non-Hermitian symmetries, we provethat $Q_{xy}$ can be quantized to $0$ or $0.5$. Considering the disordereffect, we find the disorder-induced phase transition from normal insulator tonon-Hermitian HOTAI. Furthermore, we elucidate that the real space topologicalinvariant $Q_{xy}$ is also applicable for systems with the non-Hermitian skineffect. Our work enlightens the study of the combination of disorder andnon-Hermitian HOTI.

Real space topological invariant and higher-order topological Anderson insulator in two-dimensional non-Hermitian systems