The consistency between eigenvalues calculated under open and periodic boundary conditions, named as { \it bulk-bulk correspondence} ($\mathcal{ BBC}$), can be destroyed in systems with non-Hermitian skin effect (NHSE). In spite of the great success of the generalized Brillouin zone (GBZ) theory in clean non-Hermitian systems, the applicability of GBZ theory is questionable when the translational symmetry is broken. Thus, it is of great value to rebuild the $\mathcal{ BBC}$ for disorder samples, which extends the application of GBZ theory in non-Hermitian systems. Here, we propose a scheme reconstructing $\mathcal{ BBC}$, which can be regarded as the solution of an optimization problem. By solving this optimization problem analytically, we reconstruct the $\mathcal{ BBC}$ and obtain the modified GBZ theory in several prototypical disordered non-Hermitian models. The modified GBZ theory gives a precise description of NHSE, which predicts the intriguing disorder-enhanced and disorder-irrelevant NHSEs.
Categories: cond-mat.dis-nn