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Non-Hermitian systems can exhibit extraordinary sensitivity to boundaryconditions, where the localized topological boundary modes and thenon-Hermitian skin effects can coexist or individually appear, leading to thenon-Hermitian four-fold phase transitions. Several efforts have been made toreconstruct the non-Hermitian bulk-boundary correspondence, but it is stillchallenging to present the four-fold phase diagram more quickly. Here, wedirectly give the exact expression of four-fold phase-transition point withoutany numerical calculation in the parameter space, through the combination ofexact solution approach and non-Hermitian winding numbers. Our exact solutionapproach not only characterizes the energy and penetration length of skinmodes, but also unveils the intrinsic difference between the topological modesand skin modes. To illustrate the general benefits of our methods, a morecomplicated non-Hermitian Su-Schrieffer-Heeger chain with long-range couplingsand a modified Lee model without sub-lattice symmetry are investigated,providing a general routine to study various non-Hermitian phases.

Multi-fold non-Hermitian phase transitions and exact solution of non-Hermitian skin effects