Origin and classification of fractional dynamical topology in non-Hermitian quantum systems

Dynamical quantum phase transitions (DQPT) provide a fundamental framework for characterizing nonequilibrium critical phenomena, and have recently been extended to non-Hermitian systems. Half-integer jumps of the dynamical topological order parameter can arise in both Hermitian and non-Hermitian systems, yet their origin in non-Hermitian settings remains unclear. Here, we investigate DQPTs in a two-band non-Hermitian model and show that these fractional jumps originate from the discontinuity in the complex energy spectrum. We demonstrate that this behavior is intrinsically linked to the steady-state phase diagram of the quench Hamiltonian, and also classify the occurrence of DQPT. We further extend our analysis to the multiband Hamiltonian and the two-dimensional model, providing a unified understanding of fractional dynamical topological signatures.