The dynamical quantum phase transitions (DQPTs) and the associated winding numbers have been extensively studied in the context Hermitian system. We consider the non-Hermitian analog of $p$-wave superconductor, supporting Hermitian gapless phase with complex hopping, in presence of on-site or superconducting loss term. This allows us to investigate the effect of non-Hermitian gapless phases on the DQPTs in addition to the Hermitian gapless phases. Our findings indicate that contour analysis of the underlying Hamiltonian, enclosing the origin and/or exceptional points, can predict the occurrences of DQPTs except the quench within the gapless phases. For the Hermitian case with initial and final Hamiltonians both being Hermitian, we find nonmonotonic integer jump for the winding number as the hallmark signature of the gapless phase there. For the hybrid case with initial and final Hamiltonians being Hermitian and non-Hermitian respectively, winding number exhibits integer spike in addition to the nonmonotonic integer jumps. For the non-Hermitian case with initial and final Hamiltonians both being non-Hermitian, the winding number show half-integer jumps for lossy superconductivity that does not have any Hermitian analog. On the other hand, the integer jumps in winding number is observed for lossy chemical potential. We understand our findings by connecting them with the profile of Fisher zeros and number of exceptional points and/or origin.

中文翻译：

---

具有扩展无间隙相的非厄米系统中动态量子相变的异常

动态量子相变 (DQPT) 和相关的绕组数已在 Hermitian 系统的背景下进行了广泛的研究。我们考虑非厄米模拟$p$-波超导体，在存在现场或超导损耗项的情况下，支持具有复杂跳跃的 Hermitian 无间隙相。这使我们能够研究除 Hermitian 无间隙相之外的非 Hermitian 无间隙相对 DQPT 的影响。我们的研究结果表明，包含原点和/或异常点的潜在哈密顿量的轮廓分析可以预测 DQPT 的发生，但无间隙相内的淬火除外。对于初始和最终哈密顿量都是 Hermitian 的 Hermitian 情况，我们发现绕组数的非单调整数跳跃是那里无间隙相的标志性特征。对于初始和最终哈密顿量分别为厄米特和非厄米特的混合情况，除了非单调整数跳跃外，绕组数还表现出整数尖峰。对于初始和最终哈密顿量都是非厄米特的非厄米特情况，绕组数显示半整数跳跃，用于没有任何厄米特模拟的有损超导。另一方面，对于有损化学势，观察到绕组数的整数跳跃。我们通过将它们与费舍尔零点的轮廓以及异常点和/或起源的数量联系起来来理解我们的发现。