Quench dynamics in nonreciprocal Aubry-Andr\'e-Harper model

Abstract

The critical phase of a non-Hermitian quasicrystal can support stronger transport than its surrounding delocalized phase. We demonstrate this anomalous behavior in the one-dimensional nonreciprocal Aubry-Andr\'e-Harper model through a combined study of dynamical quantum phase transitions (DQPTs) and wavepacket diffusion. Using a parity-sorted energy-spectrum classification that directly encodes the generalized $\mathcal{PT}$ symmetry, we find that DQPTs in this system are energy-resolved, in contrast to the energy-independent DQPTs of Hermitian quasicrystals. The energy-resolved features are most pronounced when the initial and final Hamiltonians belong to different phases (localized or extended), and they are tied to the even-odd index structure of the spectrum, which we exploit to organize the quench-dynamical landscape. For wavepacket dynamics after a single-site quench, the diffusion exponent $\beta$, extracted from the long-time power-law scaling of the root-mean-square displacement $\sigma(\tau)$, partitions the phase diagram into four distinct regimes. In the Hermitian limit the extended phase is ballistic ($\beta=1$), the critical phase is normally diffusive ($\beta=0.5$), and the localized phase yields $\beta\to 0$. Nonreciprocity reverses this hierarchy: the extended phase becomes normally diffusive, while the critical phase turns ballistic. We trace the anomalous $\beta=1$ at criticality to the self-similar multifractal structure of the critical eigenstates, whose nodal positions are organized by the golden ratio. A finite-size scaling ansatz built on the wave-front propagation yields $\sigma(\tau)\propto\tau$. The parity-resolved DQPTs and the $\beta$-phase diagram establish two complementary dynamical diagnostics of nonreciprocal quasicrystals, in which nonreciprocity promotes transport at the critical point and suppresses it in the delocalized phase.