In the present study, the interplay among interaction, topology, quasiperiodicity, and non-Hermiticity is studied. The hard-core bosons model on a one-dimensional lattice with asymmetry hoppings and quasiperiodic onsite potentials is selected. This model, which preserves time-reversal symmetry (TRS), will exhibit three types of phase transition: Real-complex transition of eigenenergies, topological phase transition, and many-body localization (MBL) phase transition. For the real-complex transition, it is found that the imaginary parts of the eigenenergies are always suppressed by the MBL. Moreover, by calculating the winding number, a topological phase transition can be revealed with the increase of potential amplitude, and we find that the behavior is quite different from the single-particle systems. Based on our numerical results, we conjecture that these three types of phase transition occur at the same point in the thermodynamic limit, and the MBL transition of quasiperiodic system and disordered system should belong to different universality classes. Finally, we demonstrate that these phase transitions can profoundly affect the dynamics of the non-Hermitian many-body system.

In this paper, we investigate the driven dynamics of the localization transition in the non-Hermitian Aubry-André model with the periodic boundary condition. Depending on the strength of the quasiperiodic potential $\lambda$, this model undergoes a localization-delocalization phase transition. We find that the localization length $\xi$ satisfies $\xi \sim {\varepsilon }^{-\nu }$ with $\varepsilon$ being the distance from the critical point and $\nu =1$ being a universal critical exponent independent of the non-Hermitian parameter. In addition, from the finite-size scaling of the energy gap between the ground state and the first excited state, we determine the dynamic exponent $z$ as $z=2$. The critical exponent of the inverse participation ratio for the $n\mathrm{th}$ eigenstate is also determined as $s=0.1197$. By changing $\varepsilon$ linearly to cross the critical point, we find that the driven dynamics can be described by the Kibble-Zurek scaling (KZS). Moreover, we show that the KZS with the same set of the exponents can be generalized to the localization phase transitions in the excited states.

In this paper, the interplay of the non-Herimiticity and the cascade of delocalization transition in a quasiperiodic chain are studied. The study is applied in the non-Hermitian interpolating Aubry-André-Fibonacci (IAAF) model, which combines the non-Hermitian Aubry-André (AA) model and the non-Hermitian Fibonacci model through a varying parameter, and the non-Hermiticity in this model is introduced by nonreciprocal hopping. In the non-Hermitian AA limit, the system undergoes a delocalization transition by tuning the potential strength. At the critical point, the spatial distribution of the critical state shows a self-similar structure with the relative distance between the peaks being the Fibonacci sequence, and the finite-size scaling of the inverse participation ratios $\left(\mathrm{IPRs}\right)$ of the critical ground state with lattice size $L$ shows that ${\mathrm{IPR}}_g\propto L^{-0.1189}$. In the non-Hermitian Fibonacci limit, we find that the system is always in the extended phase. Along the continuous deformation from the non-Hermitian AA model into the non-Hermitian Fibonacci model in the IAAF model, the cascade of the delocalization transition is found, but only a few plateaux appear. Moreover, the self-similar structure of spatial distribution for the critical modes along the cascade transition is also found. In addition, we find that the delocalization transition and the real-complex transition for the excited states happen at almost the same parameter. Our results show that the non-Hermiticity provides an additional knob to control the cascade of the delocalization transition besides the on-site potential.

We study the behavior of the out-of-time-ordered correlator (OTOC) in a non-Hermitian quantum Ising system. We show that the OTOC can diagnose not only the exceptional point in the states with lowest real parts of Hamiltonian spectra (LRHS), which host the Yang-Lee edge singularity, but also the exceptional point in states with higher real parts of Hamiltonian spectra (HRHS). We find that the evolution of the OTOC in the parity-time symmetric phase can be divided into two stages: In the short-time stage, the OTOC oscillates periodically, and when the parameter is near the exceptional point of the LRHS, this oscillation behavior can be described by both the scaling theory of the zero-dimensional ($0\mathrm{D}$) quantum Yang-Lee edge singularity and the scaling theory of the $1\mathrm{D}$ quantum Ising model, while in the long-time stage the OTOC increases exponentially, controlled by the exceptional point of the HRHS. Possible experimental realizations are then discussed.

In this paper, we study the critical behaviors in the non-Hermitian disorder Aubry-André (DAA) model, and we assume the non-Hermiticity is introduced by nonreciprocal hopping. We employ the localization length $\xi$, the inverse participation ratio ($\mathrm{IPR}$), and the energy gap $\mathrm{\Delta }E$ as the characteristic quantities to describe the critical properties of the localization transition. By performing scaling analysis, the critical exponents of the non-Hermitian Anderson model and the non-Hermitian DAA model are obtained, and these critical exponents are different from their Hermitian counterparts, indicating that the Hermitian and non-Hermitian Anderson and DAA models belong to different universality classes. The critical exponents of the non-Hermitian DAA model are remarkably different from both the pure non-Hermitian AA model and the non-Hermitian Anderson model, showing that disorder is an independent relevant direction at the non-Hermitian AA model critical point. We further propose a hybrid scaling law to describe the critical behavior in the overlapping critical region constituted by the critical regions of the non-Hermitian DAA model and the non-Hermitian Anderson localization.

The conventional Kibble-Zurek scaling describes the scaling behavior in the driven dynamics across a single critical region. In this paper, we study the driven dynamics across an overlapping critical region, in which a critical region (Region A) is overlaid by another critical region (Region B). We develop a hybridized Kibble-Zurek scaling (HKZS) to characterize the scaling behavior in the driven process. According to the HKZS, the driven dynamics in the overlapping region can be described by the critical theories for both Region A and Region B simultaneously. This results in a constraint on the scaling function in the overlapping critical region. We take the quantum Ising chain in an imaginary longitudinal field as an example. In this model, the critical region of the Yang-Lee edge singularity and the critical region of the ferromagnetic-paramagnetic phase transition overlap with each other. We numerically confirm the HKZS by simulating the driven dynamics in this overlapping critical region. The HKZSs in other models are also discussed.