By studying the topological invariance and Berry phase in non-Hermitian systems, we reveal the basic properties of the complex Berry phase and generalize the global Berry phases $Q$ to identify the topological invariance for non-Hermitian systems. We find that $Q$ can identify topological invariance in two kinds of non-Hermitian model, the two-level non-Hermitian Hamiltonian and the bipartite dissipative model. For the bipartite dissipative model, an abrupt change of the Berry phase in the parameter space reveals a quantum phase transition and is related to the exceptional points. These results give the basic relationships between the Berry phase and the quantum and topological phase transitions of non-Hermitian systems.

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.

We explore the signatures of Majorana fermions in a nanowire based topological superconductor-quantum dot-topological superconductor hybrid device by charge transport measurements. At zero magnetic field, well-defined Coulomb diamonds and the Kondo effect are observed. Under the application of a finite, sufficiently strong magnetic field, a zero-bias conductance peak structure is observed. It is found that the zero-bias conductance peak is present in many consecutive Coulomb diamonds, irrespective of the even-odd parity of the quasi-particle occupation number in the quantum dot. In addition, we find that the zero-bias conductance peak is in most cases accompanied by two differential conductance peaks, forming a triple-peak structure, and the separation between the two side peaks in bias voltage shows oscillations closely correlated to the background Coulomb conductance oscillations of the device. The observed zero-bias conductance peak and the associated triple-peak structure are in line with Majorana fermion physics in such a hybrid topological system.

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 driven dynamics across the critical points of the Yang-Lee edge singularities (YLESs) in a finite-size quantum Ising chain with an imaginary symmetry-breaking field. In contrast to the conventional classical or quantum phase transitions, these phase transitions are induced by tuning the strength of the dissipation in a non-Hermitian system and can occur even at finite size. For conventional phase transitions, universal behaviors in driven dynamics across critical points are usually described by the Kibble-Zurek mechanism, which states that the scaling in dynamics is dictated by the critical exponents associated with one critical point and topological defects will emerge after the quench. While the mechanism leading to topological defects breaks down in the YLES, we find that for small lattice size, the driven dynamics can still be described by the Kibble-Zurek scaling with the exponents determined by the ($0+1$)-dimensional YLES. For medium finite size, however, the driven dynamics can be described by the Kibble-Zurek scaling with two sets of critical exponents determined by both the ($0+1$)-dimensional and the ($1+1$)-dimensional YLESs.

A scheme is proposed for the generation of a three-dimensional entangled state for two atoms trapped in a cavity via quantum Zeno dynamics. Because the scheme is based on the resonant interaction, the time required to produce entanglement is very short compared with the dispersive protocols. We show that the resulting effective dynamics allows for the creation of robust qutrit-qutrit entanglement. The influence of various decoherence processes such as spontaneous emission and photon loss on the fidelity of the entangled state is investigated. Numerical results show that the scheme is robust against the cavity decay since the evolution of the system is restricted to a subspace with null-excitation cavity fields. Furthermore, the present scheme has been generalized to realize N-dimensional entanglement for two atoms.