| In 1980,it was found experimentally(von Klitzing,Dorda and Pepper)that under the condition of low temperature and strong magnetic field,two-dimensional electron gas would have an integer Hall conductance,known as the integer quantum Hall effect.The quantized Hall conductance is closely related to the topological invariants of the energy bands.In the following years,it is found that the manifestation of topology in physics is not just limited to integer quantum Hall effect,which led to the discovery of topological insulators and topological superconductors.In the classical GinzburgLandau phase transition theory,the phase transition is characterized by the local order parameters of the system.However,this picture is no longer true in topological phase transitions,where no symmetry is broken.With the development of the study of topological matter,we have gained deep understanding of this field.In recent years,quantum simulation platforms such as ultra-cold atoms and photonics have developed rapidly.Thanks to their powerful quantum simulation capabilities,synthetic topological systems can be realized,and topological phenomena in open quantum systems have attracted widespread attention.Open systems are ubiquitous in the real world,because coupling between systems and their environment is often unavoidable.Under appropriate conditions,nonHermitian Hamiltonians can provide a convenient description for open systems.In recent years,topology in non-Hermitian quantum systems has been studied extensively,and the rich physical phenomena predicted by them have been confirmed on several experimental platforms.Bulk-boundary correspondence is one of the most important principles of topological matter.It dictates that topological edge states under open boundaries can be predicted by topological invariants calculated in Bloch band theory.However,in a large class of non-Hermitian quantum systems,the traditional bulk-boundary correspondence is no longer valid.In such systems,the non-Hermite skin effects play a key role.One needs to introduce the concept of generalized Brillouin zone and the topology number obtained from the generalized Brillouin zone can perfectly predict the number of topological edge states.As a result,the non-Bloch band theory is proposed,through which we can get the band spectrum correctly and restore the bulk-boundary correspondence.In this paper,we focuses on the frontier problem in the field of topological states,and mainly studies quantum open systems with non-Hermitian skin effect,The main results are as follows1.We construct a two-dimensional,discrete-time quantum walk exhibiting nonHermitian skin effects under open-boundary conditions,and restore the non-Hermitian bulk-boundary correspondence.As a confirmation of the non-Hermitian bulk-boundary correspondence,we show that the emergence of topological edge states are consistent with Floquet winding numbers calculated by using a non-Bloch band theory invoking time-dependent generalized Billouin zones.Further,the non-Bloch topological invariants associated with quasienergy bands are captured by a non-Hermitian local Chem marker in real space,defined through local biorthogonal eigen wave functions of the non-unitary Floquet operator.2.We study the quench dynamics of non-Hermitian topological models with nonHermitian skin effects,and demonstrate how the non-Bloch topological invariants can be revealed from the dynamics.Adopting the non-Bloch band theory and projecting quench dynamics onto the generalized Brillouin zone,we find that emergent topological structures,in the form of dynamic skyrmions,exist in the generalized momentumtime domain,and are correlated with the non-Bloch topological invariants of the static Hamiltonians.The skyrmion structures anchor on the fixed points of dynamics whose existence are conditional on the coincidence of generalized Brillouin zones of the preand post-quench Hamiltonians.Global features of dynamic skyrmions,however,persist well beyond such a condition,constituting signatures for a general dynamic detection scheme of non-Bloch topology in the presence of non-Hermitian skin effects.Applying our theory to an experimentally relevant,non-unitary quantum walk,we explicitly demonstrate how the non-Bloch topological invariants can be revealed through the nonBloch quench dynamics.3.We discuss the systematic engineering of quasicrystals in open quantum systems where quasiperiodicity is introduced through purely dissipative processes.While the resulting short-time dynamics is governed by non-Hermitian variants of the Aubry-AndréHarper model,we demonstrate how phases and phase transitions pertaining to the nonHermitian quasicrystals fundamentally change the long-time,steady-state-approaching dynamics under the Lindblad master equation.Our schemes are based on an exact mapping between the eigenspectrum of the Liouvillian superoperator with that of the nonHermitian Hamiltonian,under the condition of quadratic fermionic systems subject to linear dissipation.4.We investigate the localization and topological properties of non-Hermitian topological Anderson insulators and show the competition between Anderson localization and non-Hermitian skin effect.Disorder and non-Hermiticity dramatically impact the topological and localization properties of a quantum system,giving rise to intriguing quantum states of matter.The two distinct localization mechanisms prompt a nonmonotonous change in profile of the Lyapunov exponent,which we reveal through dynamic observables.We then study the disorder-induced topological phase transitions,and demonstrate their biorthogonal criticality. |
