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Register a new userLocalization, $mathcal{PT}$-Symmetry Breaking and Topological Transitions in non-Hermitian Quasicrystals
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According to the topological band theory of a Hermitian system, the different electronic phases are classified in terms of topological invariants, wherein the transition between the two phases characterized by a different topological invariant is the primary signature of a topological phase transition. Recently, it has been argued that the delocalization-localization transition in a quasicrystal, described by the non-Hermitian $mathcal{PT}$-symmetric extension of the Aubry-Andr{e}-Harper (AAH) Hamiltonian can also be identified as a topological phase transition. Interestingly, the $mathcal{PT}$-symmetry also breaks down at the same critical point. However, in this article, we have shown that the delocalization-localization transition and the $mathcal{PT}$-symmetry breaking are not connected to a topological phase transition. To demonstrate this, we have studied the non-Hermitian $mathcal{PT}$-symmetric AAH Hamiltonian in the presence of Rashba Spin-Orbit (RSO) coupling. We have obtained an analytical expression of the topological transition point and compared it with the numerically obtained critical points. We have found that, except in some special cases, the critical point and the topological transition point are not the same. In fact, the delocalization-localization transition takes place earlier than the topological transition whenever they do not coincide.
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We propose a general analytic method to study the localization transition in one-dimensional quasicrystals with parity-time ($mathcal{PT}$) symmetry, described by complex quasiperiodic mosaic lattice models. By applying Avilas global theory of quasiperiodic Schrodinger operators, we obtain exact mobility edges and prove that the mobility edge is identical to the boundary of $mathcal{PT}$-symmetry breaking, which also proves the existence of correspondence between extended (localized) states and $mathcal{PT}$-symmetry ($mathcal{PT}$-symmetry-broken) states. Furthermore, we generalize the models to more general cases with non-reciprocal hopping, which breaks $mathcal{PT}$ symmetry and generally induces skin effect, and obtain a general and analytical expression of mobility edges. While the localized states are not sensitive to the boundary conditions, the extended states become skin states when the periodic boundary condition is changed to open boundary condition. This indicates that the skin states and localized states can coexist with their boundary determined by the mobility edges.
We investigate localization-delocalization transition in one-dimensional non-Hermitian quasiperiodic lattices with exponential short-range hopping, which possess parity-time ($mathcal{PT}$) symmetry. The localization transition induced by the non-Hermitian quasiperiodic potential is found to occur at the $mathcal{PT}$-symmetry-breaking point. Our results also demonstrate the existence of energy dependent mobility edges, which separate the extended states from localized states and are only associated with the real part of eigen-energies. The level statistics and Loschmidt echo dynamics are also studied.
The emergence of the mobility edge (ME) has been recognized as an important characteristic of Anderson localization. The difficulty in understanding the physics of the MEs in three-dimensional (3D) systems from a microscopic picture promotes discovering of models with the exact MEs in lower-dimensional systems. While most of previous studies concern on the one-dimensional (1D) quasiperiodic systems, the analytic results that allow for an accurate understanding of two-dimensional (2D) cases are rare. In this Letter, we disclose an exactly solvable 2D quasicrystal model with parity-time ($mathcal{PT}$) symmetry displaying exact MEs. In the thermodynamic limit, we unveil that the extended-localized transition point, observed at the $mathcal{PT}$ symmetry breaking point, is of topological nature characterized by a hidden winding number defined in the dual space. The 2D non-Hermitian quasicrystal model can be realized in the coupling waveguide platform, and the localization features can be detected by the excitation dynamics.
Time-periodic driving fields could endow a system with peculiar topological and transport features. In this work, we find dynamically controlled localization transitions and mobility edges in non-Hermitian quasicrystals via shaking the lattice periodically. The driving force dresses the hopping amplitudes between lattice sites, yielding alternate transitions between localized, mobility edge and extended non-Hermitian quasicrystalline phases. We apply our Floquet engineering approach to five representative models of non-Hermitian quasicrystals, obtain the conditions of photon-assisted localization transitions and mobility edges, and find the expressions of Lyapunov exponents for some models. We further introduce topological winding numbers of Floquet quasienergies to distinguish non-Hermitian quasicrystalline phases with different localization nature. Our discovery thus extend the study of quasicrystals to non-Hermitian Floquet systems, and provide an efficient way of modulating the topological and transport properties of these unique phases.
We investigate the localization and topological transitions in a one-dimensional (interacting) non-Hermitian quasiperiodic lattice, which is described by a generalized Aubry-Andr{e}-Harper model with irrational modulations in the off-diagonal hopping and on-site potential and with non-Hermiticities from the nonreciprocal hopping and complex potential phase. For noninteracting cases, we reveal that the nonreciprocal hopping (the complex potential phase) can enlarge the delocalization (localization) region in the phase diagrams spanned by two quasiperiodical modulation strengths. We show that the localization transition are always accompanied by a topological phase transition characterized the winding numbers of eigenenergies in three different non-Hermitian cases. Moreover, we find that a real-complex eigenenergy transition in the energy spectrum coincides with (occurs before) these two phase transitions in the nonreciprocal (complex potential) case, while the real-complex transition is absent under the coexistence of the two non-Hermiticities. For interacting spinless fermions, we demonstrate that the extended phase and the many-body localized phase can be identified by the entanglement entropy of eigenstates and the level statistics of complex eigenenergies. By making the critical scaling analysis, we further show that the many-body localization transition coincides with the real-complex transition and occurs before the topological transition in the nonreciprocal case, which are absent in the complex phase case.
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