Study of Localization Transition in Non-Hermitian Quasi Periodic System

Abstract

Disorder-driven delocalization-localization (DL) transition and its impact on the energy spectrum have been extensively studied in diverse physical systems. In Hermitian systems with random disorder, this transition is famously known as the Anderson localization (AL) transition. It is well established that only with the random disorder the AL transition takes place in a three-dimensional (3D) lattice system. However, with quasiperiodic (QP) potential, DL transition can take place even in one-dimensional (1D) lattice systems. Interestingly, in recent years it emerged that such DL transitions can also take place in the non-Hermitian systems, with a much more fascinating set of associated phenomena, compared to their Hermitian counterpart. The non-Hermitian systems can broadly be classified into two classes; one class without PT symmetry and the other class with PT symmetry. The absence or presence of such symmetry drastically alters the nature of the eigenspectrum across the DL transition. In this thesis, we investigate the DL transition and its associated changes in the energy spectrum in the context of non-Hermitian lattice systems with QP potential. In the first part of our study, we have proposed a generalized non-PT symmetric 1D non-Hermitian QP lattice. Well-studied non-Hermitian models, namely the Aubry-André-Harper (AAH) model and the Hatano-Nelson (HN) model with QP potential can be obtained from our proposed Hamiltonian as limiting cases. In our generalized Hamiltonian, the non-Hermitian behavior arises due to asymmetry in the hoppings and the complex nature of the QP potentials. We demonstrate that the interplay between these two leads to more diverse and intricate phases. For identical modulation of the real and the complex parts of the QP potential, we obtain the analytical expression of the critical point that precisely captures the DL transition for systems with periodic boundary conditions (PBC). Our numerical investigations reveal that the critical point remains unchanged even with open boundary conditions (OBC). One particularly fascinating aspect of our findings is the emergence of a mixed phase between the delocalized and localized regions in systems with non-identical modulation of the real and complex parts of the QP potential. This mixed phase presents a remarkable coexistence of skin modes and localized states for systems with OBC, while systems with PBC exhibit a coexistence of delocalized and localized states within the mixed phase. To provide further insights into the underlying physics, we construct comprehensive phase diagrams, shedding light on the crucial role of various parameters in a broad range of non-Hermitian QP lattices. In the second part of our investigation, we turn our attention to the PT symmetry class of non-Hermitian systems. In particular, we have focussed on one such system, namely the well-known non-Hermitian AAH model, that can be obtained from our preceding generalized Hamiltonian as a limiting case. The nature of the DL transition is extensively studied in this system. Recently, in the contemporary realm of spintronic device development, spin-orbit interaction has garnered considerable attention from researchers due to its potential impact on electronic systems. This has led to the exploration of the influence of Rashba spin-orbit (RSO) on DL transition in the Hermitian AAH model. However, a similar investigation in the context of non-Hermitian lattice systems has not been taken up so far. Motivated by this, we investigate the impact of RSO coupling on the DL transition and the energy spectrum in the non-Hermitian AAH model. Incorporation of RSO coupling into the PT symmetric AAH model does not alter its symmetry. Employing computational techniques and analytical methods, we scrutinize the alterations induced by RSO coupling on the DL transition. We observed consistent quantitative changes in the DL transition point for both PBC and OBC cases. However, in PBC, the breaking of PT symmetry aligns with the DL transition point, while in the OBC case, the symmetry remains broken regardless. There is a crucial difference between the DL transition in the Anderson model and the systems addressed in the previous sections. The DL transition in the Anderson model is associated with the existence of a mobility edge, while is absent in both the systems addressed in the previous chapters. However, some studies have revealed that the introduction of short-range hopping gives rise to a mobility edge in the 1D non Hermitian PT symmetric AAH Hamiltonian. In the third and final phase of our investigation, we address the fate of the mobility edge and PT symmetry in such systems with RSO coupling. Interestingly, our results show that the presence of RSO can significantly modify the mobility edge identically in both PBC and OBC systems; in certain cases to the extent of almost completely suppressing it. However, in systems with PBC, the breaking of PT symmetry matches the point where the mobility edge occurs. On the other hand, in systems with OBC, the symmetry remains broken regardless, which is similar to what we observed in the non-Hermitian AAH model we examined earlier with RSO coupling. In summary, this thesis provides a comprehensive exploration of the DL transition and its implications for the energy spectrum in non-Hermitian 1D QP systems with and without PT symmetry. Our findings offer valuable insights into the fundamental aspects of DL transitions which could potentially benefit various fields, ranging from condensed matter physics to quantum information science.