| 研究実績の概要 |
In this year, I have made progress on two tracks related to the dynamical problem at the exceptional point. While the original intention was to focus on the exceptional point dynamics at the level of the Liouvillian, I have found a new discovery that redirects my focus somewhat. I have found an extension of a simple model for a topological insulator that gives rise to an exceptional point with unique properties.
Usually the model for a topological insulator is finite and exhibits edge states (or zero-energy modes) that have nearly zero energy eigenvalue and act as conducting surface states despite that the bulk of the system behaves as a conductor. These states are partially protected against disorder. I have found that by taking a semi-infinite extension of the Su-Schrieffer-Heeger (SSH) model, which has alternating couplings along a 1-D lattice, I can obtain an edge state with eigenvalue exactly zero such that the protection against disorder is maximized. Further, by introducing an impurity at the endpoint of the system, I can show that two new parameter regimes appear that have no correspondence in the uniform lattice. Further, all of the eigenvalues appear inside the bulk gap in these two regions, which are separated from the 'trivial' parameter space by an exceptional point that has topological properties.
On a separate track, I have also made some preliminary progress on the problem of writing the Lindblad equation for a simple system and will consider how to extend this to incorporate quantum jumps at exceptional point in future work.
|
| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
Although the focus has changed somewhat from expectations, at least at this early stage, the preliminary findings are rather promising. We expect to be able to observe unique Markovian and non-Markovian dynamics at the topological exceptional point in our semi-infinite lattice system, even working at just the Hamiltonian level. The eventual extension to the Liovillian formalism should be interesting as well.
|
| 今後の研究の推進方策 |
In future work, I plan to analyze the survival probability dynamics at the topological exceptional point as well as in the two 'non-trivial' (topological) regions of the parameter space in which the discrete spectrum is confined to the bulk band gap of the semi-infinite lattice system. In each of these cases, I expect that the dynamics should experience some degree of protection from disorder due to the inherited topological properties from the semi-infinite SSH chain. I also plan to analyze the impulsive dynamics that can be explored through the local density of states. Preliminary analysis indicates this quantity should be significantly enhanced at the exceptional point. Further, one should eventually consider the extension to the Liouvillian picture.
|
