Progress in geometric mechanics and topological study of quantum systems

2016 Fiscal Year Final Research Report

• Project/Area Number: 26400068
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Kyoto University
• Principal Investigator: Iwai Toshihiro 京都大学, 情報学研究科, 名誉教授 (10021635)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Keywords: 多様体上の最適化問題 / エネルギーバンド / チャーン数 / ディラック方程式 / 境界条件 / スペクトル流

Outline of Final Research Achievements

Geomtric machanics is applied to optimization problems on Grassmann manifolds, which are realted to eigenvalue problems for real symmetric matrices. Topological study is made on Hamltonians which describe generalized spin-orbital angular momentum coupling. In accordance with the band rearrangement for the quatum Hamiltonian against a parameter, change in Chern number of the eigen-line bundle is observed in the corresponding semi-quantum system. The change is shown to be counted by means of the respective mapping degrees assigned to singular points at which the Hamiltonian is linearized through homotopy deformation. Conversely, the linearized simi-quantum Hamiltonian is quantized to give a Dirac operator, for which the eigenvalue problem can be solved under boundary conditions to exhibit energy band rearrangement against a parameter. In practice, 2D Dirac equations have been solved under the APS (Atiyah-Patodi-Singer) and the chiral bag boundary conditions.

Free Research Field

幾何学的力学系理論