2003 Fiscal Year Final Research Report Summary
The standard realization of crystal lattices and spectra of magnetic transition operators
| Project/Area Number |
14540057
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Geometry
|
|---|
| Research Institution | Tohoku University |
|---|
Principal Investigator |
KOTANI Motoko Tohoku University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (50230024)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
FUJIWARA Koji Tohoku University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (60229078)
SHIOYA Takashi Tohoku University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (90235507)
SUNADA Toshikazu Meiji University, School of Science and Technology, Professor, 理工学部, 教授 (20022741)
OHNITA Yoshihiro Tokyo Metropolitan University, Faculty of Science, Professor, 大学院・理学研究科, 教授 (90183764)
IZEKI Hiroyasu Tohoku University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (90244409)
|
|---|
| Project Period (FY) |
2002 – 2003
|
| Keywords | crystal lattice / magnetic transition operators / central limit theorem |
|---|
| Research Abstract |
A crystal lattice is an abelian covering infinite graph of a finite graph. The integer lattices, the triangular lattice, the hexagonal lattice are examples of crystal lattices. We define magnetic transition operators to describe electron transfer on a crystal lattice under periodic magnetic field. The definition is justified by the central limit theorem : Namely, we show that the semigroup generated by the magnetic transition operators converges to the semigroup generated by a magnetic Laplacian of the Euclidean space with the Albanese metric. Magnetic fields on a crystal lattice are defined in terms of the second group cohomology. Next we construct a C^*-algebra associated with the magnetic field and show the magnetic transition operator belongs to the C^*-algebra. By using this, we show the spectra of the magnetic transition operators is a Lipschitz continuous function in magnetic field.
Without magnetic field, electrons behave like random walks. We show large deviation principle holds for random walks on a crystal lattice. By letting lattice spacing smaller, a crystal lattice converges to a finite dimensional vector space with a Banach norm in the Gromov-Hausdorff topology. This Banach norm is characterized in terms of the rate function appearing in the large deviation.
|
