Condensed Matter Seminar
物性論セミナーMarch 15 (Thurs.), 2012, 15:00 -17:00
総合研究棟B棟1階: 108
http://rhodia.ph.tsukuba.ac.jp/~hatsugai/modules/pico/PDF/Map.pdf

Electron in the field of magnetic charge : Tight binding
solution and mapping on a realistic physical system
Prof. Yshai Avishai
Ben Grion University, Israel
Prof. Avishai has been staying with us at the University of Tsukuba about
two month until the end of March. In this opportunity, he is willing to give atalk on his recent research. The talk is divided into two. The first part isintroductory and we welcome very much for non-specialists such as noncondensed matter graduate students. After taking a short break, he willcontinue to present the detailed results for people interested in. The
surprising connection between the two parts comes at the end of the secondpart.
The detail of the talk is described in the next page.
今回イスラエルから筑波大学の国際連携プロジェクトの一環で筑波大学に滞在されているAvishai先生にセミナーをお願いいたしました。セミナーは2部構成で前半部分は、非専門の学生の皆さんにもよくわかるようなお話をお願いし、詳しい議論は後半部分にとお願いいたしました。学部および大学院学生の皆さん、特に物性理論以外の皆さんの参加を歓迎いたします。
Contact : Y. Hatsugai 初貝安弘 Tel:029-853-4204Email: hatsugai@sakura.cc.tsukuba.ac.jp
Supported by Grants-in-Aid for Scientific Research, # 23340112, # 23654128,
TIMS & “学位システム国際化経費(物理学専攻)”

Electron in the field of magnetic charge: Tight binding solution and mapping on arealistic physical system
Yshai Avishai (in collaboration with Jean Marc Luck)
Seminar at the physics department, Tsukuba University, March 2012

It was shown by Dirac about 80 years ago that if there is a magnetic charge g leading to a central
magnetic field B = g rr̂2 , then g must be quantized as 2eg = n!c (n = 0, 1, 2 . . . is the monopolenumber). The corresponding ”hydrogen atom problem” (a spinless electron in the field of a magneticcharge) was solved by Igor Tamm just a few months after Dirac’s paper. Here I approach thisproblem from a ”condensed matter point of view” using a tight binding model. The motivationis threefold: First, the physics is rather beautiful and involves interesting relations with sphericalgeometry and the theory of graphs. Second, the notion of magnetic monopole is quite relevant incondensed matter physics. Among others, it serves as a useful tool for constructing translation
invariant many electron wave functions in the FQHE (such as Laughlin’s and Moore Read’s Nelectron wave functions). Third, I will show that under some conditions, this seemingly inaccessible
system can be mapped on a realistic physical system. When the sites upon which the electron
resides and hops form a highly symmetric object, the energy spectrum is calculated analytically asfunction of n and displays a beautiful pattern, which is entirely distinct from that of the Hofstadterbutterfly. The systematics of level degeneracy is unusual and poses some challenges to the theoryof point symmetry groups. The spectrum of an electron hopping on the sites of a Fullerene reveals
a set of magic (monopole) numbers ni .
Is this system realistic? Surprisingly the answer is affirmative. To show this, I nvestigate within
the same geometry, a seemingly completely different system: that of a spin-full electron subject to anelectric field of a point charge E = q rr̂2 . The sole effect of this field is to generate a Rashba type spinorbit interaction. The energy spectrum is calculated analytically as function of the (dimensionless)spin-orbit strength and displays rich and beautiful pattern with some unexpected symmetries inwhich physics and geometry interlace. This mission accomplished, I then expose a remarkable
relation between the two distinct physical problems: The energy spectrum in the second system
at a certain symmetry point is identical with the energy spectrum in the first system at monopole
number n = 1. Thus, it is principally possible to test the physics of an experimentally inaccessible
system (electron in the field of magnetic monopole) in terms of an experimentally accessible one (anelectron subject to spin-orbit force induced by central electric field).