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Electron in the field of magnetic charge: Tight binding solution and mapping on a realistic physical system

Speaker: 
Yshai Avishai (Ben-Gurion University, Israel)
Date: 
Thu, 01/08/2013 - 1:30pm to 2:30pm
Location: 
S13-M01-11 (Physics Conference Room)
Host: 
Shaffique Adam
Event Type: 
Seminars

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 r/r^3, then g must be quantized as 2eg = n c h/(2pi) (n = 0, 1, 2 . . . is the monopole number). The corresponding ”hydrogen atom problem” (a spinless electron in the field of a magnetic charge) was solved by Igor Tamm just a few months after Dirac’s paper. Here I approach this problem from a ”condensed matter point of view” using a tight binding model. The motivation is threefold: First, the physics is rather beautiful and involves interesting relations with spherical geometry and the theory of graphs. Second, the notion of magnetic monopole is quite relevant in condensed 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 N electron 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 as function of n and displays a beautiful pattern, which is entirely distinct from that of the Hofstadter butterfly. The systematics of level degeneracy is unusual and poses some challenges to the theory of point symmetry groups. The spectrum of an electron hopping on the sites of a Fullerene reveals a set of magic (monopole) numbers ni .

Within the same tight-binding geometry, this problem is now confronted with that of a spin-full electron subject to an electric field of a point charge E = q r/r^3 which is responsible for Rashba type spin-orbit interaction. The spectrum is calculated analytically as function of the (dimensionless) spin-orbit strength and displays rich and beautiful pattern with some unexpected symmetries in which physics and geometry interlace. 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 (an electron subject to spin-orbit force induced by central electric field).

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