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MUCH OF “THEORETICAL” mathematics finds application in other areas of science, as seen here yesterday in Math Words Part 1. More tidbits gleaned from AAAS Science magazine, September 20, 2019, follow today in Part 2.

Magnetic Weyl Semimetal Phase in a Kagomé Crystal. D.F. Liu and colleagues studied Co3Sn2S2, Kagomé crystals, and found that these magnetic Weyl semimetals “may serve as a platform for realizing phenomena such as chiral magnetic effects, unusually large anomalous Hall effect and quantum anomalous Hall effect.”

Crystal structure of Co3Sn2S2, showing the stacked layers. Image from Science September 20 2019.

Among other applications, Hall-effect devices have plenty of automotive uses, in speedometers, fuel flow sensors, and ignition systems.

In the same Science issue, Noam Morali et al. write about “Fermi-arc Diversity on Surface Terminations of the Magnetic Weyl Semimetal Co3Sn2S2. Physicist Enrico Fermi won the 1938 Nobel Prize in Physics. He also appeared here at SimanaitisSays in “Is Anyone Out There?”

Non-Abelian Band Topology in Noninteracting Metals. QuanSheng Wu and colleagues observe “Noncommutative topological charges…” are associated with the crystalline solids of their study.

Neils Henrik Abel, 1802–1829, Norwegian mathematician. Honored by establishment of the Abel Prize, first awarded in 2002.

It was Neils Abel who characterized mathematical operations in which the order of operands does not matter: a ∎ b = b ∎ a. For example, this property of commutativity is satisfied by ordinary addition and multiplication, but not subtraction nor division.

Wu and his colleagues also cite that the phenomena studied are “quaternion charges.” Quaternions, first described by Irish mathematician William Rowen Hamilton in 1843, are a mathematical extension of complex numbers: A complex number has the form a + bi, where a and b are real numbers and i2 = -1. A quaternion has the form a + bi + cj + dk, with a, b, c, and d real numbers; i, j, and k are quaternion units with special properties that ij = k, ji = –k, and thus ij = –ji.

Image from Wikipedia.

Just as complex numbers can be graphed in the two dimensional plane, quaternions reside in four-space. A product of quaternion units represents a 90-degree rotation. And, by the way, notice that quaternion multiplication is not abelian.

As a math major at Worcester Poly, I fooled with quaternions as an exercise in learning Fortran programming. At the time, I never thought I’d see them again.

The Moral of All This. Hermann Weyl proposes quasiparticles in 1929, and in 2015 along come Weyl semimetals. The early nineteenth-century Henrik Abel imagines abelian groups and in Science, September 20, 2019, researchers report non-abelian band topology in noninteracting metals. And calculus devised by Sir Isaac Newton and Gottfried Wilhelm Leibniz more than 300 years ago appears on t-shirts today. ds

© Dennis Simanaitis, SimanaitisSays, 2019

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