#symmorphism

[materialsscienceandengineering]

New chromium-based superconductor has an unusual electronic state

When certain materials are cooled below a critical temperature they become superconductors, with zero electrical resistance. An international research team observed an unusual electronic state in new superconductor chromium arsenide. This finding could prove useful in future superconductor research and material design. The study was published on June 5 in Nature Communications.

These discoveries were made by a research team at the Chinese University of Hong Kong in collaboration with Associate Professor KOTEGAWA Hisashi (Kobe University Graduate School of Science) and other researchers from Kobe University and Kyoto University.

Well-known superconductors include high-temperature copper-oxide superconductors and iron-based superconductors. These have two-dimensional layered crystal structures. In contrast, chromium arsenide has a "non-symmorphic" crystal structure formed by zigzag chains of chromium. The relationship between this crystal structure and its superconductivity has drawn attention from scientists.

Read more.

[mathematicianadda]

Semidirect product action and its geometry

I'm going by the maxim

Groups, like men, are known by their actions

This naturally leads one to ask "given groups $G, H$ which act on sets $S, T$ and the semidirect product $G \rtimes H$, how does one visualize the action of $G \rtimes H$? What does it act on? Some combination of $S$ and $T$? ($S \times T$ perhaps?)

I know some elementary examples, likr $D_n \simeq \mathbb Z_n \rtimes \mathbb Z_2$. However, given an unknown situation, I am sure I cannot identify whether it is a semidirect product that is governing the symmetry.

The best responses on similar questions like intuition about semidirect product tend to refer to this as some kind of "direct product with a twist". This is shoving too much under the rug: the twist is precisely the point that's hard to visualize. Plus, not all "twists" are allowed --- only certain very constrained types of actions turn out to be semidirect product. I can justify the statement by noting that:

the space group of a crystal splits as a semidirect product iff the space group is symmorphic --- this is quite a strong rigidity condition on the set of all space groups.

The closest answer that I have found to my liking was this one about discrete gauge theories on physics.se, where the answer mentions:

If the physical space is the space of orbits of $X$ under an action $H$. Ie, the physical space is $P \equiv X / H$. Then, if this space $P$ is acted upon by $G$. to extend this action of $G \rtimes H$ onto $X$ we need a connection.

This seems to imply that the existence of a semidirect product relates to the ability to consider the space modulo some action, and then some action per fiber. I feel that this also somehow relates to the short exact sequence story(though I don't know exact sequences well):

Let $1 \rightarrow K \xrightarrow{f}G \xrightarrow{g}Q \rightarrow 1$ be a short exact sequence. Suppose there exists a homomorphism $s: Q \rightarrow G$ such that $g \circ s = 1_Q$. Then $G = im(f) \rtimes im(s)$. (Link to theorem)

However, this is still to vague for my taste. Is there some way to make this more rigorous / geometric? Visual examples would be greatly appreciated.

from Hot Weekly Questions - Mathematics Stack Exchange from Blogger https://ift.tt/322ROo8

[arxt1]

Polarization tensor for tilted Dirac fermions: Covariance in deformed Minkowski spacetime. (arXiv:1904.13277v1 [cond-mat.mes-hall])

The rich structure of solid state physics provides us with Dirac materials the effective theory of which enjoys the Lorentz symmetry. In non-symmorphic lattices, the Lorentz symmetry will be deformed in a way that the null energy-momentum vectors will correspond to on-shell condition for tilted Dirac cone dispersion. In this sense, tilted Dirac/Weyl materials can be viewed as solid state systems where the effective spacetime is non-Minkowski. In this work, we show that the polarization tensor for tilted Dirac cone systems acquires a covariant from only when the spacetime is considered to be an appropriate deformation of the Minkowski spacetime. As a unique consequence of the deformation of the geometry of the spacetime felt by the electrons in tilted Dirac cone materials, the Coulomb density-density interactions will generate corrections in both longitudinal and transverse channels.

from gr-qc updates on arXiv.org http://bit.ly/2V5frMX

[sisiad]

Nexus fermions in topological symmorphic crystalline metals

http://dlvr.it/P6BMPk