Understanding Inversion Symmetry and Space Symmetry Breaking

[fyw]

1. P. Marder ever said that there is no special symmetry results in two dimensional oblique lattice. But it still possesses inversion symmetry. r → -r
How to understand r → -r?

2. Many book ever states that space symmetry broken by atomic displacement can bring ferroelectricity. But why this kind of displacement breaks the space symmetry?

 

[M Quack]

The definition of the inversion symmetry operator I that it transforms a vector into a different vector of same magnitude but antiparallel orentation. This can be written in many ways, e.g. I(r) = -r, or r --> -r, where r is a vector. All "naked" Bravais lattices have inversion symmetry (=they are invariant under inversion symmetry).

Special symmetry elements in 2D are mirror axes and 60, 90 or 180deg rotation symmetry.

http://en.wikipedia.org/wiki/Bravais_lattice

Electric polarization is a vector. Therefore I(P) = -P. If the crystal is invariant under inversion symmetry, then P=0 and the crystal cannot be ferroelectric.

 

[fyw]

The definition of the inversion symmetry operator I that it transforms a vector into a different vector of same magnitude but antiparallel orentation. This can be written in many ways, e.g. I(r) = -r, or r --> -r, where r is a vector. All "naked" Bravais lattices have inversion symmetry (=they are invariant under inversion symmetry).

Special symmetry elements in 2D are mirror axes and 60, 90 or 180deg rotation symmetry.

http://en.wikipedia.org/wiki/Bravais_lattice

Electric polarization is a vector. Therefore I(P) = -P. If the crystal is invariant under inversion symmetry, then P=0 and the crystal cannot be ferroelectric.

Dear Quack,

Thank you for your answer. I've got it.
Additionally, can you explain the time symmetry broken induced magnetism and gauge symmetry broken induced superconductor or superliquid?

 

[M Quack]

Time reversal inverts linear momentum (p) and therefore angular momentum, L. It also inverts the spin, S and therefore the magnetic moment.

In a magnetically ordered material, there are well-defined expectation values of the magnetic moment. For example, in a ferromagnet there is a macroscopically observable magnetic moment. Time reversal inverts that.

One approach to systematically investigate the possible arrangements of magnetic moments is the classification into 1651 Shubnikov groups (black-and-white space groups) that are an extension of the 230 crystallographic space groups. Here one moment direction is represented by the color white, and the opposite by black. Time reversal exchanges black and white. Depending on the moment direction, this may also happen for some "normal" space group operations, e.g. a 180-deg rotation about an axis that is perpendicular to the moment direction.

 

[DrDu]

... and gauge symmetry broken induced superconductor or superliquid?

A good explanation can be found in this article:
http://adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1986PThPS..86...43W&link_type=EJOURNAL