Topological question from Ashcroft-Mermin

[hagopbul]

TL;DR

about chapter 2

Hello :

doing some reading in physics and some of it is in solid state physics , in Ashcroft- mermin book chapter 2 page 33 you read

" Thus if our metal is one dimensional we would simply replace the line from 0 to L to which the electron were confined by a circle of circumference L. In three dimensions the geometrical embodiment of the boundary condition , in which three pairs of opposite faces on the cube are joint , becomes topologically impossible to construct in three dimensional space "

the above is not very clear to me could some one provide references to above paragraph or a short explanation

Best Regards
HB

 

[hutchphd]

I believe they are talking about "periodic" boundary conditions. In 1D it is easy to picture in real space as described. For 3D periodic boundary conditions no such real structure should be contemplated. The construct however still works and gives the correct density of states. Just don't try to picture it in your head, you will get agita.

 

[Keith_McClary]

It is like:

two pairs of opposite sides on the square are joint , becomes topologically impossible to construct in two dimensional space

because that is topologically a torus, which can be constructed in 3D but not 2D.

 

[hagopbul]

could you prove it is tours ? it is 3cubes why tours not hexagon or other 3d object ?

 

[Keith_McClary]

could you prove it is tours

No, I can't even prove the 2D case.

 

[hutchphd]

Summary:: about chapter 2

" Thus if our metal is one dimensional we would simply replace the line from 0 to L to which the electron were confined by a circle of circumference L. In three dimensions the geometrical embodiment of the boundary condition , in which three pairs of opposite faces on the cube are joint , becomes topologically impossible to construct in three dimensional space. Nevertheless the analytic form of the boundary condition is easily generalized "

I have added the next line from Ashcroft and Mermin. Your question is not salient to the physics of Born-von Karman boundary conditions. Not to worry.

 

[PAllen]

Joining opposite faces of a cube produces a 3-torus, a perfectly well defined topology. The limitation is simply that it cannot be represented (embedded) in 3-dimensional Euclidean space. However, it is easily embedded in a higher dimensional Euclidean space. This is similar to the situation of a Klein bottle. Trying to represent it 3-space requires self intersection, which is not feature of the actual Klein bottle (which can be embedded with no problems in 4-space).

Also, note that something as simple as 3-sphere cannot be embedded in Euclidean 3-space, so there is no problem here other than that our intuitions are guided by Euclidean 3-space.