Why are Topological Insulators Called Topological?

[tayyaba aftab]

why are topological insulators called TOPOLOGICAL insulators?
what factor of topology apperas in the phenomenon

 

[Physics Monkey]

Hi tayyaba aftab,

Here are some reasons: Topological insulators may be classified in terms of homotopy classes of mappings, often from the Brillouin zone to some suitable space of gapped Hamiltonians. Also, these systems have semi-protected edge states which have a topological character. For example, the boundary action of a 3 dimensional topological insulator may contain a topological Chern-Simons term. Topological defects in these materials may also carry interesting quantum numbers.

 

[tayyaba aftab]

dear physics monkey
can u please suggest some research paper or book to understand the phenomenon.

 

[xepma]

Try this one for starters:

 

[xepma]

And this review paper was put on the arxive today

http://arxiv.org/abs/1002.3895

 

[element4]

If you want to understand how topology comes in, I recommend you to start with the so called TKNN number in the time-reversal breaking topological insulators (such as Integer Quantum Hall Effect). Mathematically speaking, this is nothing but the first Chern number of a U(1) principal bundle on a Torus (if you haven't studied fiber bundles, don't worry you can go on without it).

For an accessible and physically appealing introduction see http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WB1-4DF4YV5-XT&_user=10&_coverDate=04%2F01%2F1985&_rdoc=1&_fmt=high&_orig=search&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=862b779c8a126c90aa172210416526a3".

The ideas behind the new time-reversal invariant topological insulators are similar, though more complicated. Also take a look at http://arxiv.org/abs/1001.1602" . (There are many review papers, I can cite more if you want).

 

[tayyaba aftab]

what is z2 topological order?
@ element 4 can u please recommend something to read?
i am unable to get the idea of it:(

 

[element4]

Hi tayyaba aftab

I can try to find some relevant papers for you. But the question is which aspects are you interested in, the basic physical ones or the more mathematical ones? And are you familiar with some basic notions in Lie group theory, differential geometry/topology and algebraic topology? Or would you avoid such papers?
And did you read the Kohmoto paper and got an idea of how the first Chern number came into the picture? This is the easiest way to get an feeling for the z_2 number.

In you are more interested in the physical consequences of the z_2 order, you should read completely different kind of papers.

We cannot help you unless you give some more information about what you are interested in.

 

[tayyaba aftab]

thanku
yah i read kohomoto paper and got an idea of chern numbers
i am physics student so don't have much knowledge of topology and maths
but as my work is based on theoretical research i have to do some basic mathematics:(
i want papers with more physical insight but mathematics too.
and i don't have idea of Lie group theory, differential geometry/topology and algebraic topology:(

now please suggest some papers:(

thanking you
tayyaba aftab

 

[atyy]

http://arxiv.org/abs/1001.1602
The quantum spin Hall effect and topological insulators
Xiao-Liang Qi, Shou-Cheng Zhang

 

[genneth]

I strongly suggest becoming educated about topology --- differential geometry and algebraic topology. You don't need to understand them like a mathematician --- just like you don't need to understand numbers as far as number theory, but you need to be able to do arithmetic. These tools are going to become increasingly a standard part of a physicist's mathematical toolbox.

 

[tayyaba aftab]

thnx to all:)
i actually want to know a pathway which lead me towards understanding topology of topological insulators.and i don't know which path to follow:(

thats why i am asking for papers so that they can lead me towards my ultimate goal.
i have read http://arxiv.org/abs/1001.1602
i want some paper more mathematical than this paper

 

[genneth]

Have you read http://arxiv.org/abs/1002.3895 ? That is a very nice review. Follow the references for more details.