Why are Topological Insulators Called Topological?

In summary: Also http://arxiv.org/abs/0907.2993 is a nice pedagogical review. If you want more mathematical material, you need to be much more specific about what you are interested in. There is a large literature out there.In summary, topological insulators are materials that can be classified in terms of homotopy classes of mappings and have semi-protected edge states with a topological character. The factor of topology appears in the phenomenon through the boundary action of a 3 dimensional topological insulator, as well as in the topological defects found in these materials. To understand this phenomenon, research papers and books such as "Topology and Physics" by M. Nakahara and "Quantum Field Theory
  • #1
tayyaba aftab
20
0
why are topological insulators called TOPOLOGICAL insulators?
what factor of topology apperas in the phenomenon
 
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  • #2
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.
 
  • #3
dear physics monkey
can u please suggest some research paper or book to understand the phenomenon.
 
  • #4
Try this one for starters:

 
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  • #6
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" [Broken]. (There are many review papers, I can cite more if you want).
 
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  • #7
what is z2 topological order?
@ element 4 can u please recommend something to read?
i am unable to get the idea of it:(
 
  • #8
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.
 
  • #9
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
 
  • #10
  • #11
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.
 
  • #12
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
 
  • #13

What are topological insulators?

Topological insulators are materials that are electrically insulating in their interior, but have conducting surface states that are protected by the material's topology. This means that the surface states are robust against impurities, defects, and other external disturbances.

What makes topological insulators unique?

Topological insulators are unique because they exhibit a phenomenon called topological order, which is a type of order that cannot be described by traditional symmetry-breaking methods. This topological order is what gives rise to the protected surface states.

What are the potential applications of topological insulators?

Topological insulators have potential applications in quantum computing, as the robustness of their surface states makes them ideal for carrying and manipulating quantum information. They could also be used in next-generation electronic devices with lower power consumption and higher speed.

How are topological insulators different from other materials?

Topological insulators are different from other materials because of their unique electronic properties. Traditional insulators have a large band gap between the valence and conduction bands, while topological insulators have a small band gap and conductive surface states. This makes them distinct from both insulators and conductors.

Are there any challenges in studying topological insulators?

Yes, there are some challenges in studying topological insulators. One challenge is that they are often only stable at very low temperatures, making it difficult to study their properties at room temperature. Another challenge is that their surface states can be easily affected by impurities and defects, making it challenging to control and manipulate them in experiments.

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