Non-trivial Topology: Definition & Explanation

[Fixxxer125]

Hi there, I've come across the term 'non-trivial topology' or 'non-trivial surface states' when researching topological superconductors and really need a bit of help as to exactley what this means? I've tried google but no-one seems to give a definition?
Many thanks for checking this out

 

[DimReg]

Wouldn't it just be a topology that isn't trivial?

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

 

[Fixxxer125]

Yea I was just hoping someone could explain it a bit nicer!

 

[DimReg]

Oh ok then, I'll try to do better. I do have to warn you that I'm not particularly knowledgeable in topology or it's application to superconductors.

In the definition of a topology, one specifies a set, and a collection of subsets. Roughly speaking, the specified subsets tell you the abstracted geometry of the set. So for example, in the real number line you have the familiar definition of open set as the specification of the subsets. These open set can always be made smaller, and this is an important feature of the real numbers. Then consider the discrete topology, where every single point set is an open set. In this case you cannot always make an open set smaller. This would be a good choice of topology for the integers, for example.

So again, roughly speaking the topology carries some information on the "shape" of the set. The trivial topology, on the other hand, can be imposed on any set. So clearly, the trivial topology fails to tell you this kind of information. If this isn't clear, I'll make another example. If you try to put the same topology of the real numbers on the integers, you'll end up with the discrete topology( (-a,a) will eventually only contain 0 as you make a smaller). However, you could easily put the trivial topology on both sets.

So non trivial topologies are topologies that have enough structure to tell you something about the set.

I'm not sure what a "non trivial surface state" is. Hopefully someone will come along who knows more about that than I.

 

[Fixxxer125]

Thats very helpful, thanks!

 

[Fredrik]

It's hard for us to guess what exactly "non-trivial topology" is in this context. A topology on a set X is a collection of subsets of X that satisfies a number of properties. A member of the topology is called an open set. These open sets are useful because we can use them to assign a meaning to statements like "this sequence converges to this point" or "this function is continuous".

The two simplest ways to define a topology on a given set X is to choose the topology to be either "all subsets of X" or "just the sets ∅ and X". These choices are considered trivial, and as you can guess, they are pretty useless. For example, what's the point of the concept of continuity if all functions are continuous? I don't think that your "non-trivial topology" has anything to do with this.

If X is a set and T is a topology on X, then the pair (X,T) is called a topological space. If (X,T) and (Y,S) are topological spaces, it may or may not be true that there exists a bijection from X into Y that's continuous with respect to the topologies T and S. If such a function exists, then we describe this situation with phrases like "X and Y are topologically equivalent". (Actually, "homeomorphic" is the correct technical term).

When two spaces are not topologically equivalent, there's typically something fundamentally different about their shapes. An example that's often brought up is that a donut is topologically equivalent to a coffee mug, but not to a sphere. This is because the former two shapes both have a hole in them, and a sphere doesn't.

My guess is that your non-trivial topology has something to do with this sort of thing. For example, a surface that's "like" a donut rather than "like" a sphere.

 

[DrDu]

Yes, there is a big hype around topological insulators and the like.
This article is at least remotely comprehensible and Kane is one of the fathers of the concept:
http://arxiv.org/pdf/1002.3895