Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Is edge state of topological insulator really robust?

+ 6 like - 0 dislike
1322 views

I am a little confused! Some people are arguing that the gapless edge state of Topological insulator is robust as long as the time reversal symmetry is not broken,while other people say that it is not stable for lack of topological order。Please help me out!

This post imported from StackExchange Physics at 2014-04-05 04:35 (UCT), posted by SE-user Zac.Dummy
asked Mar 18, 2013 in Theoretical Physics by Zac.Dummy (55 points) [ no revision ]

1 Answer

+ 7 like - 0 dislike

I see how that can be confusing. Unfortunately understanding how to reconcile these statements will require a lot of background. I will try to answer this as concisely as I can (hopefully) without relying on concepts that are too advanced.

Well, topological insulators do not possess a so-called intrinsic topological order. It means that the bulk states of a topological insulator are not entangled quantum mechanically over a long range. Topological insulators are, in fact, short-range entangled just like trivial insulators. However, topological insulators and trivial insulators are clearly not the same phases. Therefore short-range entangled phases are further broken down into subcategories. Two such subcategories are: symmetry protected topological phases (topological insulators) and symmetry-breaking phases (trivial insulators).

The reason the word “topological” appears in the distinction between of topological insulators and trivial insulators is that they can be assigned a distinct “topological invariant.” The notion of a topological invariant comes from topology. For example a sphere and a torus have different topological invariants. Just as you cannot deform a torus into a sphere without cutting it, in the same way you cannot deform the band structure of a topological insulator into that of a trivial insulator without closing the bulk gap. As a consequence of this subtle difference in the two types of band structures the number of edge states will either be even (trivial insulators) or odd (topological insulators). Now this is where time reversal symmetry comes in. If any kind of perturbation, which itself obeys time reversal symmetry, acts on these edge states then it can destroy these edge states only in pairs. Therefore if you had odd number of edge states to begin with then you will end up with at least one edge state even if the perturbation destroys all the remaining edge states (in pairs). Hence time reversal symmetry is responsible for the protection of these edge states in topological insulators. You can find a more detailed explanation here:

What conductance is measured for the quantum spin Hall state when the Hall conductance vanishes?

Just scroll all the way down until you see the question in the block quote “Also: Why is there only a single helical edge state per edge? Why must we have at least one and why can't we have, let's say, two states per edge?” To give the above analogy with topology a firm footing I suggest you take a look at Berry curvature and the Chern number (if you haven't already). The topological invariants are closely connected to these.

So to summarize, gapped phases of matter can be divided into two categories: long-range entangled (with intrinsic topological order) and short-range entangled (without intrinsic topological order). Two subcategories of short-range entangled phases are: symmetry protected topological phases (topological insulators) and symmetry-breaking phases (trivial insulators).

In case you are wondering about long-range entangled phases and what it means to have (intrinsic) topological protection then I recommend a little more background reading on the principle of emergence, the fractional quantum Hall effect, string-net condensation (in that order). There are some excellent posts on physics stackexchange on the topic of string-net condensation. Some of them are even answered by Prof. Xiao-Gang Wen who, as a matter of fact, developed the theory of string-net condensation along with Michael Levin (I don’t know if he’s here).

This post imported from StackExchange Physics at 2014-04-05 04:35 (UCT), posted by SE-user NanoPhys
answered Mar 18, 2013 by NanoPhys (360 points) [ no revision ]
Thanks a lot for your answer!You mentioned that systems with intrinsic topological order are entangled quantum mechanically over a long range. But how can we know a system is long range entangled or not. Could you tell me some articles to follow?

This post imported from StackExchange Physics at 2014-04-05 04:35 (UCT), posted by SE-user Zac.Dummy
The best place to start is wikipedia! LOL! en.wikipedia.org/wiki/Topological_order. According that article intrinsically topologically ordered phases will have: emergent gauge theory, emergent fractional charge and fractional statistics. Like I said, there's a lot of ground to be covered, starting from the principle of emergence to string-net condensation

This post imported from StackExchange Physics at 2014-04-05 04:35 (UCT), posted by SE-user NanoPhys
Here's a nice and short review: rmp.aps.org/abstract/RMP/v77/i3/p871_1

This post imported from StackExchange Physics at 2014-04-05 04:35 (UCT), posted by SE-user NanoPhys
Wen's work is terrific! But is too hard for me. So, is there something much easier to follow than wen's paper and more detail than the wikipedia?

This post imported from StackExchange Physics at 2014-04-05 04:35 (UCT), posted by SE-user Zac.Dummy
@Zac.Dummy: Actually Wen's paper was the easiest I could find! Maybe I haven't looked everywhere (for something simpler than that). Anyways, here are some very elementary articles you might find interesting and manageable: arxiv.org/abs/1207.6433 and pnas.org/content/97/1/28.full. You also might want to take a look at the seminal article by P.W. Anderson: "More is different"

This post imported from StackExchange Physics at 2014-04-05 04:35 (UCT), posted by SE-user NanoPhys
The above articles don't talk directly about topological order. But they build of the necessary background in understanding topological order better.

This post imported from StackExchange Physics at 2014-04-05 04:35 (UCT), posted by SE-user NanoPhys
Thanks again!I think the question that confusing me is not the one that I ased ! Thanks for your great answer and patient!

This post imported from StackExchange Physics at 2014-04-05 04:35 (UCT), posted by SE-user Zac.Dummy

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysics$\varnothing$verflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...