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！
$\begingroup$ The short answer: the fact you need time-reversal symmetry means it is not stable against all perturbations---that is what people mean by "lack of topological order". Instead, topological insulators are examples of symmetry-protected topological phases (with possibly the exception of the integer quantum Hall and Kitaev chain, which are protected by the 'unbreakable' fermion parity symmetry). $\endgroup$– Ruben VerresenFeb 7, 2021 at 16:27
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).
$\begingroup$ 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? $\endgroup$ Mar 18, 2013 at 5:56
$\begingroup$ 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 $\endgroup$ Mar 18, 2013 at 6:05
$\begingroup$ Here's a nice and short review: rmp.aps.org/abstract/RMP/v77/i3/p871_1 $\endgroup$ Mar 18, 2013 at 6:21
$\begingroup$ 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? $\endgroup$ Mar 18, 2013 at 8:00
$\begingroup$ @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" $\endgroup$ Mar 18, 2013 at 20:12