# Why are topological materials/phases "exotic"?

From what I understand, when a system has topological order, any local perturbation doesn't change the phases and thus its properties. This would suggest that it should be really easy to find experimentally such phases, since even with disorder/at non-zero temperature we should find their topological invariants, which usually manifest in optic/electronic/thermal transport properties.

However, this is a concept which has become important only in the last few decades, even if we've been studying mesoscopic systems for longer than that. I would think that this would be one of the first things we discovered, where different "non clean" samples show exactly identical properties.

However, this is not the case, which leads me to think that topological phases are actually exotic and not so easily realized in nature. This brings me to my question, why is this case? Why didn't we discover topological phases earlier?

I'm just a novice in this field so of course I may be missing some big step here, any clarification would be great.

The answer lies in a refinement of your claim that phases with topological order are robust to all local perturbations. The more precise statement is:

A (zero-temperature) phase of matter with (intrinsic) topological order is robust to all local perturbations which do not close the energy gap above the ground state.

If we close the energy gap, critical modes appear which destroy the topological order, and on the 'other side' of the critical point we can then have a 'boring' phase (such as a symmetry-breaking magnet etc).

Hence, the energy gap (defined as the smallest amount of energy necessary to create a quasiparticle excitation) is paramount. It turns out that in many cases, the gap $$\Delta$$ is rather small, making these phases hard to see. First, it means we have to achieve temperatures $$T \ll \Delta$$ such that we will effectively see the quantum regime (remember that topologically ordered phases are really only well-defined at zero temperature). Second, a small gap $$\Delta$$ means that these phases are typically constrained to live in a small region of parameter space, since perturbations (e.g., an external field $$h\gtrapprox\Delta$$) can close the gap and trivialize the phase.

Ultimately, then, we arrive at the question: why is the gap so small? In principle, the gap can be large. Many solvable models are known where the energy gap is of the same order as the interaction strength of the model. A paradigmatic example is the Kitaev toric code model. However, these models involve interactions where multiple constituents interact at once (these are call multi-body interactions); this is often necessary to give rise to the exotic correlations that characterize topological phases. But of course physical systems tend to have at most two-body interactions. It has been well-appreciated that for such two-body interactions, one can still realize an effective multi-body interaction by using perturbation theory. However, this naturally means that the energy gap will be quite small (since, after all, we are relying on some perturbative approximation). I think this is ultimately the reason why we are not being buried in examples of topologically ordered materials.

There is, however, a celebrated instance of intrinsic topological order in solid-state materials: the fractional quantum Hall states. How do these evade the aforementioned issue? Here, very large magnetic fields are used to create well-separated Landau levels. Due to the non-triviality of these levels (and due to the magic of Pauli exclusion), it turns out that the two-body Coulomb interactions are sufficient to deform (and split) these Landau levels into topological phases with exotic correlations---one does not need to rely on higher-order perturbation theory. (For further reading in this direction, you can look up the Haldane pseudo-potential.) In this case, the gap is thus rather favorable (at least relatively speaking).

Let me mention that in addition to the issue of the energy gap, there is a second subtlety that makes it hard to find experimental realizations of topological order. Even if you were lucky and found such a material, it is very non-trivial to devise a measurement that actually tells you that you have such a phase of matter. Indeed, almost by definition local probes cannot give you conclusive evidence. Hence, it might very well be that many known materials exhibit topological phases, but that they have been overlooked for this reason. Again, the quantum Hall states are an exception since in that case we can detect them through the very accessible quantized Hall conductance, which is sadly not a generic feature of other topological phases.

• I assumed implicitly that perturbations did not close the gap, since in my mind "perturbation = infinitesimal variation". Of course real perturbations and gaps are both finite, which I forgot to take into account. This makes it a bit more clear why it may be difficult to find such phases. Apr 5 at 11:01
• However, why then are ordinary insulators usually not topological even if they have a gap? From the other answer, citing an article by Wen, it would seem that this is the case because the mechanisms themselves for the phases are exotic. This seems to reduce the argument for why topological phases are exotic to "because they are". Apr 5 at 11:01
• I'm thinking that maybe a renormalization group argument, if applicable, might justify why phases are often unstable. But I don't know if there is such a general argument. Apr 5 at 11:02
• @AnotherUser "why then are ordinary insulators usually not topological even if they have a gap?" They can be stabilized by Hamiltonians with two-body interactions (sometimes even single-body), and so one does not need to rely on perturbation-theory arguments and hence one does not expect a small gap. To say it differently: the 'stabilizers' (i.e., the operators which leave the state invariant) of topological states are typically involving more than two sites, and this is not the case for trivial phases, explaining why the latter can have larger gaps. Apr 5 at 18:08
• @AnotherUser Correct, it lies in the physical origin of the gap. I would not characterize it as interacting-vs-non-interacting, I think that is a bit too simplistic. (Certain phases of matter require interactions but they nevertheless do not exhibit topological order.) Apr 7 at 2:05

It seems like your understanding is that many or most materials can realize topological phases, and since topological observables are robust, we should be able to find them easily if we look for them. However, the premise is flawed. Generically, we expect that complex electronic systems will be sensitive to all perturbations. After all, why shouldn't they be? If we change the hamiltonian, we should expect to change the observables. Topological phases require a very special conspiracy among the underlying electrons in order to keep the topological observables robust against local pertubrations. (Xiao-Gang Wen refers to topological phases as "choreographed entangle[ment] dances.")

Having said that, the first topological phase was actually discovered experimentally much earlier than many people realize, because superconductors are topologically ordered.

• I imagined that some of my premises were wrong, this is now clear. I had also forgot about the fact that superconductors have topological order (I already "read" the article - with limited understanding - but forgot about it while I was thinking about this question). This gives me another question, is there a "simple" reason for why topological order requires such a "conspiracy"? I have already "read" through Wen's article, but even if it is pedagogical, it requires a much richer background than I have. Is it simply something that will be clearer when I have all the pieces/entered the field? Apr 1 at 7:38
• @AnotherUser The simple reason is basically what I said in my answer: if you change the hamiltonian, you change the eigenstates and dynamics of the system to change, and you would expect this to change the values of the observables. A small perturbation may change the observables very little, but if you make precise enough measurements you should be able detect the perturbation. It might help to play around with perturbing some simple toy models where you can actually do calculations, like the harmonic oscillator, many coupled oscillators, systems of coupled spins, etc...
– d_b
Apr 1 at 16:24
• ... Add some generic perturbations and try to work out how the observables change. You should see that it's not really possible to perturb the hamiltonian by some $\epsilon$ without affecting the observables in some way parametric in $\epsilon$.
– d_b
Apr 1 at 16:27
• I know that's the case, but in my mind it's still not clear why this is the rule rather than the exception (a priori, without knowing explicit hamiltonians or experimental results). Apr 2 at 8:05
• @d_b I'm not sure I am understanding your statement. It is known that (intrinsic) topological order is robust to all perturbations if they are sufficiently small. More precisely, if we represent a particular model with topological order as a point in (multi-dimensional) parameter space, then there is always an open ball in parameter space which has topological order. If your point is that in many physical circumstances of interest, this open ball often tends to have a small radius, then I agree. But nevertheless, the radius is nonzero. Apr 2 at 23:39