Magnetic solitons and topological insulators have “opposite” protected regimes
A cute thought occurred to me today. In the study of magnetic textures, we can have topologically protected spin configurations which belong to nontrivial classes of some homotopy group. Generally speaking, such a configuration assigns a vector on the Bloch sphere to each point in space, m : R² → S², which is equivalent to an endomorphism on S² (by compactifying through the point at infinity). Such a map m, then, is classified by the second homotopy group π2(S²). A [baby] skyrmion belongs to the charge ±1 sector of this group.
Topological protection is so-called because the integral classification of a topological phenomenon cannot be continuously modified by a weak perturbation. What seems to follow: once in a topologically nontrivial sector, always in a topologically nontrivial sector.
But we can of course create and destroy skyrmions in the lab. How is this possible? This question was (mythically) answered by Alexander III of Macedon, namely Alexander the Great, in the fourth century BCE. When faced with the problem of untying the untieable Gordian knot in ancient Phrygia, he famously slashed the knotted rope in half with his sword making him, by a retroactive prophecy, the King of Asia.
Alexander’s key insight was that the topological protection of the Gordian knot could be circumvented by confronting a key assumption: that the rope is a continuous manifold, a prerequisite condition for homotopical classification. At the length scale of a blade’s edge, and the energy scale of human strength, this assumption about rope breaks down, and so Alexander could remove the knot without, strictly speaking, untying it.
This same principle carries over to skyrmions. When the continuum approximation is respected (the Gordian regime), skyrmions are indeed topologically protected. But when sufficiently energetic perturbations occur at small enough length scales (on the order of the lattice constant), the continuum approximation fails to accurately describe our spin system. In this Alexandrian regime, there is no notion of topological protection, and it is here that skyrmions may be created or destroyed.
As promised, I had a cute thought today, and here it is: in a certain sense, the Alexandrian and Gordian take on opposite roles in two main schools of condensed matter topology. I have outlined a bit about spin textures above, which are conserved due to a real space configuration. But the more famous topological insulators are protected by homotopies of the d-torus, which arises as the [crystal] momentum space. In that case, it turns out that the discretization of the lattice is a necessary precondition for nontrivial topology: if the lattice constant were taken to zero, the indices labeling the atomic orbitals would end up being real valued rather than integer valued. Instead of the torus, then, the we would end up with a momentum space given by the Pontryagin dual to the reals—which is just the always uninspiring real numbers themselves, where there are no circles to wind and no homotopy to classify.
Perhaps the inverted role of the continuum approximation in these two cases has something to do with the inverted physical units or the conjugate nature of real and momentum space, but I’ve not figured it out quite that far quite just yet.