Subjective Gravity

une tasse de coucher de soleil, pour le tisserand dormant

[subjectivegravity-blog]

Today I had to dig up and old program of mine for imaging spin data where the underlying lattice isn't square---and therefore doesn't provide a simple mapping to a square array of pixels. I ended up finding a suitable solution to the problem, but along the way I tried some weird stuff, and I found these various test images, nominally of Monte Carlo simulation data, scattered throughout my source directory.

They had enough character that I thought they were sharable. They're, at the least, pretty to gaze upon.

[subjectivegravity-blog]

A property of fast modes that’s “too good to be true”

In studying spin waves (though this should hold for any semiclassical wave phenomenon), it's convenient to use the following fact: terms of odd order in the spin wave fluctuations can be ignored.

The way to do this is to apply a coarse graining procedure in the time domain. Supposing that the magnon period is much smaller than the other time scales in the system (such as the mean free path of the magnons, say, or the velocities of spin textures that interact with the spin waves), we can apply the following operation to the Lagrangian/equations of motion/dynamical-whatever f(t):

A[f](t) = (1/T) ∫[t-T/2,t+T/2] f(t') dt'

for T the spin wave period. To the degree that the linear approximation holds locally for any time t' for all the slow times scales, this operation is just the identity on those terms (since the mean value of f(t') on t±T/2 is just f(t) for linear f).

So for terms of the dynamical equation at zeroth order in the spin wave fluctuations, A[...] is the identity. It turns out that for reasonable assumptions about the spin wave fluctuations, A[...] is usually the identity on terms quadratic in the spin wave flucutations too (since, roughly speaking, sin² + cos² = 1). But terms linear in the spin wave fluctuations lie in the kernel of A[...], since they will average out to zero over a single period.

This is only a rough sketch, of course, and things can go wrong. Terms that depend only on the magnitude, but not the direction, of spin wave fluctuations will not average to zero. And sometimes terms can mix in ways that A[...] applied to quadratic terms picks up a scalar factor other than unity.

But it turns out that A[...] really does work, at least anecdotally, for most vanilla Lagrangians.

So here's what seems, to me, "too good to be true": it just so happens that (linear) equations of motion quadratic in the variables can be pulled back to a bilinear Hamiltonian. We can rewrite such systems as a matrix equation, like Dψ = Hψ or what have you, and then import all the various tools that we know about for solving quantum mechanical systems in various regimes. But we can only pull out a Hamiltonian matrix precisely because our terms all happen to be quadratic.

What's more, it turns out that if you had tried staying quantum mechanical (say with a Holstein-Primakoff transformation) from the get-go instead of falling back to semiclassical spins and applying A[...], the Hamiltonian you'll end up with is exactly the same.

This isn't really surprising, because in both cases the same approximations are being made about the smallness of spin wave fluctuations. But in the H-P case, you never apply A[...]. So what is A[...]? Where does quantum mechanics get away with applying it implicitly; or perhaps more appropriately, what is it that the semiclassical spin model does wrong that needs to be corrected by A[...]?

Edit: Now that I've written that all down, it almost feels like I'm maybe doing some kind of requantization procedure without realizing it. Anyone know anything about requantization that doesn't involve linking me to some high level arcane runes on n-lab?

[subjectivegravity-blog]

fixing at higher orders

In reminding myself about the fixed-point combinator yesterday, I wrote some sample functions in Haskell such as fib, prod, and sum. Eventually I convinced myself that I could write a fold but I only got so far. I have that

import Data.Function (fix) foldGenerator :: (b -> a -> b) -> b -> ([a] -> b) -> ([a] -> b) foldGenerator f b g (x:[]) = b `f` x foldGenerator f b g (x:xs) = g xs `f` x myFold :: (b -> a -> b) -> b -> [a] -> b myFold f b = fix (foldGenerator f b)

This is all well and good, but what I really like is to find some f such that fix f = fold. I would be fine even with just getting a fold1. But everything seems so much higher order that I'm not sure if it's possible. Does anyone know if it is? Can anyone point to a reference that does this sort of thing, or even just let me know if it's possible in principle?

[subjectivegravity-blog]

The beautiful thing about Haskell is that it provides a convenient cultural bridge from commonplace programming to the depths of type theory and modal logic that I could never have reasonably approached by any other route.

This post inspired by the Introduction (which is all I've read so far) of Kenneth Foner's thesis, which I'm sure the remainder of which can be nothing but a fun read.

[subjectivegravity-blog]

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.