The defect that needs two laps

The circuit that does not close makes a defect into a crystallographic object by walking round it: the walk comes back to the wrong lattice point, and the amount by which it misses is quantised because the lattice is. How many dislocations a lattice has then counts the distinct ones, by taking orbits of those misses under the point group.

Both of those are arguments about a lattice, and the interesting thing about the circuit argument is that it never needed one. What it needed was a space for the order parameter to take values in, and a loop in that space. Change the space and the same argument classifies a completely different set of defects, with no lattice anywhere.

The classification is a fundamental group #

Take a medium described by an order parameter — a direction, a lattice, a set of moments — and let G be the symmetry the medium had before it ordered and H the symmetry it kept. The values the order parameter may take are the cosets G/H, because two elements of G give the same state exactly when they differ by something H fixes.

Now go round a line defect. Far from the core the medium is locally in some ordered state, so the circuit traces a closed loop in G/H. If the loop can be shrunk to a point without leaving G/H, the order parameter can be continued into the core and there is no defect there. If it cannot, there is. So the kinds of line defect are the loops of G/H up to deformation, which is the fundamental group π₁(G/H).

One more step gives the classification its final form. A loop has a base point, and moving the base point conjugates the class. Two defects related that way are the same defect looked at from somewhere else, so what a medium actually distinguishes is the conjugacy classes of π₁(G/H) and not its elements. This is the same reduction how many dislocations a lattice has makes when it takes orbits rather than vectors, and it is made for the same reason: a classification is of physical objects rather than of the labels a description happened to give them.

Two things about this deserve to be said before any group is written down, because both are easy to lose. The first is that the loop has to be taken far from the core. Close in, the order parameter is doing whatever it does — it may be melted, or rotated out of the manifold entirely — and the classification says nothing about that. What it classifies is the boundary condition the core is obliged to match, which is why the answer is the same for a thin core and a fat one and for a core made of any material at all.

The second is that the argument gives a complete invariant rather than a necessary condition. Two defects whose loops are freely homotopic can be deformed into each other through configurations that are legal everywhere outside the core; two whose loops are not, cannot, by anything. A great many arguments in this collection produce a condition that must hold and leave open whether it is enough — Neumann’s principle is the standing example — and this is not one of them.

Why the answer is twice as big as the point group #

Everything above is general. It becomes arithmetic as soon as G is the rotation group and H is finite.

The rotation group is not simply connected. A loop that turns through a full turn about a fixed axis cannot be shrunk to a point inside SO(3), and a loop that turns twice can — which is the fact behind the plate trick, and the fact two turns to come back turns into a group. The double cover SU(2) is simply connected, and covering-space theory then gives the answer directly: π₁(SO(3)/H) is the preimage of H in SU(2).

That preimage is the binary group H*, of order 2|H|. It is exactly the object the double groups are made of — the quaternions closing on a set of generators, with the full turn −1 always in — so nothing new has to be constructed here. What is new is the reading. In the spin problem the extra element is what makes a half-integer state change sign; in the defect problem it is what makes a loop fail to close. The same minus one does both jobs, and the two problems have no other contact.

It is worth being explicit about what “the preimage” means computationally, because it is what turns a topological statement into a table. Each rotation of H has two unit quaternions above it, q and −q, and the preimage is the set of all of them. Closing a set of quaternion generators under multiplication produces exactly that set, and the closure automatically contains −1 because some product of generators lands there. So the construction is: pick generating rotations, write them as quaternions, multiply until nothing new appears. Eleven proper classes go in and eleven groups of twice the order come out, and no covering space is ever built.

The last column of that table is where the physics is, and it is worth reading against the third. A medium whose residual symmetry is 432 has forty-eight loops to classify and only seven kinds of defect, because the group is large and its conjugacy is coarse. A medium whose residual symmetry is 6 has twelve loops and eleven kinds, because a cyclic group’s classes are its elements. Higher symmetry gives fewer distinguishable defects, not more, which is the opposite of what the number of loops would suggest.

One check on the whole scheme is available for nothing and is worth taking. The number of conjugacy classes of a group is at least the number of its abelian quotient’s elements, and for these covers it is usually far more than that, so a count of classes that came out smaller than the number of classes of the point group itself would be evidence of an error. It never does: every row of the table has at least as many classes as the class it covers, and the two extra are the identity and the full turn. That is not a proof of the construction, but it is a statement that would fail loudly if the closure had missed elements.

One circuit, drawn #

The half-turn disclination is the case worth seeing, because it is the one where the counting and the picture come apart.

A headless direction is unchanged by a half turn, so the configuration drawn is continuous and single-valued all the way round. There is no seam and no discontinuity to point at. What there is instead is a loop in the order parameter space that cannot be contracted — and the only way to see that is to ask what the loop does in the group, which is exactly what the classification is for. This is the sense in which a defect is a topological object rather than a place where something is broken: the neighbourhood is perfectly well behaved and the obstruction is global.

Going round twice traces a loop that can be contracted, so two half-turn disclinations of the same kind annihilate. That is not a statement about how they are brought together or about what the core does; it is the statement that the corresponding element of the binary group squares to the identity, and it holds however messy the intermediate configurations are.

Which defects are their own antidefect #

The order of a class is how many identical defects have to be merged before the result can be undone, and it is the cheapest thing the group says.

The full turn is the entry to notice. Every one of the eleven binary groups contains −1, and −1 is central, so it is a conjugacy class on its own in every case. It is a genuine defect — a loop that goes round the core twice as the order parameter goes round once — and it is undoable, in the sense that two of them combine to nothing. Media in which it is the only defect are the ones with trivial residual symmetry, and there the classification says a medium can have exactly one kind of line defect and it is topologically the plate trick.

There is a second reading of the order column which is worth having, because it is the one an experiment would use. A defect of order n is one that n copies of can be brought together and removed. So a medium whose defects all have order two is one in which every line defect is its own antidefect and any two like defects annihilate; a medium with defects of order three or four has lines that can only be cleared in bundles. That is a statement about what a relaxing sample can do to itself, and it comes out of a multiplication table rather than out of any dynamics.

A class that is not its own inverse is a defect with a distinguishable antidefect: bringing two of them together does not annihilate, and which of them is “the” defect is a convention rather than a fact. Whether that happens is decided entirely by whether the class contains the inverse of its own representative, and in the cyclic cases it usually does not — a third-turn disclination and a minus-third-turn disclination are genuinely different objects.

Defects that do not commute #

The sharpest consequence of the whole classification is invisible in any count, because it is a statement about pairs.

If two defects a and b have aba⁻¹ ≠ b, then carrying a round b turns b into something else. Two such lines therefore cannot cross: dragging one through the other leaves a third line joining them, because the loop that used to enclose only b now encloses aba⁻¹b⁻¹ as well. In a medium with a non-abelian defect algebra the lines get stuck on each other, and that is a property of the symmetry breaking and not of the material’s stiffness or its viscosity.

The argument for why they cannot cross is worth writing out, because it is short and it is the whole reason the distinction matters. Take a loop enclosing b alone, and drag the line a through the plane that loop bounds. Before the crossing the loop sees b; after it, the loop has been dragged round a and back, so what it sees is aba⁻¹. If the algebra commutes those are the same defect and nothing happened. If it does not, the loop’s class has changed, and a class cannot change continuously — so something discontinuous happened, which is a new line joining the two. That third line has class aba⁻¹b⁻¹, the commutator, and it is exactly the obstruction the group’s non-commutativity measures.

The smallest crystallographic instance is 222, whose binary group is the quaternion group of order eight. Its three half-turn disclinations lift to ±i, ±j and ±k, and any two of those anticommute. It is the case biaxial media are known for and it is entirely elementary once the group is in hand.

The merge is where the non-commuting shows up as something a measurement could see, and it shows up as an ambiguity rather than as an exotic outcome.

An entry naming two classes is saying something quite strong: two experiments that prepared the same two kinds of defect and brought them together can get different answers, and neither experiment did anything wrong. The classification is complete and the merge is still not a function of it. That is not a gap in the theory; it is what it means for the group to be non-abelian, and a classification that hid it would be worse rather than tidier.

A word is owed on the media this describes, since none of them is a crystal in the sense the rest of this collection uses. A crystal breaks translations, and it is the translations that carry its defect classification. The media whose residual symmetry is a bare point group acting on orientations are the ordered fluids — the nematics and their biaxial relatives, and the ordered phases of some magnets when only the moment directions are considered. What they have in common is that orientation is ordered and position is not, which is why the order parameter space is a quotient of the rotation group alone.

That is also why the eleven proper classes are the right list rather than the thirty-two. An improper operation reverses handedness, and a continuous path in the rotation group cannot reach one — so a residual symmetry containing a mirror does not sit inside the group being covered, and the binary construction has nothing to lift. The machinery refuses such a request rather than silently answering a different question, which is one of the three refusals below.

What this shares with the lattice version, and what it does not #

The lattice case is the same construction with a different G. There the broken symmetry is the group of translations, H is trivial, and G/H is a torus whose fundamental group is the lattice itself — so π₁ is ℤ³ and the defects are lattice vectors, which is the Burgers vector exactly. Translations commute, so the algebra is abelian, so dislocations pass through each other freely and the merge of two is decided by their two kinds. Every convenient thing about dislocations comes from that one fact.

What the two cases do not share is the residual symmetry’s role. In the crystal it is a point group acting on the classification from outside, cutting ℤ³ down to orbits — the reduction how many dislocations a lattice has performs. In the rotational case it is the classification: the group being classified is the residual symmetry, doubled. So a crystal has fewer distinct dislocations when it is more symmetric, and a rotationally ordered medium has fewer distinct disclinations when it is more symmetric, and the two facts have quite different proofs.

There is one more asymmetry between the two cases and it runs the other way. The crystal’s classification is infinite — ℤ³ has infinitely many elements and a crystal has infinitely many distinct Burgers vectors — while every rotational case here is finite, with at most eleven kinds. So a crystal’s defects are graded by magnitude, and the reason only the shortest ones are seen is energetic rather than topological. In the rotational media there is no magnitude to grade by: the classification is a finite list and every entry on it is as topological as every other.

Neither of them is about energy. A defect that is topologically allowed may cost so much that no sample ever contains one, and a defect that is topologically forbidden cannot be made cheap by anything. That division is the same one an order parameter is a representation draws between which order parameters a symmetry permits and which of them a material actually chooses, and it is drawn in the same place: symmetry says what is possible, and nothing here says what happens.

The first of those is the mistake worth naming, because it gives the right answer often enough to survive. For a cyclic class the point group and its binary cover have the same number of conjugacy classes as they have elements, so counting classes of the point group and doubling gets the total right by accident. It fails on the first class with two axes: 222 has four conjugacy classes and its cover has five, not eight, and the missing structure is exactly the part that makes the defects fail to commute.

Where this stops #

Two things this collection does not compute are worth naming rather than leaving implied. The first is point defects, which are classified by the second homotopy group of the same space, and for a discrete residual symmetry that group is trivial — so a medium with a point-group residual symmetry has no topologically stable point defects at all, and demonstrating that needs machinery about covering spaces this collection does not carry. The second is what happens when H is not discrete: a uniaxial medium keeps a continuous rotation about its own axis, G/H is a projective plane rather than a coset space of finite index, and its fundamental group has order two. That is a smaller answer than any row of the table above and it is arrived at another way.

What is computed here is the finite case in full: eleven residual symmetries, their covers built from quaternions, their conjugacy classes counted, and the six algebras in which two defects cannot pass. Which magnetism a class permits and how many domains a transition makes are the same style of question one and two steps below this one — the zeroth homotopy group counts domains, the first counts lines — and it is worth seeing that the domain count and the defect count come from one construction, differing only in which loop is being asked about.