The point defect whose charge has no sign

The defect that needs two laps classifies line defects by what happens to the order parameter round a loop drawn through the material. This essay changes the loop for a sphere, and almost everything changes with it.

The change is not a generalisation. A loop and a sphere ask different questions, they are answered by different groups, and a medium can have a rich answer to one and no answer at all to the other.

What a sphere reads #

A line defect runs along a curve, so a small loop can be drawn round it and the order parameter followed along that loop. A point defect sits at a point, so what encloses it is a sphere, and the order parameter is not a path but a map: every direction from the defect carries a value, and the sphere of directions is carried into the space of possible values.

When that space of values is itself a sphere, such a map has a degree — the number of times it covers the target, counted with sign. That integer is the defect’s charge, and it is what everything below computes.

One condition on the sphere is easy to leave unstated and is doing real work: the order parameter must be defined and continuous everywhere on it. A sphere that a disclination line passes through has points on it where the director is undefined, the map is not a map, and the degree is not merely hard to compute but meaningless. So the sphere has to enclose the point defect and avoid every line defect, which in a real sample is a constraint on where the sphere may be drawn rather than a formality — and it is why the classification is of isolated defects.

The computation is an integral rather than a count of preimages, and the reason is robustness. Counting preimages of a chosen target value works and requires the value to be regular, which has to be checked; adding signed areas works for any map and needs nothing checked. The signed area of a spherical triangle with vertices a, b, c is

E = 2 · atan2( a · (b × c), 1 + a·b + b·c + c·a )

and the sign comes out of the triple product without a separate orientation test. Summing E over a triangulation of the domain, with each triangle’s image taken in the same order as the triangle, gives 4π times the degree.

The sign is the whole of it. A map that covers the target twice one way and once the other has degree one, and the unsigned areas would add to three. Every fold in the map has to cancel, and it does, because the triple product changes sign exactly when the image triangle is traversed the other way round.

What makes the integer a classification rather than a measurement is that it does not move. A degree is a homotopy invariant: deform the field continuously, by any amount, and the number is unchanged — which is exactly the property a defect charge must have, since a real material’s field is never the idealised one. That is tested here rather than assumed. Each of three fields is perturbed forty times by a randomly oriented sinusoidal twist of amplitude nearly one radian, large enough to make the field unrecognisable, and the degree is recomputed each time. It comes back unchanged in all one hundred and twenty cases. A deformation that did change it would have to pass through a field that is not defined somewhere, which is to say through the creation or destruction of another defect — and there is nowhere for one to hide inside a smooth perturbation.

The cross-section is worth reading carefully, because it shows how little the sign is visible. The hedgehog appears as a source: every segment radiates from the core. The hyperbolic defect appears as a saddle: the segments point outward along one direction and inward along the perpendicular one. Those two pictures are as different as two pictures of point defects get, and the charges they carry differ by two — which is the largest difference two unit-magnitude charges can have and the smallest amount by which a nematic’s charges can differ meaningfully at all. So the visible difference between a source and a saddle is doing real work, and the invisible difference between the two lifts is doing none.

One construction is worth recording because getting it wrong is easy and silent. Writing the sphere as the plane plus a point at infinity, the map z ↦ zᵏ has degree k for positive k. The obvious guess for degree −k is z ↦ z⁻ᵏ, and it is wrong: z ↦ 1/z is a Möbius transformation, therefore holomorphic, therefore orientation-preserving, so z ↦ z⁻² has degree plus two. A holomorphic map never has negative degree. The map of degree −k is the conjugate one, z ↦ z̄ᵏ, and the difference between the two is invisible in any picture of the field.

The sign that is a choice #

A nematic’s order parameter is a director: an axis with no arrowhead, so n and −n are the same physical state. That single fact removes the sign of every charge computed above.

A field of directors has two continuous lifts to a field of vectors — pick a sign at one point and carry it round, or pick the other. Both are valid, neither is preferred, and they differ everywhere by a minus sign. Composing a map with the antipodal map multiplies its degree by the degree of the antipodal map, which is −1.

That is not an ambiguity in the calculation; it is a property of the object. A hedgehog of charge +1 and one of charge −1 are the same defect described with the two available conventions, and there is no measurement that distinguishes them — because the thing that would distinguish them, the arrowhead, does not exist.

The physical form of this is more striking than the arithmetic. Carry a point defect once round a disclination line and bring it back. The disclination is a place where the lift flips, so the transport has changed which lift is in use, and the charge that comes back is the negative of the one that went out. Nothing was done to the defect. What changed is the route home. That is the action of the fundamental group on the second homotopy group, and it is why a charge is a statement about a defect together with a path rather than about the defect alone.

There is a consequence worth naming. In a medium with disclinations, a defect and its antidefect are not distinguishable kinds, so the rule that a defect can only annihilate with its antidefect loses its content: any two point defects of equal magnitude can annihilate, provided one of them is walked round a line first. A conservation law that a symmetry of the medium can flip is a bookkeeping convention rather than a constraint.

What the loop still says #

The loop question does not go away, and running it alongside is what makes the two computations comparable.

Walking the sign round a circuit that encloses a disclination of strength s and comparing the end with the start decides the line defect’s class, and the answer is: it closes when s is a whole number and flips when s is a half. There are two outcomes and no others, so the space of directors has a fundamental group with two elements — computed by walking, rather than quoted.

Comparing that walk with the one the circuit that does not close performs is instructive, because the two look identical and measure different things. A Burgers circuit steps through a lattice and accumulates a translation failure, and the answer is a lattice vector. This walk carries a sign round a continuous loop and the answer is one of two labels. The first is a defect of a periodic structure and lives in the translation group; the second is a defect of an orientational order and lives in a homotopy group. Five copies and the gap they leave is the intermediate case, where a rotational closure failure is measured on a structure that has both. The three together are the reason “defect” is a family of objects rather than a single one.

Then the whole-strength lines can be removed, and the removal is explicit. Tilt the director out of the plane, letting the tilt angle run from zero on the axis to a right angle at the boundary. The result matches the disclination’s boundary condition, is a field of unit vectors everywhere, and at the core every direction of approach arrives at the same vertical vector — so there is no singularity and no defect. That is the escape into the third dimension, and what makes it possible is precisely that a whole-strength line has a single-valued vector field to tilt.

A half-strength line has none. Going once round returns the opposite vector, so there is no vector field on the punctured disc at all, and nothing to tilt. The escape is not merely harder there; the object it operates on does not exist. That is the difference between a defect that is stable and one that is a temporary arrangement, and it is decided by whether a number is whole.

Which media have them #

The last question is the one that brings this back to crystals, and the answer is a flat negative.

Point defects are classified by maps from a sphere into the space of order parameter values, and that classification is unchanged by passing to a universal cover. A medium whose residual symmetry is a finite point group has SO(3) modulo that group for its space of values, and the defect that needs two laps computes those covers: each is the three-sphere, carrying a binary group of twice the point group’s order. The three-sphere admits no non-trivial map from a two-sphere. So a medium ordered by a crystal class has line defects, has as many kinds of them as its binary group has conjugacy classes, and has no point defects whatever.

The three-sphere in that sentence is the same three-sphere two turns to come back is about, arrived at from a different direction. There it is the double cover of the rotation group, the reason a spinor needs two full turns to return to itself. Here it is the universal cover of the order parameter space of an orientationally ordered medium, and the reason that medium’s line defects are counted by a binary group rather than by the point group itself. One space, two uses, and the second follows from the first: a medium whose states are frames is a medium whose states are rotations up to a finite group, and the cover of the rotations is what everything is computed on.

A nematic does, and the reason is the opposite property: it keeps a continuous symmetry, the rotations about its own director, so its space of values is two-dimensional rather than three, and a two-dimensional cover has maps from a sphere to count. The same is true of a ferromagnet, whose values are the directions of the magnetisation.

There is a physical reading of the same dimension count. A medium keeps a continuous symmetry exactly when its order is incomplete — a nematic has settled on an axis and not on a frame, so a rotation about that axis costs nothing. The most of an icosahedron a crystal can keep is about the other end of the same scale, where a medium wants more order than a lattice will carry. Point defects live at the incomplete end, and the more thoroughly a medium orders itself the fewer kinds of defect it has room for — which is the reverse of the intuition that a more structured thing has more ways to go wrong.

So the division is between media that break the rotations completely and media that leave some of them. It is not between simple media and complicated ones, and it is not between crystals and liquids. It is a statement about the dimension of what is left over, and a crystal class leaves three dimensions of it, which is one too many.

That has a consequence for how the word “defect” is used about crystals. How many dislocations a lattice has counts the line defects a lattice permits, and the point defects a crystallographer names — a vacancy, an interstitial, a substituted atom — are not topological objects at all. They are local departures from a structure, removable by moving a finite number of atoms, and nothing about them is protected. The distinction matters because it says which defects can be annealed out in principle and which cannot, and for a crystal the answer is that every point defect can.

The instrument that lies quietly #

The last of those refusals is about the computation rather than the physics, and it changed how everything here is measured.

The natural guard on a numerical integral is the residual: the degree must be an integer, so the distance from the raw integral to the nearest integer measures how well the computation is doing. That guard fails here completely. A field varying faster than the mesh is sampled at the vertices and the sum lands on a different integer, exactly, with a residual of 6 × 10⁻¹⁴.

The measurement is not approximately right and it is not visibly wrong. It is the correct degree of a different map — the one the sampled vertices actually describe — which is why it is so clean. Aliasing produces confident wrong answers, and a check that asks “is this answer clean?” will approve every one of them.

So every degree in this essay is read on three successively finer meshes and required to give the same integer on all three. That is the only guard that catches it, and it costs three times as much. The general lesson is worth carrying past this essay: a self-consistency check that a wrong answer also passes is not a check, and the residual here is exactly that. The same trap is available to any figure in this collection that integrates something over a sampled domain.

Where this stops #

The classification here is of isolated defects in an infinite medium, which is the case where a sphere can be drawn round one thing. Two defects close together are classified by a sphere round both, and the charge is then additive — but which pairs can actually merge, and along which paths, is the question the fundamental group’s action makes complicated, and nothing here computes it.

Additivity is the first thing that goes. Draw a sphere round two defects instead of one and the degree of the resulting map is the sum of the two individual degrees, which is the statement that charge is conserved when defects merge. In a nematic that statement has to be qualified by everything above: the signs are conventions, so “the sum” depends on having carried both charges back to one place along specified paths, and different paths give different sums. What survives is the sum modulo two, which is the only part no route can change. So a nematic conserves a charge and conserves an integer only locally, and the descent of symmetry is the general setting for that pattern — a quantity that is well defined until the medium’s own symmetry is allowed to act on it.

Nor are the media exhaustive. Biaxial nematics, superfluids and the ordered phases of liquid crystals with more structure all have larger order parameter spaces, and several of them have both point and line defects with a non-abelian interaction between them. The defect that needs two laps computes the non-commuting line algebras for the crystal classes; the corresponding statement about points is empty there, for the reason above, and non-empty elsewhere.

And the topology is quoted rather than derived. That the three-sphere admits no non-trivial map from a two-sphere, and that homotopy classes are unchanged by passing to a universal cover, are standard results and are used here as inputs. What is computed is everything downstream of them: the degrees, the two lifts, the loop that flips, the field that removes a whole-strength line, and the mesh that lies.