The crossing at the corner

The honeycomb is the smallest net in this collection with two vertices in its cell: two vertices, three edges, nine integers, and every hexagonal mesh in nature is a drawing of it. Its symmetric operator is therefore a two-by-two matrix at each wavevector, which is small enough that everything about it can be written out.

What comes out is the cleanest exact statement in this ladder. At the corner of the zone the two levels are equal — not nearly equal, not equal to the precision of the eigenvalue routine, but equal because a sum of three roots of unity is zero.

The matrix, written out #

The honeycomb’s two vertices are joined by three edges, and each edge reaches a different cell: in the description the collection already uses, the voltages are (0, 0), (0, −1) and (−1, −1). The Bloch matrix at wavevector k has zeros on its diagonal — no edge joins a vertex to itself — and off the diagonal the sum of the three phases the edges carry:

$$f(k) = 1 + e^{-2\pi i k_2} + e^{-2\pi i (k_1 + k_2)}$$

Its two levels are ±|f(k)|, symmetric about zero for the reason every bipartite structure’s are: the two vertex classes can be given opposite signs, which turns any solution into another with the opposite value.

So the whole question is where f vanishes, and it vanishes exactly where three unit complex numbers cancel.

Three phases that cancel #

At the corner of the hexagonal zone the wavevector is (⅓, ⅓), and the three phases become 1, ω and ω², the three cube roots of unity. Their sum is zero, and it is zero for the same reason the three cube roots of unity have been summing to zero everywhere in mathematics since the sixteenth century: they are the roots of x³ − 1, whose coefficient of x² is zero.

That is an identity in the ring of cyclotomic integers, which is where this collection’s phases live and which is why the arithmetic is exact. It is checked as one — the sum is computed as an integer vector and compared with the zero vector — rather than evaluated numerically and found to be small.

Three phases summing to zero requires three of them, and requires them at a hundred and twenty degrees. Both are facts about the honeycomb’s connectivity rather than about its geometry: the net has three edges per vertex, and their voltages differ by the right amounts. Nothing was measured to reach the crossing, and no drawing of the honeycomb was consulted.

Where else the function vanishes, and where it does not #

It is worth asking whether the corner is special or merely one of many places f vanishes, because the answer decides whether the crossing is a feature or a curiosity.

Three unit complex numbers sum to zero only when they are the three cube roots of unity up to a common rotation — an equilateral triangle in the plane, and nothing else. Writing the three phases as 1, $e^{i\theta}$, $e^{i\varphi}$ and requiring the sum to vanish forces θ and φ to be ±120°, which pins the wavevector to the two corners of the zone and to no other point of it.

So the honeycomb’s two levels touch at exactly two wavevectors — the two inequivalent corners — and are separated everywhere else. That is a statement about the whole zone obtained without sweeping it: an algebraic condition on three phases, solved once.

The same reasoning explains what a fourth edge would do. A vertex with four edges gives four such factors, and four unit numbers can sum to zero in a continuum of ways, so the vanishing set of a four-edge net is generally a curve rather than a pair of points. Connectivity decides the shape of the degeneracy, which is the claim the nets essays make about nets in a new setting.

Why the crossing is protected #

An exact cancellation invites the suspicion that it is a fluke of a simple model — a coincidence of the kind the third rung of the representations ladder is about, which would vanish the moment the model is made less tidy.

It does not, and symmetry says why. At the corner of the zone the little group of the honeycomb’s plane group p6m is 3m, whose representations have dimensions 1, 1 and 2. The two states at the corner carry that two-dimensional representation, so no operator commuting with the group can separate them — not a further shell of bonds, not a change of weights, not anything a crystallographer could do to a honeycomb while leaving it a honeycomb.

That is the same protection as the previous rungs describe, arriving at the smallest case in the subject. The exact cancellation of three phases is what the protection looks like in this model; the protection itself is a statement about 3m and holds for every model with that symmetry.

The gap, and what opens it #

Making the two vertices different destroys the crossing, and the amount by which it is destroyed is exact too.

Give one vertex the value +δ and the other −δ. The matrix at the corner is no longer zero: it is diagonal with entries ±δ, so the two levels are +δ and −δ and the gap is 2δ. Nothing else changes anywhere, because the off-diagonal part is untouched.

The crossing was protected by the two vertices being alike, which is to say by the operations of p6m that exchange them. A structure whose two vertices differ has a smaller group — p3m1 rather than p6m, since the six-fold rotation exchanged them — and in that group the corner’s little group is 3, which is abelian and forces nothing.

How the levels leave the corner, which is the second exact statement #

That f vanishes at the corner says the two levels meet. How they part on the way out is a separate question, and it has an equally exact answer, because f is a sum of three exponentials and differentiating it is one line.

Write the wavevector as the corner plus a small displacement q. The two partial derivatives at (⅓, ⅓) are −2πiω and 2πi, neither of them zero and neither a multiple of the other, so f vanishes to first order and no faster:

$$f \approx 2\pi i\,(q_2 - \omega q_1), \qquad |f| \approx 2\pi\sqrt{q_1^2 + q_1 q_2 + q_2^2}$$

The two levels are ±|f|, so they separate linearly in the distance from the corner. That is a cone rather than the paraboloid a level ordinarily has where it turns round, and the difference is visible in the hero figure as a corner in the curve rather than a smooth minimum.

The quadratic form under the square root is worth a second look, because it is not an accident of the coordinates. q₁² + q₁q₂ + q₂² is the metric of the hexagonal lattice — the squared length of q₁b₁ + q₂b₂ when the two reciprocal vectors are equal in length and sixty degrees apart. So the cone is circular, in the reciprocal space the lattice actually has, even though the coordinates it was computed in are oblique. A cone that came out elliptical would have meant an error in the metric rather than a feature of the net.

The gap refines the same statement. With the vertices made unequal the matrix has ±δ on the diagonal and f off it, so the levels are ±√(δ² + |f|²): exactly 2δ apart at the corner, as before, and approaching the same two straight lines far from it. The cone is not destroyed by the gap, it is rounded off — the asymptotes belong to the net and the rounding belongs to δ.

That is easier to believe at a small δ than at a large one, because at a small one the rounding is confined to a neighbourhood of the corner and the straight sides are visible on either side of it. The picture is the same picture at three settings of one number, and what changes across them is only how far from the corner the difference between the two vertices is still doing anything.

One consequence follows immediately and is worth naming because it is the only statement here about counting rather than about position. Near the corner the set of wavevectors with |level| < E is a disc of radius proportional to E, so its area grows as E² and the number of levels below E grows the same way. A level that turned round smoothly would give an area growing as E and a count that did not vanish at zero. The linear parting is therefore not a detail of the picture; it changes the leading behaviour of every quantity computed by summing over levels near the crossing.

Two things about that derivative are worth keeping separate, because the essay’s own accounting depends on it. The coefficients are exact: −2πiω and 2πi are values of the same cyclotomic integers the cancellation was written in, and no floating-point arithmetic enters. The approximation is the truncation — dropping the second-order terms — and it is a statement about a neighbourhood of the corner rather than about the whole zone. A figure drawn across the entire zone shows the curvature the linear term omits, and the two descriptions do not conflict: one is the tangent to the other, and the cone is exactly what the tangent is.

There is also a case the expansion quietly rules out. Had one of the two partial derivatives vanished, |f| would have grown linearly in one direction and quadratically in another, and the crossing would have been a cone in one direction and a trough in the other. Nothing in the identity 1 + ω + ω² = 0 forbids that on its own — it is a statement about the value of f, not about its gradient — so the linearity had to be computed rather than inferred. It is the six-fold symmetry that makes the answer round once it is known to be non-degenerate, and the computation that makes it non-degenerate.

The two corners, and the one operation that connects them #

A hexagonal zone has two corners that no operation of a hexagonal group carries onto each other. The star of one of them has two members: the corner itself and the other one.

That has a consequence worth stating, and it is where the antiunitary bookkeeping of the previous rungs reappears. The operation carrying one corner to the other is the two-fold rotation — inversion, in the plane — which p6m contains and p3 does not. In a group containing it, the two corners are related and their levels are equal by symmetry. In a group without it, they are unrelated, and their levels need not agree.

But an operator with real matrix elements is unchanged by complex conjugation, which sends k to −k, and −K is the other corner. So the two corners of a real model are always related, whether or not the group relates them — by an antiunitary operation rather than by a motion of the plane. The distinction has consequences a measurement can see, and it is the reason a chiral hexagonal crystal is not the same as an achiral one at its zone corners.

What the little group of p3 does at that corner is worth seeing beside the honeycomb’s own crossing, because it is the case where nothing is forced. 3 is cyclic and abelian, so all its representations are one-dimensional and its characters predict three separate levels; the measurement finds three separate levels; and a model with that symmetry at that wavevector has no degeneracy the group accounts for. The honeycomb’s pair at the corner is not a fact about the wavevector, and it is not a fact about hexagonal lattices. It is a fact about 3m — the mirror is what makes the two-dimensional representation available — and a structure that keeps the three-fold and loses the mirror keeps the corner and loses the pairing.

What this is a model of, and what it is not #

The honeycomb’s crossing has a large literature, because the honeycomb is graphene’s net and the crossing is where graphene’s remarkable electronic behaviour comes from. Nothing about that literature is this collection’s business, and the boundary is worth drawing sharply.

What is claimed here is that a symmetric operator on the honeycomb net has two levels that coincide at the corner of the zone, that the coincidence is exact and follows from an identity between cube roots of unity, that it is protected by a two-dimensional representation of 3m, and that making the two vertices different opens a gap of exactly twice their difference.

What is not claimed is anything about electrons, energies, velocities or materials. The operator here is the least committal one that commutes with the group — a hopping of one between neighbours — and it is an instrument for exhibiting a group-theoretic fact, in exactly the way the collection’s patterns are instruments for exhibiting groups. A real material has interactions, several orbitals per atom, a substrate and a temperature, and none of them is here.

The reason the boundary matters is that the group-theoretic half is the half that transfers. Any structure on this net with this symmetry has the crossing; the crossing is not a property of carbon. Boron nitride, whose two sites differ, has the gap; that is not a property of boron either.

One consequence of all this is a warning about how a calculation is sampled, and it is worth putting where the boundary between model and material is being drawn. The crossing lives at two isolated wavevectors of the zone, and everywhere else the two levels are apart. A calculation that visited only the interior — a grid whose denominator is not a multiple of three, say — would find a gapped pair of levels and would be reporting its own sampling rather than the net. The wider picture, the levels followed along a whole path through the zone rather than along the one line that reaches a corner, is what makes that visible: most of the path shows levels comfortably separated, and the meeting is a single point on it.

An accounting worth keeping: three quantities, all exact #

The pieces of this rung can be listed with what each one is, because the mixture of exact and measured runs through the whole ladder and this is where it is cleanest.

The wavevector is a pair of rationals with denominator three. Exact, and stored as such.

The three phases are cube roots of unity, stored as integer vectors in the twelfth cyclotomic ring. Their sum is compared with zero as an integer identity, not as a small number.

The gap is 2δ, a difference of two entries of a diagonal matrix, exact for any δ.

The only measured quantity anywhere in the essay is the shape of the two curves away from the corner, which is drawn by taking square roots of a modulus at a hundred sample wavevectors. Nothing in the argument depends on it, and the figure says so.

That accounting is the reason this collection prefers small cases to impressive ones. A model with fifty sites would produce a picture just as convincing and no statement that could be checked in a line.

The kagome comparison #

The next essay is about the kagome net, whose behaviour at the same kind of question is entirely different, and the contrast is worth planting here.

The honeycomb has two vertices and three edges, and its two levels touch at one point of the zone. The kagome net has three vertices and six edges, and one of its three levels does not depend on the wavevector at all — it is flat across the whole zone, for a reason that is also exact and is entirely unlike this one.

Both facts come from the same nine or twenty-one integers, both are checked by integer arithmetic, and neither needs a drawing. That is the argument the nets essays make about nets in general, arriving in reciprocal space: what survives when the distances are thrown away includes a great deal more than a first look suggests.

The two facts sit beside a third that this collection already had. Counting the vertices outwards from one vertex — the coordination sequence, which is the invariant the nets essays use to tell nets apart — separates the honeycomb, the kagome net and the square net immediately, because they grow at different rates. So does the reciprocal-space behaviour: a crossing, a flat level, and neither. Both separations come out of the same handful of integers by different arithmetic, and neither needs a drawing or a distance. That is as close as this collection comes to a general claim about nets: the quotient graph with its voltages is a small object, and a surprising amount of what a structure does is decided in it.

Why the smallest case is worth an essay #

Two reasons, and both are about what a small case is for.

It can be checked by hand. Three phases, one identity, two levels. A reader with a pencil can verify the central claim of this rung in a minute, which is not true of any of the census tables. The machinery here computes it as an integer identity, and the fact that the same result is available by hand is what makes the machinery trustworthy rather than merely convincing.

It separates the exact from the protected. The cancellation is exact in this model; the degeneracy is protected in every model with this symmetry. Those are different statements and the small case makes the difference visible: a reader who saw only the cancellation might think a more complicated honeycomb would spoil it, and a reader who saw only the representation might not believe a computation as small as this one.

Where this ladder goes #

The k-symmetry anchor ends here, with the machinery built and tested on the smallest interesting object. Its consequences carry on in two directions.

Into the applied field, where the same nets are read as frameworks of rigid bars and the same wavevector-by-wavevector analysis counts mechanisms rather than levels — and where a flat level and a line of mechanisms turn out to be the same statement about a cycle in a graph.

And into the finite box, where a calculation cannot hold every wavevector and the ones it cannot hold are not approximated but absent. The corner of a hexagonal zone has denominator three, so a box of four cells cannot see this crossing at all — which is a fact about arithmetic that anybody computing on a finite block has to know before choosing its size.