The level that does not move

The kagome net is three vertices and six edges — the midpoints of the triangular net’s edges, joined where they meet — and it is the second-simplest net this collection draws. Its symmetric operator is a three-by-three matrix at each wavevector, so it has three levels, and two of them do what levels normally do: rise and fall across the zone.

The third does nothing. It is a horizontal line at the value −2, across the entire zone, and it stays there whatever the wavevector. That is a strange thing for a level to be, because a level is supposed to say how the operator’s value depends on the wave, and this one has no dependence at all.

The state, and why it is exactly an eigenvector #

Take one hexagon of the kagome net — six vertices in a ring — and put amplitudes +1, −1, +1, −1, +1, −1 round it, alternating, with zero on every other vertex of the infinite net.

Apply the adjacency operator, which replaces each amplitude by the sum of its neighbours’ amplitudes. On a vertex of the hexagon, the two neighbours inside the ring both carry the opposite sign, so the sum is −2 times its own value. On a vertex outside the ring the sum is over whichever ring vertices it touches, and in the kagome net every such vertex touches exactly two of them with opposite signs, so the sum is zero.

So the state is an exact eigenvector with eigenvalue −2, and both halves of that sentence are integer arithmetic with no rounding in them. The machinery here checks it in a finite supercell: build the adjacency matrix, place the alternating state, multiply, and require the result to equal −2 times the state exactly. The residual is zero, not small.

The parity that makes it possible #

Two properties of the ring are doing the work, and it is worth separating them because only one of them is about the kagome net.

The ring must be even. A sign that alternates round a cycle contradicts itself at the join if the cycle has odd length. The kagome net’s shortest cycles are triangles, and a triangle cannot carry such a state at all; the search that finds the ring here is therefore for the shortest cycle of even length, and it steps past the triangles to the hexagons.

Every outside neighbour must touch the ring an even number of times, with cancelling signs. That is a fact about how the kagome net’s hexagons and triangles fit together, and it is what makes the state vanish outside the ring rather than merely being small there.

Why a localised state means a flat level #

Here is the connection that makes the flatness inevitable rather than surprising.

A level’s dependence on the wavevector is a statement about how a state spreads: a wave that extends across the crystal has a phase that advances from cell to cell, and the value of the operator on it depends on how fast. A state confined to one hexagon has no such phase — there is nothing outside the hexagon for a phase to advance across — so there is no wavevector in it, and no wavevector can change its value.

And there is one such state per hexagon, so there are as many of them as there are cells. A whole level’s worth. When those states are combined into waves — which is what a decomposition by wavevector does — every combination is still an eigenvector at −2, whatever the phases, and the level is flat.

That argument is why flat levels and localised states are two descriptions of one thing. It is also why a flat level is fragile in a way an ordinary one is not: anything that lets the state leak out of its hexagon gives it somewhere to spread, and the line acquires a curvature.

Nothing in the invariants a net wears on its sleeve announces any of this. The kagome net’s coordination sequence — the count of vertices at each distance from a chosen one, which is what the nets essays use to tell nets apart — is perfectly ordinary, and so is its degree, and so is the size of its cell. It took an eigenvector to find the flat level, and the only way to know whether the eigenvector is a fact about this net or a fact about the construction is to run the construction on nets where it should not work.

Finding the ring rather than recognising it #

A reader looking at a drawing of the kagome net sees the hexagons immediately, and it would be easy to write a demonstration that begins “consider one of the hexagons”. This collection’s habit is to make the machinery find them, and the reason is not fastidiousness.

The search here is for the shortest cycle of even length through a vertex, run on the adjacency matrix of a finite supercell with no coordinates anywhere. What it returns for the kagome net is a six-cycle, and the essay’s argument then proceeds from a cycle the computation located rather than a shape a reader pointed at.

The difference shows up the moment the same routine is asked about another net. On the square net it returns a four-cycle, and the alternating state there is not an eigenvector, because a vertex outside a square touches only one of its corners rather than two. On the honeycomb it returns a six-cycle, and the state fails for the same reason — a honeycomb hexagon’s outside neighbours touch it once each, and nothing cancels. A demonstration built on recognition would not have noticed that the kagome net’s hexagons are special among hexagons; a demonstration built on a search has to explain why the other cases fail, which is where the second condition above came from.

Running it on seven nets rather than one turns that from a remark into a table, and the table says two things. Most nets fail, with an integer residual of one or two rather than something small — so a residual of zero is a result and not the only answer the routine knows how to give. And the kagome net is not alone: the five-coordinated net, whose two vertices have degree five, carries the same state on a four-cycle and is exact at every supercell size. Whatever the property is, it is not a peculiarity of hexagons or of the kagome net’s particular symmetry.

The three columns are the correction, and it is the sort that is easy not to make. The natural way to run this check is on the smallest supercell that holds the cycle, because it is the cheapest; and the smallest supercell is exactly where the wrapping creates adjacencies the infinite net has not got. At three cells across, the square net reports a residual of zero and the triangular net reports one at four cells across, and both are artefacts of the torus rather than facts about the nets. Neither is visible from inside a single run. What makes them visible is asking the same question at three sizes and requiring the answer not to change, which is the same discipline as pinning a numerical sweep at the wavevectors where an exact answer is available.

The states are not independent, and the count says by how much #

A hexagon per cell suggests one state per cell, which would be exactly a level’s worth. The states are not linearly independent, though, and the shortfall is the sort of thing this collection likes to count rather than wave at.

Add the alternating states of all the hexagons in a finite supercell with periodic boundaries, with a suitable choice of signs, and they cancel: the sum is zero. So the states span a space of dimension one less than the number of hexagons — or two less, depending on how the supercell wraps — and a level’s worth of states is very slightly over-counted by the naive argument.

That is not a defect. It is the same accounting as the Euler relation for a net folded onto its torus: the cycles of a graph on a torus satisfy one relation for each independent way round, so a family indexed by faces has a small deficiency against a family indexed by cells. The deficiency shows up in the level count as the flat level touching the others at exactly one wavevector, which is what the figure above shows at the centre of the zone.

What the kagome net does that the honeycomb does not #

The previous essay followed the honeycomb, whose two levels touch at the corners of the zone and are apart everywhere else. The kagome net’s behaviour is entirely different and comes from the same kind of arithmetic.

Both facts are decided by connectivity: the honeycomb’s by three phases that cancel only at a corner, the kagome net’s by a cycle that supports an alternating state. Neither needs a coordinate, both are checked exactly, and neither could be guessed from a drawing.

The comparison is worth making because it answers a question a reader might reasonably have about the whole nets ladder: if a net is only a graph, what can a graph decide? These two essays are an answer. A graph decides where levels touch, whether any of them is flat, how many independent motions its framework has, and — by way of the barycentric placement — its whole symmetry group.

The same fact, one ladder along: a line of mechanisms #

The rigidity essays found something about the kagome net that reads very differently and is the same statement.

Read as a framework of rigid bars, the kagome net has one mechanism in a cell of one, two in a cell of two, three in a cell of three — a count that grows with the size of the cell it is looked for in, while Maxwell’s count sits at one throughout. That was recorded as a finding and left as one, with the note that a mechanism count quoted without its cell is half a result.

The reason is now available. The kagome framework’s zero modes lie along lines in reciprocal space, and a larger cell samples a line at more places; the mechanism count is the number of sampled points, not a property of the framework alone. And the mechanism is the same alternating pattern round a hexagon: three bars twist one way and three the other, and the motion dies outside the ring for the same cancellation that made the state an eigenvector here.

Where a flat level is not flat #

Two honest limits, both of which a reader meeting the kagome net elsewhere will need.

The flatness is a property of this operator. A nearest-neighbour adjacency with one weight is what makes the outside cancellation exact. Adding a second shell of neighbours, or giving the three vertices different values, generally destroys it: the state can leak out of its hexagon and the line acquires curvature. The flat level is not protected by the symmetry in the way a two-dimensional representation protects a degeneracy, and this is one of the clearest cases where the two kinds of robustness differ.

The value is a convention. The level sits at −2 because the operator is an adjacency with unit weights; scale the weights and the value scales with them. What is invariant is that the level does not move with the wavevector, not the number it does not move to.

Both limits are the sort of thing that vanishes when a result is quoted and matters when it is used, and both are checkable on the machinery here by changing an option.

What a physicist wants from a flat level, and what this is not #

Flat levels have a large literature and it is worth marking the boundary of what is claimed here, since the words are borrowed.

A level with no dispersion means states that do not travel, and a system whose states do not travel behaves very differently from one whose states do — in a solid, interactions between the particles occupying such states dominate everything else, because there is no kinetic energy to compete with them. Whole research programmes rest on the fact, and the kagome net is one of their standard hosts.

None of that is here. This essay claims that a certain integer vector is an exact eigenvector of a certain integer matrix, that the corresponding level is independent of the wavevector, and that the reason is a parity argument about a cycle. Everything about what a material made this way would do belongs to a different subject, and this collection’s rule is to compute what its own arithmetic reaches and to say where that stops.

What does belong here, and is the reason the essay exists, is that the flatness is a fact about a graph. It survives every drawing of the net, needs no metric, and is decided by the same twenty-one integers that decide the net’s symmetry group. That is the thesis of the nets essays tested in a place it was not built for.

What does belong here, and is worth following one step, is the other net the ring test found. The looped net carries the same alternating state and therefore the same flat level, and its framework does not behave like the kagome one: its zero modes sit at a single wavevector rather than along a line. So a flat level and a line of mechanisms are not the same statement after all — they are two consequences of the same cancellation, and how many wavevectors each of them occupies is decided separately. That is a refinement of the parallel this essay draws, and it came out of running a test on a net nobody had asked about.

Why the exact route was worth building #

A flat level can be found by sweeping the zone and noticing that a curve does not move. That is a measurement, and it is subject to the usual doubt: is the level flat, or flat to within the resolution of the sweep and the precision of the eigenvalue routine?

Exhibiting the eigenvector removes the doubt entirely. A vector of integers, an adjacency matrix of integers, a product computed exactly, a residual of zero. There is no tolerance in it and nothing to interpret, and the claim afterwards is a theorem about this net rather than an observation about this computation.

That is the same preference the collection has everywhere: a pattern’s group is rediscovered from the point set rather than measured off a picture, a net’s group is recovered from its incidences, and a degeneracy is exhibited rather than sampled. The sweep is still drawn, because a reader wants to see the line; the argument does not rest on it.

Every drawing in this essay is made at the placement a net has when nobody chooses one: every vertex at the average of its neighbours. That matters here only as a disclaimer. The flat level and the mechanism are both statements about the graph, so neither depends on the picture; the picture is what makes the hexagons and triangles visible to a reader, and the argument never consulted it.

The honeycomb framework is worth one last look beside the kagome one, because it is the third behaviour and it completes the set. The honeycomb has three bars per two joints against the kagome net’s six per three, so it is under-constrained everywhere rather than at particular wavevectors, and its zone is not a set of lines or a pair of points but the whole of itself.

The kagome net is a line graph, and every line graph does this #

The alternating state was found by a search rather than recognised, which is the right way round. There is also a structural reason it exists at all, and it says the kagome net is not special so much as a member of a family.

The kagome net is the line graph of the honeycomb: its vertices are the honeycomb’s edges, and two of them are joined when the edges they came from share a honeycomb vertex. That relation has an algebraic consequence. Write B for the matrix recording which edges meet which vertices in the honeycomb; then the adjacency operator of the line graph is

$$A = B^{\mathsf{T}}B - 2I.$$

B^T B is a square, so it is never negative, and therefore every eigenvalue of a line graph is at least −2. The value −2 is attained exactly on the null space of B, which is the set of edge-weightings summing to zero at every vertex — precisely the alternating states this essay is about, since an alternating sign round an even cycle is the simplest such weighting.

So the flat level is the null space of an incidence matrix, and its size is a count: the number of edges less the rank of B. For the honeycomb that is three edges against two vertices per cell, giving one state per cell — which is exactly a level’s worth, and the flat band’s existence with no case analysis.

The rank correction is the missing state. B for a bipartite graph has rank one less than its vertex count, so the null space is one larger than the naive count in a finite system — and correspondingly, in the periodic reading, the alternating states are one relation short of independent. That is the cancellation the essay reports as a curiosity, arriving as a term in a rank formula.

The rest of the family #

Once the mechanism is a property of line graphs, the other examples announce themselves and it is worth naming two, because they are the ones a reader will meet.

The pyrochlore net — corner-sharing tetrahedra, the three-dimensional analogue of the kagome — is the line graph of the diamond net, and it has a flat level for the same reason, at the same value, with a multiplicity the same rank formula gives.

The checkerboard and the dice nets and several others in the same literature are line graphs or close relatives, and every one of them was found to have a flat level before anybody noticed they had a construction in common.

That is the useful reading of this essay. A flat level is not a coincidence to be searched for net by net; it is a signature of a net being built from another net’s edges, and the census of small nets can therefore be sorted by that property rather than swept. The search that found the alternating ring is still the right thing to run — it is what makes the claim about this net a computation rather than a citation — but what it finds is an instance of a rule.

Where this goes #

The next essay takes the picture of mechanisms across the zone seriously and makes it quantitative: the rank of the rigidity matrix wavevector by wavevector, the identity that turns a sum over wavevectors into a supercell’s mechanism count, and the explanation of a finding the framework essays recorded without one.

And the census of nets asks how special the kagome net is by enumerating the nets nobody chose — which turns out to be the only way to find out that almost none of them has any symmetry at all.