A coincidence the group did not ask for

Two quantities with the same value are two quantities with the same value. Nothing about the number says whether they were obliged to agree or merely happened to, and a table of measurements cannot be interrogated on the point.

Symmetry can answer it, and the answer is a procedure rather than an inspection. A degeneracy the group forces is one that cannot be removed by anything respecting the group; a coincidence has no such protection. So the test is to change what the symmetry does not decide and watch what survives — which is an experiment in the ordinary sense, performed here on an operator rather than on a crystal.

What “forced” means, precisely #

An operator H commuting with every element of a group cannot mix the pieces the group treats separately. Restricted to one irreducible piece it must be a multiple of the identity — Schur’s lemma, which is the whole reason representations were introduced — so the d states of a d-dimensional irreducible representation share one value. That is a forced degeneracy: it holds for every H commuting with the group, whatever else is true of it.

Nothing in that argument says two different irreducible pieces cannot happen to receive the same value. They are separate objects and their values are separate numbers; if the numbers agree, they agree. Such a coincidence is called accidental, which is a poor name for something with a reason and a good one for something the group does not decide.

The experiment, and why it is legitimate #

The instrument in these figures is deliberately uncommitted. Take an orbit of points under a group; consider the space of functions on it; build a symmetric matrix whose entry between two points depends only on the orbit of the pair under the group. Such a matrix commutes with the group by construction, whatever weights are chosen, and the weights are exactly the freedom the symmetry does not constrain.

So moving the weights moves everything a symmetric operator is free to be, and nothing else. A degeneracy that survives every such move is one the group requires; a degeneracy that vanishes at the second choice was a property of the first.

That is a genuine test with a genuine failure mode, and the failure mode is worth naming: a degeneracy could survive several weight schemes by luck. The remedy is more schemes, and the figures here use four; there is no proof in it, only a measurement whose confidence rises with the number of trials. This collection’s usual position — claims here are decidable and exact — does not extend to this one, and the essay says so rather than letting the exactness elsewhere carry it.

Why a coincidence is not rare #

A first reaction to an accidental degeneracy is that it should never happen: two real numbers computed from unrelated pieces of a matrix have no reason to agree, and a coincidence in a continuum is an event of probability zero.

That reasoning is right about a random operator and wrong about the operators anybody writes down. The weights in a model are chosen for tidiness — one, a half, a quarter — and tidy numbers collide. In the specimen below, two levels meet at exactly zero when the shells are weighted 1 and ½, and separate as soon as the ratio is anything else. Nothing was contrived; the collision is what happens when the arithmetic is simple enough to have solutions.

The same applies with more force to a physical model. A nearest-neighbour model with one hopping constant is not a generic operator: it has more structure than the symmetry requires, and the extra structure has consequences that look like symmetry and are not. Bipartite lattices are the standard example — a lattice whose sites fall into two sets with bonds only between them has a spectrum symmetric about zero, which is a property of the bond pattern rather than of any group of motions, and it produces coincidences in quantity.

So the honest position is that an accidental degeneracy in a computed spectrum is ordinary, that it usually records a simplification rather than a fact, and that the only way to find out is to remove the simplification and look again.

The specimen: pgg at the centre of the zone #

The one accidental case in the census is worth following, because it is small enough to see all of.

The model is an orbit of four sites under pgg, with hoppings out to the second shell of neighbours. At the centre of the zone the little group is the full point group 2mm, which is abelian, has four classes and therefore four one-dimensional representations: the prediction is four single levels and no degeneracy at all.

The measurement at the first weight scheme is a single, a double, and a single. The character table has been contradicted — or, more carefully, something has happened that the character table does not describe, since a table forbids nothing about coincidences.

Changing the weights settles it, and the sweep is the same one the figure at the top of this page performs on a point group. At the second scheme the doubled level is two singles a third of a unit apart; at the third and fourth the same. The coincidence was a property of the first choice of weights, in which two levels of different symmetry happened to arrive at zero together — and the model’s weights were 1 and ½, which is the whole of why it happened at all.

The doubling that looks accidental and is not #

The census contains a second kind of surplus degeneracy, and confusing it with the first is the commonest error in this part of the subject.

In the groups p3, p4 and p6 — the chiral ones, with a rotation and no mirror — the point group is abelian, so every representation is one-dimensional and the prediction is no degeneracy anywhere. The measurement doubles half the levels, at every weight scheme tried. It is robust, so it is not a coincidence; and the characters do not require it, so it is not forced by the unitary operations.

What forces it is complex conjugation. The character table of p4’s point group contains i, so two of its representations are complex conjugates of one another rather than real. An operator with real matrix elements is unchanged by conjugation, and conjugation carries each of those representations to the other, so the two levels they label cannot be separated by any real operator. The pairing is systematic, predictable and computable — the Frobenius–Schur indicator says exactly which representations it applies to — and it is not in the group.

The moral generalises past this collection. An unexplained degeneracy that survives perturbation is a symmetry nobody has written down yet. Sometimes it is antiunitary, as here; sometimes it is a hidden conserved quantity, as with the accidental degeneracies of the hydrogen atom, which turned out to record a symmetry larger than the rotations. The response to a robust surplus is to look for the missing operation rather than to call it luck.

Running the test on every plane class at once turns that from an observation into a census, and the census closes the argument, because the surplus does not appear in a scattering of places.

That agreement is the point. Neither computation is told what the other found: one measures spectra and one sums characters, and they name the same three classes. A surplus appearing anywhere else would be a coincidence the sweep would then have to move; a surplus missing from one of the three would mean the conjugation argument was wrong.

Three rows the machinery leaves open #

The census above marks three rows unaccounted, and this is the place to be precise about them, because a table with a category called “unexplained” is either honest or lazy and the difference is in the detail.

All three are zone-boundary wavevectors of non-symmorphic groups — pg at two points, pgg at one — where the levels double, robustly, and the little group’s characters do not require it. The unitary explanation is genuinely absent: at those points the factor system that makes the Bloch operators multiply strangely can be removed by rephasing, so what is left is an ordinary representation of a group whose characters are real.

What is not implemented here is the antiunitary bookkeeping for that case — Herring’s criterion, which turns on the sign of a sum over the antiunitary coset and distinguishes three possibilities where the naive conjugation argument sees one. The exact fact underneath is computed and is stated in the glide essay: the glide’s operator squares to minus the identity at those wavevectors. Naming the gap is not the same as filling it, and this collection’s practice is to say which one has happened.

The exact fact those three rows rest on is worth stating in one line, because it is integer arithmetic and the gap above it is not. A glide applied twice is a lattice translation, and a translation acts on a wave as its own phase — which at the edge of the zone is −1. So the glide’s operator squares to minus the identity there, as an exponent modulo twelve rather than as a number near −1, and no operator with that property acts on a one-dimensional space and survives conjugation. What is missing is not that fact but the bookkeeping that turns it into a verdict.

What the test costs, and what it cannot do #

Two limitations, both of which the figures obey and neither of which is a defect in the idea.

Moving the weights moves the levels, so the levels cannot be compared between columns. A degeneracy is a statement about two levels of one spectrum, and each column is a different spectrum. What is compared is the pattern — how many levels there are and how many states each holds — which is why the figures print a multiset of multiplicities rather than a set of values.

A degeneracy protected by something other than the group survives the test and is reported as forced. That is the antiunitary case above, and it is a feature rather than a fault: the test asks whether anything invariant can lift the degeneracy, and the answer for those pairs is no. What the test cannot do is say what the protecting symmetry is. It reports that one exists, which is the useful half, and finding it is then a question about the model rather than about the group.

Near-degeneracy, which is a different question again #

There is a third thing two levels can do, and it is neither of the above: they can be close. A near-degeneracy is a measurement with an error bar and has no group-theoretic status at all, which is the same position this collection took about near-symmetry — a pattern is either invariant or it is not, and a pattern that is nearly invariant is a measurement.

The figures here read multiplicities off a numerical spectrum at a stated gap, which makes every “degenerate” verdict a claim about that gap. Two levels a thousandth apart are reported as two levels; the same pair at a coarser tolerance would be reported as one. The gap is carried with the result for exactly that reason, and it is the only place in this ladder where a tolerance appears at all — the characters, the classes and the multiplicities above them are integer arithmetic with nothing to tune.

What a crystallographer does with this #

The distinction is not decorative. Three places where it decides an interpretation.

A vibrational spectrum. Two modes at the same frequency may be one two-dimensional representation, or two one-dimensional ones that happen to coincide. Changing the crystal — isotopic substitution, pressure, temperature — moves the second and not the first, which is exactly the experiment the figures here perform on weights.

A diffraction pattern with more symmetry than the crystal. Diffraction adds a centre whether or not the structure has one, so a symmetry seen in intensities is not automatically a symmetry of the arrangement. That is a forced coincidence of a third kind — forced by the measurement rather than by the group — and the essays on Friedel’s law are about how to break it.

A phase transition. A degeneracy that lifts as a crystal cools is a symmetry lowering itself. A degeneracy that persists through a transition was never forced by the symmetry that changed. Reading which is which off a spectrum requires knowing which representations the levels belong to, which is the decomposition this ladder computes.

The habit this is an instance of #

Every argument in this collection that separates a fact from an artefact has the same shape: change the thing that should not matter, and see whether the answer moves.

A pattern’s group is rediscovered from its point set rather than asserted, because a caption cannot check itself. A net’s group is asked of all five metrics, because declaring the lattice type is how a net gets reported as smaller than it is. A lattice’s distance from hexagonal is measured after reduction, because an unreduced form gives a number about the basis. And a degeneracy is perturbed, because a coincidence and a theorem look identical in one column.

None of those is a general method for finding truth. Each is the same small idea applied to a specific freedom: identify the thing the answer must not depend on, vary it, and see. What makes it work here is that the freedom is exactly characterisable — an invariant operator is any weighting of the orbits of pairs — so varying it is a finite, honest sweep rather than a hopeful poke.

How much tuning a coincidence needs #

The essay says an accidental degeneracy is ordinary in a model and would be surprising in a random operator. That contrast can be made precise, and the precise version is a count of parameters.

Take two levels and ask what has to happen for them to coincide. Restricted to the two states involved, a symmetric operator is a two-by-two matrix, and two levels coincide when it is a multiple of the identity — which is two conditions if the matrix is real, since the off-diagonal entry must vanish and the two diagonal entries must agree, and three if it is complex, the off-diagonal entry then being two real numbers.

So a degeneracy between levels of different symmetry is not a single condition. It is a set of codimension two or three in whatever space of parameters the operator is drawn from — the von Neumann–Wigner counting — and hitting it requires tuning two or three quantities at once.

That is why a model produces them and a random operator does not. A model has few parameters and they are tied together: nearest-neighbour hoppings all equal, further ones set to zero, a lattice with more symmetry than the group being used. Every such simplification is a constraint, and constraints are exactly what a codimension-two set is met by. Set the further hoppings free — which is what the weight sweep on this page does — and the coincidence has nothing holding it.

It also says what the test is really measuring. Moving the weights moves the operator through the space of symmetric operators, and a degeneracy of codimension two will not survive a generic path through that space, while a degeneracy forced by the group is not a codimension condition at all — it is a property of every point of the space.

Where the count of parameters shows up #

The same counting explains a feature of the reciprocal-space essays that would otherwise look like a separate fact.

In a band structure the parameters being varied are the components of the wavevector, and in the plane there are two of them. A real symmetric model therefore has degeneracies on a set of codimension two in a two-dimensional zone — which is a point. That is exactly what the honeycomb’s crossing is: an isolated wavevector, not a line, and isolated for a reason that has nothing to do with the honeycomb.

Raise the codimension and the points disappear. A model with complex entries and no symmetry relating them needs three conditions, which cannot be met by tuning two components, so its levels never cross in the plane at all unless something forces them to.

And a degeneracy along a whole line or surface is therefore evidence of something the counting does not include — a symmetry, an antiunitary operation, or a factor system no rephasing removes. That is a useful diagnostic to have in advance: count the parameters, count the conditions, and anything left over is being held by something that has not yet been named.

Where this ladder goes #

The instrument built here is used throughout the reciprocal-space ladder, where degeneracies arrive from three directions at once: the little group’s characters, the sign a glide leaves at the zone edge, and the reality of the operator. And the shell that splits applies the decomposition to something a reader can hold — the neighbours of an atom — where the multiplicities are the count of independent combinations before any of them is computed.