Seven points that looked like a switch

The constant that belonged to one net measured how two coupled bands distort one another, and found that the quotient of the excess gap by the band’s shape departure is one constant times the separation — exactly, and to a fiftieth of a per cent — when both bands hop on a square net, and no constant at all when both hop on a triangular one. It named the coupling graph as the deciding object and offered an explanation: the first-order term is invisible because $\langle\varepsilon^3\rangle$ vanishes for a bipartite graph.

That explanation makes a prediction two nets cannot test. If the third moment is the mechanism, then the behaviour should track the third moment rather than the coordination or the dimension, and nets of the same coordination with different triangle counts would separate them.

Seven wrapped nets span three dimensions and four coordinations. One column sorts them completely.

The census #

Four of the seven have a third moment of exactly zero: a chain, a honeycomb net, a square net, a cubic structure. Every one of them returns a gap exponent within 0.03 of −2 and a quotient constant to better than four parts in a thousand.

Three have a third moment that does not vanish: a chain reaching to its second neighbours, a chain reaching to its third, a triangular net. Every one of them returns an exponent near −1 and a quotient whose spread across the same six separations is 128 per cent.

The four in the first group are the four bipartite ones, and that is not a coincidence: a closed walk of three steps needs an odd cycle, and a bipartite graph has none. So the column being read is a statement about the graph before it is a statement about any spectrum, which is why a width that is a count of neighbours can be exactly right about the second moment and silent about this one.

Neither dimension nor coordination survives as a candidate, and they fail in the strongest available way — not by correlating weakly, but by taking the same value in both groups. Coordination six occurs at −2.0277 on a cubic structure and at −1.1054 on a triangular net. Coordination four occurs at −2.0118 on a square net and at −1.0644 on a chain of reach two. One dimension occurs in both groups; two dimensions occur in both groups.

Plotted against the skewness the seven points do not lie on a curve. They lie on two horizontal lines. The three skewed nets span a factor of 1.63 in skewness — 0.7500, 0.8165, 1.2247 — and their exponents differ by 0.043, which is the scatter of a fit rather than a trend.

Read on its own, that is a switch: the third moment sorts the nets as a predicate rather than as a quantity, and the size of it makes no difference at all. It is also the natural reading, because the alternative — a quantity so steeply non-linear that a factor of 1.63 in it is invisible — is not the sort of thing a census suggests.

One pair, one difference #

A census of seven is a correlation. What settles a mechanism is a pair that differs in one thing.

A square net and a chain reaching to its second neighbours both have coordination four, so their second moments are both exactly 4 — that much is a theorem already shown not to be the whole story. What is not required by anything is that their kurtoses agree as well: both are 2.250000, to every figure computed. That is the shape a band takes on its way to a bell curve, reached here by two routes that share no dimension.

They are in different dimensions, and it changes nothing. They differ in the third moment, and it changes a whole power.

Two shapes agreeing is the useful accident. A comparison in which the fourth moment also moved would leave two candidate causes; this one leaves one. And it is worth noticing that the kurtosis is the quantity the measurement itself is built on — the shape departure being measured is a change in kurtosis — so the pair also rules out the possibility that the exponent is somehow inherited from the reference shape.

Turning the variable on #

Seven points are still seven points, and the third moment in them takes one of two kinds of value: zero, or something of order unity. Nothing in the census distinguishes the third moment vanishes from the graph is bipartite, and nothing in it can see what happens in between, because there is no wrapped net in between.

So the variable is made continuous. A square net whose diagonal bonds carry a strength $\lambda$ is a square net at $\lambda = 0$ and a triangular net at $\lambda = 1$, and its third moment is proportional to $\lambda$ in between — one triangle per diagonal, each weighted by the one weak bond in it.

Every net in the family is scaled so that its second moment is exactly 4. That matters more than it sounds: switching a diagonal on adds width as well as skewness, and a width change moves every exponent in the problem. Holding $\langle\varepsilon^2\rangle$ fixed leaves the shape as the only thing moving, and it is checked to nine figures at every setting. The coupling runs along this net’s own bonds at their own strengths, as it has in every measurement so far, so what is being turned on is the skewness of the whole arrangement rather than of one part of it — which is a confound this family cannot separate and a separately coupled one could.

The exponent does not jump. It slides from −2.0118 at $\lambda = 0$ through −1.9786, −1.7621, −1.4518, −1.2581 and −1.1027 to −1.0835, monotonically, and it is half-way at a skewness of 0.0300.

The smallest non-zero skewness any of the seven nets has is 0.7500 — twenty-five times that. Every net in the census sits far past the crossover, on the flat right-hand end of the curve, and the two horizontal lines the census showed are two clusters of points on one continuous function.

So the switch was an artefact of the sample, and the sample was not chosen carelessly: it is every wrapped net in the census, and there is no reason a lattice should have a small third moment. A regular graph with any odd cycles at all has plenty of them. The region of the curve where the crossover lives is a region no lattice occupies, and it took a family of graphs built to occupy it to see the shape.

That is worth separating from the finding it corrects. The bipartite mechanism is right, and the census confirms it: the third moment is the deciding quantity, dimension and coordination are not, and the controlled pair holds. What is wrong is the reading of how it decides, and the wrong reading is the one the evidence supported.

Where a fit would have hidden it #

A single exponent fitted over six separations could hide a crossover in the other variable too. If the −2 behaviour held at small separations and the −1 behaviour took over at large ones, a fit would return something in between and would look like a smooth interpolation without there being one.

At the two ends of the family there is no such thing to hide. At $\lambda = 0$ the local slope runs from −2.028 to −2.004, settling on −2 from below as the separation grows; at $\lambda = 1$ it runs from −1.145 to −1.046, settling on −1. Both are approaching an integer from outside, and the fitted values of −2.01 and −1.08 are finite-separation artefacts rather than the exponents themselves.

In the middle of the family it is a different picture. At $\lambda = 0.02$ the local slope is not a constant at all: −1.604, −1.529, −1.473, −1.411, −1.328 across the six separations, still drifting at the end. That is what a system genuinely between two powers looks like, and it is why the fitted −1.4518 is a summary rather than an exponent. A crossover in the separation and a crossover in the skewness are the same crossover seen along two axes, and the middle of the family is where both are visible at once.

What the amplitude does meanwhile #

The exponent turns over between skewnesses of 0.003 and 0.1. The size of the effect does something else entirely.

The quotient falls from 2.362 to 0.0856 — a factor of 27.6 — monotonically, across three decades in $\lambda$, with nothing happening at the place where the exponent changes character. So one number is doing two jobs on two different schedules: it sets how large the effect is, over the whole range, and it sets which power the effect follows, over one decade in the middle.

There is a second reading of that curve worth separating out, because it says the family is measuring what it claims to. The quotient falls as the net becomes more triangular. If the third moment simply made the effect bigger, the amplitude would rise; instead the effect is largest where the third moment is smallest and non-zero. Both the excess gap and the shape departure grow with $\lambda$, and the shape departure grows faster — which is what makes the quotient a diagnostic of the response’s shape rather than of its size. That combination was chosen because it came out constant on a square net, and it turns out to be the combination that isolates the crossover as well.

What was computed, and how #

Each net is a wrapped graph with unit hops, diagonalised exactly. The moments are $\langle\varepsilon^k\rangle$ over its own spectrum, so the second moment is the coordination as a theorem rather than as a check, and the third moment counts closed three-step walks — twice the number of triangles a site sits in.

The coupled measurement is the original one, unchanged. Two copies of one net on the same sites, separated by $\Delta$, coupled along the lower graph’s own bonds at a strength of 0.06; the upper band’s kurtosis is compared with the isolated net’s, and the excess gap is the measured gap minus the four band edges written down from the isolated spectra rather than obtained from a second calculation. The exponent is a least-squares slope on logarithms across six separations from 12 to 48.

The interpolating family adds a per-edge strength, so a diagonal bond can be weaker than a square one, and the two-band construction now reads each graph’s own edge strengths rather than assuming they are equal — including on the coupling, so that a weak diagonal is weak as a coupling bond too. Every figure drawn before this is unaffected, because every other net here carries no strengths at all and falls back to one.

Two refusals had to hold and both were available. The family at $\lambda = 0$ must reproduce the plain square net’s exponent to two decimals, which it does — otherwise the construction would be measuring itself rather than the net. And dimension and coordination are required to occur in both groups of the census, so a version of this that had accidentally chosen a set of nets in which the third moment happened to correlate with one of them would fail rather than pass.

The monotonicity of the family is checked rather than observed, which matters because a crossover established from a curve with a wobble in it is a curve with noise in it.

Where the model stops #

Everything here is one hopping matrix and no repulsion, which is this collection’s standing caution about a band. It bites in a particular way on this result: a third moment that is exactly zero is a statement about a graph, and a real solid’s third moment is zero only to the extent that its Hamiltonian is a bipartite hopping matrix. Second-neighbour hopping, a site-energy difference, anything that breaks the sublattice symmetry, puts a non-zero third moment back.

The crossover says how much of that is tolerable, and the answer is: very little. A skewness of 0.03 is enough to take the exponent half-way, and 0.03 is what a second-neighbour hop a few per cent of the first would produce. So the bipartite behaviour is not robust; it is the fragile case, in the way a gap that is not a band width is not fragile at all — one is an inequality between quantities and the other is a cancellation between them.

The exponents are measured over a factor of four in the separation, which is enough to distinguish −1 from −2 and not enough to establish either to three figures. The local slopes are quoted for exactly that reason: they say where the values are going, which the fits alone would not, and they are what shows that the fitted number in the middle of the family is not an exponent at all.

And the amplitudes are not compared between different nets. The quotient’s mean is 2.442 for a square net and 0.086 for a triangular one, and those are numbers about two systems rather than two values of one function — only the $\lambda$ family, where the second moment is held, gives an amplitude that can be read as a trend.

The generalisation #

A census of the cases that exist is not a scan of the variable that decides. Seven nets, chosen because they are the seven a lattice can be, put no point anywhere near the crossover — and the resulting two clusters are the strongest possible evidence for a dichotomy that is not there. The repair is not more cases of the same kind; it is a family built to take intermediate values, even if no material does. The same shape turns up wherever a discrete set of real systems is used to establish a continuous law: four molecules with force fields fitted a coefficient that a fifth molecule would have been free to break.

And a power and an amplitude are different kinds of quantity with different schedules. The amplitude here moves over three decades and the exponent over one, and neither is a proxy for the other. Anyone reading a fitted exponent as a smooth diagnostic of a coupling is reading a quantity that is nearly constant over most of the range and does all its moving in a place nothing samples — which is the same trap as an exponent that is a switch rather than a scale, arrived at from the other side.

Who found it, and when #

The vanishing of odd spectral moments for a bipartite graph is elementary and old: a closed walk of odd length needs an odd cycle, and a bipartite graph has none. Its use as the organising fact in tight-binding moment expansions is Cyrot-Lackmann’s, from the nineteen-sixties, and the whole recursion-method tradition is built on the idea that a few moments settle a band’s shape — which is both useful and limited, since two structures agreeing on three moments still differ in binding.

What that tradition does not have occasion to say, because it is asking about shapes rather than about exponents, is how steeply the first forbidden moment controls a power. The census above is a demonstration of an old fact. The $\lambda$ family is what turns it from a classification into a curve, and it is the part that could have come out otherwise — a linear response would have been the natural expectation and is not what happens.

Still open: coupling on a different graph, and moving the crossover #

The obvious open question is what happens when the coupling moves rather than the bands. In every measurement here the coupling has run along one of the bands’ own graphs, so the band’s graph and the perturbation’s graph have never been separate objects, and no measurement so far can say which of them the third moment has to belong to. Coupling two bipartite bands along a non-bipartite set of bonds separates them in one step, and the two readings disagree about the answer.

The nearer question is where the crossover sits as the coupling strength changes. Everything above is at one coupling, 0.06, and the crossover is presumably a competition between a first-order term and a second-order one — in which case its position in the skewness should scale with the coupling in a way that can be written down and then checked. Three couplings would say whether it does, and a crossover whose position is predictable is a crossover that can be used as a measurement rather than merely reported.