The disc was a band with holes in it

A plane spends the same reach on a disc moved the argument about defect bands from a chain into a square net and changed one thing: the count of candidate twins within a distance. Everything else was the chain’s. The coupling between two runs of six low sites — its decay length of two sites and its prefactor — came from two runs in a chain, and was spent on a disc. Half the runs had a resonant twin at 10³·⁹³ sites, against 10⁴⁰·⁸ in a chain, and thirty-seven orders of magnitude were the finding.

That essay said what it was holding fixed. The level a run holds was the chain’s. And the coupling was taken to depend on the distance between two runs and not on its direction, which it called false in detail — a twin directly above is reached by a different path from one along the row — and harmless in sum: a factor of order one in the prefactor, moving the half-way length by a fraction of a decade.

Whether that is so can be computed rather than estimated, and the computation needs no finite net at all.

A host with no edge #

The model is the chain’s in every number. Host sites sit at an energy of four in units of the hopping, sites of a run at zero, and every nearest-neighbour pair is joined by unit hopping. A run of six is six adjacent low sites in one row.

What changes is the method. A finite net has edges, and two runs in a net of a few thousand sites have a splitting that depends on how far the edges are. The host by itself, though, is an infinite square net with no defects, and its Green’s function — the response at one site to a disturbance at another, at an energy below the host’s band — has an exact form: an integral, over an auxiliary time, of a product of two modified Bessel functions, one for the displacement along the row and one across it. A set of low sites binds a level exactly where the Green’s function, restricted to those sites, has an eigenvalue equal to minus one over the contrast. For one run that is a six-by-six problem, for two runs twelve-by-twelve, and the levels it gives belong to an infinite plane.

The same construction with one Bessel function instead of two is the chain, and that is the check that matters. On two runs in a chain, at every gap from one to fourteen host sites, the Green’s-function splitting agrees with the diagonalised chain to nine figures, by two routes with nothing in common but the model. In the plane, a dense diagonalisation of a small net with the two runs in it agrees with the Green’s function to within two per cent at three displacements, the residue being the small net’s edges.

The plane’s run binds fewer levels #

The first thing the plane’s Green’s function says is about one run alone.

In a chain all six of a run’s levels lie below the host’s band. In a plane only four do. The host band of a square net is twice as wide as a chain’s — from zero to eight rather than from two to six — so its lower edge sits two units nearer the run’s levels and swallows the two highest. A run of six in a plane is a different particle in a box from the same run in a chain: the box has walls above and below it as well as at its ends, and some of its states leak out.

The level the argument is about survives. The coupling between two runs is the splitting of their shallowest level, the one nearest the host’s band and therefore the one whose tail reaches furthest, and that level is bound in both: 0.248 below the band edge in the chain, 0.239 in the plane. Treating it as the chain’s level was nearly right. The coupling built on it is where the plane differs.

Two directions, two laws #

Put a second run in the same row, end to end with the first, and count the host sites between them; or stack it directly above, and count the host rows between.

Along the row, the plane’s twins couple two to five times as strongly as the chain’s at the same gap, and the coupling falls with a fitted length of 1.77 sites against the chain’s 2.00. Stacked, the coupling starts stronger still and falls faster, with a length of 1.50, crossing the row’s curve at four host sites. Neither is the chain’s law with a different prefactor; the plane’s decay is not a single exponential, because a level spread over two dimensions thins out as it travels as well as decaying, and a fitted length absorbs that.

Measured in gaps, those are differences of a few. They are not what the census counted.

What the census measured from #

The census of twins in generated nets measured the distance between the centres of two runs, which is the natural measure for a point process and for a disc. The coupling law it was combined with was fitted against the gap between them. In a chain the two differ by the length of a run, and the chain’s argument never needed to say which it meant; on a plane they differ by six sites along the row and by one across it.

At ten sites between centres, a twin in the same row couples thirty-three times as strongly as one stacked above. The row twin is four host sites from its partner; the stacked one is nine rows. A run of six is not a point to the coupling at any distance that matters, because the coupling falls across the gap and the gap along a row is the centre distance less six. The chain’s law read as a radius lies below both curves at every distance — it was fitted on gaps and spent on radii — and below the row twins by one to two orders of magnitude.

An offset has a parity #

Between the two pure directions lie the offsets: a run some rows above the other and slid along by some sites. A disc treats all offsets at one distance alike. The plane does not even treat neighbouring offsets alike.

Two rows apart, sliding the second run from directly above to one site along drops the splitting from 0.42 to 0.016, and sliding it one site further brings it back to 0.24. The coupling alternates from offset to offset by more than an order of magnitude, and three rows apart by a factor of eight, until about seven sites of slide, after which it falls smoothly. The cause is a sign. Every path from one run to the other through the host carries a sign set by the parity of its length — the square net is bipartite, and below the band the Green’s function alternates in sign from one sublattice to the other. Two runs whose sites mostly sit on matching sublattices add their paths; two whose sites mostly sit on opposite ones cancel them. It is the lattice’s own two-colouring, read in a coupling, and it is why a hundred and seventy decades of couplings were never going to be a smooth function of distance once a second dimension was allowed.

On the map at the top these are the holes: displacements inside the dashed disc whose twin barely couples, beside displacements outside it whose twin couples strongly.

Half of the runs, again #

The resonance criterion is unchanged: a run has a resonant twin when some other run of six couples to it more strongly than the level spacing of the sample — a count above a threshold rather than an average — and the spacing is the published one, falling as one over the number of sites. The only change is what replaces πs². At each spacing, the region of displacements whose twin couples above it is counted directly, displacement by displacement, from the Green’s function, and the chance a run has no twin there is the chance none of those displacements holds one.

Half the runs are resonant at 10³·⁰⁹ sites rather than 10³·⁹³. The region that does it holds 177 displacements, as it must at the half-way point, but it is not a disc of radius 7.5: it reaches eleven sites along the row and six across it, with holes. The coupling threshold there is 6.4 × 10⁻³, against the published 9.3 × 10⁻⁴, because the plane’s couplings are strong enough that half the runs clear a spacing seven times larger.

That reading treats twins as scattered independently over the region, and it can be checked without the assumption. In three generated nets of 400 by 400 sites at half filling, 1,422 runs of six far enough from the edges were each given their strongest coupling to any other run, looked up from the same Green’s-function table. The share above each threshold matches the region’s reading to within four points over the whole range, and at the measured half-way threshold 51 per cent of the counted runs have a resonant twin. The hard core that the census found costing the disc four points is already inside the region, because displacements where two runs would overlap are not displacements of two runs.

What moved, and what it was worth #

The published estimate was that direction is a factor of order one in the prefactor and worth a fraction of a decade. It is worth 0.84 of a decade, which is a fraction, and the reasoning that predicted it does not survive. The factor between directions is not of order one: it is thirty at ten sites. And most of the shift does not come from a prefactor. It comes from the run’s length, which the disc set to zero by measuring a coupling across gaps and spending it across radii, and from the plane’s couplings being stronger at every gap.

It moved in the direction that strengthens the conclusion, and it leaves the saturating rise exactly as it was: the share still climbs as one minus an exponential in a count, only the count is a measured region rather than a disc. A chain reaches half at 10⁴⁰·⁸ sites, a body the size of an asteroid; a plane, recomputed, at 10³·¹. The thirty-seven decades are thirty-eight. Everything the argument said about there being no partly resonant plane is said more strongly, because the plane’s twins couple more than a chain’s law allowed.

How it was computed #

The host Green’s function below the band is $G(m,n) = -(-1)^{m+n}\int_0^\infty e^{-\varepsilon t}\,\tilde I_m(2t)\,\tilde I_n(2t)\,dt$, with ε the depth below the band’s lower edge, $\tilde I$ the exponentially scaled modified Bessel function, and the sign the bipartite one. The integral is taken on a uniform grid in the logarithm of t, which for an integrand this smooth converges exponentially; the Bessel functions come from Miller’s backward recurrence normalised by their sum rule. The one-dimensional version agrees with its closed form to twelve figures, and the two-dimensional on-site value with its closed form, an elliptic integral, to ten.

A level is bound where δ times an eigenvalue of −G among the low sites equals one, δ being the contrast of four. Each root is found by regula falsi on the logarithm, bracketed between a quarter of the single run’s depth and the next level down. For two runs the shallowest level splits into the two weakest bound channels of the twelve, and the splitting is the difference of their depths. Displacements run over every offset up to thirty sites along a row and twenty across, 644 of them, using the net’s two mirror symmetries for the other quadrants. Displacements where two runs would overlap or touch in one row are excluded, being one run rather than two.

The checks, run wherever these figures are drawn: the Green’s function against both closed forms; the chain’s splittings by the Green’s function against the diagonalised chain at every gap from one to fourteen; three plane splittings against a dense diagonalisation of a small net; the chain’s run binding six levels and the plane’s four, with the shallowest within a twentieth of each other; a row twin coupling more than ten times a stacked one at ten and twelve sites between centres; a dip at an odd offset between two even ones three rows apart; the half-way length moving by between half a decade and a whole one; and the generated nets agreeing with the region’s reading to four points where the share is between a fifth and four fifths. The refusal is the chain: its decay length must come back from the Green’s-function route exactly as the diagonalisation gives it.

Where the clean host stops #

The host is clean. Every site that is not part of the two runs sits at the host energy. At half filling a real net’s host is half low sites, and it percolates; a level’s tail travels through a disordered landscape, not a uniform one, and both the decay and the parity pattern would be scrambled by it. The clean host is the assumption the chain’s law made too, which is what makes the comparison fair, and it is the assumption that matters most for any real material.

Very close twins are not pairs. Two runs one row apart with no host between them, or two rows apart and overlapping, couple as strongly as a level is deep, and their two levels are better described as one larger cluster’s. Every such displacement is far above any threshold used here, so nothing in the half-way length depends on what they are called.

The spacing is the published one. The level spacing of a plane’s gap states would follow from the plane’s own levels of runs of three to nine, which bind differently from a chain’s as the levels in the second figure show. Holding the spacing fixed isolates the coupling, which is the question; it leaves the spacing as a separate one.

Who found what #

Lattice Green’s functions of the square net and their Bessel-function representation are standard in solid-state physics, as is the condition that a localised perturbation binds a state where the restricted Green’s function has a particular eigenvalue — the Koster–Slater method of 1954. The sign alternation of a bipartite lattice’s Green’s function below its band is the same fact as the pairing of its spectrum, the property that also pins a vacancy’s levels at zero. What is computed here is the splitting of two runs of six at every displacement in a clean square net, the region of strong enough twins it gives, and what that region does to the half-way length.

A neighbour count carries a geometry #

The essay this continues found the dimension of a defect band in one line of combinatorics: the number of sites within a distance. The lesson here is that the line assumed a geometry it did not state. Counting twins within a distance treats both runs as points and the coupling as a function of one number, and a run of six is neither a point nor isotropic. When a quantity is exponential in a distance, which distance it is matters more than any prefactor, because a change of variables by a run’s length is a change of the exponent’s argument by six sites. That is worth more than any factor of two the prefactor could have hidden.

Still open: a host that is half low, and a cluster that is not a run #

The obvious open question is the host. Every coupling here passes through clean host, and at half filling there is none; the Green’s function of a disordered net has no closed form, but the restricted-Green’s-function condition holds for any host whose Green’s function can be computed, and a net with its low sites drawn at random can be solved by the same twelve-by-twelve problem with the host’s resolvent computed numerically. Whether the parity holes survive disorder is the sharpest version of the question, since a random site energy changes which paths cancel.

The nearer question is the one the plane essay asked first and this one sharpens: whether a plane’s defects should be runs at all. The coupling’s dependence on direction comes from the run’s shape, and a two-dimensional cluster of six sites has a level and a coupling that depend on its shape too. The level and coupling calculation accepts any set of low sites, so a census of six-site clusters, each with its own level and its own coupling map, is the same calculation with a different set of sites.