Concept

Spectral moments — where it appears

The averages of successive powers of a band's levels, taken over the band. The second counts a site's neighbours exactly, the third counts its triangles, and the fourth measures how far the shape departs from a semicircle — so a few of them describe a band without diagonalising anything.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Seven nets, and the one column that sorts them. Every wrapped net here, with its dimension, coordination, third moment and band shape, beside the exponent the coupled-band measurement returns for it. The 4 nets whose third moment vanishes all give an exponent within 0.03 of −2 and a quotient constant to a fiftieth of a per cent; the 3 that do not all give one near −1 and no constant at all. Dimension does not sort them and neither does coordination — each takes values in both groups.

Seven points that looked like a switch

A constant belongs to a square net and not to a triangular one, which points at the coupling graph. Seven wrapped nets say which property of it: the third moment, and neither the dimension nor the coordination. They also say it is a switch — and made continuous, it is a crossover that every one of the seven sits twenty-five times past.

solids · Bands in a solid

Nine combinations, and the column that sorts them is not the bands'. Two bands and a coupling, varied separately. The composite graph's third moment splits into triangles that lie inside a band and triangles that use two coupling bonds, and only the second sorts the table: every row with no gap-crossing triangle gives an exponent near −2 and a nearly constant quotient, whatever its bands are made of. Triangular bands carrying an intra-band moment of 7.296 behave exactly like square ones when the coupling is a matching.

The triangles that were never in the bands

A switch in how a gap scales is usually attributed to a band's third moment, and the attribution cannot be tested while the coupling runs along one of the bands. Separated, the bands turn out to decide nothing. Two triangular bands coupled along a matching — which cannot close a triangle across the gap — behave exactly like square ones.

solids · Bands in a solid

Named alongside it

The objects these essays reach for when they reach for this one.

Bands in a solid Band gap Closed form Coordination number Exact diagonalisation Model limit Perturbation theory Reference state Tight-binding models Convention Convergence Degeneracy

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