A 6×5 patch with one site missing

One of the figures on extended structures: Chains and rings taken far enough to behave like solids — bands, gaps, fillings, defects and ends — every one of them a finite matrix diagonalised, with no lattice anywhere in the argument.

One essay draws this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

A vacancy is not an impurity

A patch of a two-dimensional structure with one site removed. The two colours are the two halves of the bipartite structure — colour a site by whether its row and column indices sum to an even or an odd number, and every bond joins one colour to the other. The disc areas are the amplitudes of the state that sits at exactly zero, and every one of them is on the same colour.

The perfect patch: thirty levels, none of them at zero, the nearest at 0.0699. The spectrum is symmetric about the middle because the structure is bipartite, and every level is paired with one at minus its energy.

The same patch with one site removed. Twenty-nine levels, and one of them is at zero to within 8×10⁻¹⁷, which is arithmetic noise rather than a small number.