http://www.nature.com/news/paradox-at-th...=ODIxNTQyNjU3S0 Cubitt and his collaborators focused on calculating the ‘spectral gap’: the gap between the lowest energy level that electrons can occupy in a material, and the next one up. This determines some of a material’s basic properties. In some materials, for example, lowering the temperature causes the gap to close, which leads the material to become a superconductor.
The team started with a theoretical model of a material: an infinite 2D crystal lattice of atoms. The quantum states of the atoms in the lattice embody a Turing machine, containing the information for each step of a computation to find the material's spectral gap.
Cubitt and his colleagues showed that for an infinite lattice, it is impossible to know whether the computation ends, so that the question of whether the gap exists remains undecidable.
For a finite chunk of 2D lattice, however, the computation always ends in a finite time, leading to a definite answer. At first sight, therefore, the result would seem to have little relation to the real world. Real materials are always finite, and their properties can be measured experimentally or simulated by computer.
But the undecidability ‘at infinity’ means that even if the spectral gap is known for a certain finite-size lattice, it could change abruptly — from gapless to gapped or vice versa — when the size increases, even by just a single extra atom. And because it is “provably impossible” to predict when — or if — it will do so, Cubitt says, it will be difficult to draw general conclusions from experiments or simulations.
Would this not be a problem only if change could occur twice (once in each direction) between two measurements. If this were possible, how would solving the problem for an infinite lattice help? Couldn’t it change an infinite number of times between minimal and infinite lattice sizes?
