Posted on Dec 10, 2015

"Alan Turing is famous for his role in cracking the Enigma code," said Dr Toby Cubitt from UCL Computer Science. "But among mathematicians and computer scientists, he is even more famous for proving that certain mathematical questions are `undecidable' – they are neither true nor false, but are beyond the reach of mathematics. What we've shown is that the spectral gap is one of these undecidable problems. This means a general method to determine whether matter described by quantum mechanics has a spectral gap, or not, cannot exist. Which limits the extent to which we can predict the behavior of quantum materials, and potentially even fundamental particle physics."

A mathematical problem underlying fundamental questions in particle and quantum physics is provably unsolvable, according to scientists at UCL, Universidad Complutense de Madrid – ICMAT and Technical University of Munich.

It is the first major problem in physics for which such a fundamental limitation could be proven. The findings are important because they show that even a perfect and complete description of the microscopic properties of a material is not enough to predict its macroscopic behaviour.

A small spectral gap – the energy needed to transfer an electron from a low-energy state to an excited state – is the central property of semiconductors. In a similar way, the spectral gap plays an important role for many other materials. When this energy becomes very small, i.e. the spectral gap closes, it becomes possible for the material to transition to a completely different state. An example of this is when a material becomes superconducting.

Mathematically extrapolating from a microscopic description of a material to the bulk solid is considered one of the key tools in the search for materials exhibiting superconductivity at ambient temperatures or other desirable properties. A study, published today in Nature, however, shows crucial limits to this approach. Using sophisticated mathematics, the authors proved that, even with a complete microscopic description of a quantum material, determining whether it has a spectral gap is, in fact, an undecidable question.

Astronomers used the Hubble image at the top of the page to chart the invisible matter in the massive galaxy cluster Abell 1689, located 2.2 billion light-years away. The cluster's gravity, the majority of which comes from dark matter, acts like a cosmic magnifying glass, bending and amplifying the light from distant galaxies.

The Daily Galaxy via University College London