A Quantum Physics Problem Has Been Proved To Be Unsolvable

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This week scientists have discovered that quantum mechanics is even weirder than we already knew. The spectral gap problem, which has also been investigated for its potential technological applications, has been shown to be unsolvable. This discovery puts another crucial limit on our understanding of the universe.

An international team of scientists has provided a theoretical approach to the problem of the spectral gap, the difference between the lowest energy state that electrons can occupy in a material and the first excited state. In some materials, this gap can disappear completely, and when that happens, the material can become a superconductor.

In a paper published in Nature, the team looked at a simulation of a 2D lattice structure, a distribution akin to a crystal but in two dimensions. When the simulation looked at a finite chunk of this structure, the scientists were always able to obtain a definite value for the spectral gap in a finite time. When they studied an infinite 2D lattice structure, they found that there was no way to guess when the computation would end. The problem was undecidable.

People might think that this is not an issue. Real world materials are definitely finite. But the difference in behavior in the two scenarios implies that the changes can happen at any scale. A single atom might suddenly change the behavior of the material from gapless to gapped. This implies that it’s impossible to understand the macroscopic (larger) properties of a material even if you had a complete understanding of its microscopic (smaller) description.

"Alan Turing is famous for his role in cracking the Enigma code," said co-author Dr. Toby Cubitt from UCL Computer Science, in a statement. "But amongst 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."

The undecidability of the spectral gap could have a huge impact in the search for room temperature superconductors, which are a holy grail of materials science research and could bring huge technological advancements (and hoverboards!). The scientists are currently looking to test their calculations on more realistic quantum material that could be realized in the lab. 

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