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Mathematics may be the alphabet with which God has written the universe, but that doesn’t mean it’s perfect.
“Not every statement can be proved or disproved, even by maths,” says David Perez-Garcia, a mathematician at the Complutense University of Madrid and coordinator of the GAPS project, which was funded by the European Research Council. “In fact, in any system of mathematics where you can do basic arithmetic, there will always be statements that are impossible to prove or disprove."
To explain, Perez-Garcia points to the spectral gap.
The spectral gap is what physicists call the amount of energy a quantum system needs to transition from a state of low energy to an excited one. According to Perez-Garcia, this gap could help address one of the biggest challenges in quantum physics – the classification of all the possible phases of matter.
“We are familiar with the three classical phases of matter, namely solids, liquids and gases,” explains Perez-Garcia. “However, at very low temperatures, where quantum mechanics is the law that governs the physics of a system, there are many more, and much more exotic, phases of matter.”
Examples of these quantum phases of matter include superconductors, superfluids, topological spin liquids, and fractons, to name only a few. “Despite around a century of quantum mechanics, we are still far from understanding – or even discovering – the many counterintuitive phases of matter that are possible within the quantum regime,” adds Perez-Garcia.
A mathematical mission impossible
Because the spectral gap is what protects the quantum properties of a system, quantum phase transitions can only occur when that gap closes. “Determining whether or not a quantum material has a spectral gap is the key to determining the boundaries and transitions between different phases of matter,” remarks Perez-Garcia.
The EU-funded GAPS project proved that the spectral gap cannot be determined even with the most sophisticated mathematics. “Some mathematical questions are undecidable, that is, they are neither true nor false – they are simply beyond the reach of mathematics,” he says.
As Perez-Garcia explains, the GAPS project showed that, even if one knows all the microscopic properties of a material, their macroscopic properties (those that we observe with our eyes and that determine the phase of the system) cannot, in some cases, be predicted.
“This is not an issue of a lack of precise instruments or of not having powerful enough computers,” he notes. “It’s an issue of there being physical properties that we simply cannot calculate.”
In other words, the problem of determining the phase of matter of any possible given quantum material is unsolvable.
For every negative there is a positive
But all is not lost. According to Perez-Garcia, this negative result has a positive counterpart – it proved the existence of a totally new phase of matter.
“The properties of these new predicted materials depend strongly on the size of the sample, meaning that these properties change dramatically at a given critical size – a size that can be tuned to be any desired value,” explains Perez-Garcia. “We are currently working on a proposal to synthesise this type of material in a lab environment, which we’re very excited about.”
Moreover, even if determining the phase for each quantum system is impossible, it is still conceivable that one could describe all possible phases of quantum matter. According to Perez-Garcia, tensor networks seem like the right tool for doing that. “Being mathematical representations of a quantum state, tensor networks are flexible enough to represent all the relevant quantum states of nature,” he adds.
GAPS project researchers developed a mathematical theory of tensor networks to create one-dimensional renderings of every two-dimensional quantum phase of matter, making them much easier to understand – and use. “A very practical, and unexpected, outcome of this research is the use of tensor network methods to close the privacy gap found in many machine learning tasks,” notes Perez-Garcia.
However, in terms of a complete classification of all the possible quantum phases of matter – that continues to remain elusive. “Establishing a type of periodic table of quantum phases of matter could lead to an array of new materials and technologies,” he concludes. “And so our work continues.”