It’s rare for anyone to claim that something physicists have believed for a century is wrong. But this is exactly what two physicists argue in a paper recently published in the prestigious science journal Nature.
The ‘something’ is about how a certain type of particle – including the fundamental building blocks of the world – behaves. Everyone believed these particles fell into just two categories: bosons and fermions.
Bosons are considered gregarious, as they group together and often behaves as a single entity, while fermions are more ‘antisocial’ because they can’t share the same quantum state as each other.
“Now we think there may be an infinity of other possibilities,” says Prof Kaden Hazzard, one of the authors of the new research and associate professor of physics and astronomy at Rice University, in the US.
Hazzard’s PhD student made the discovery unexpectedly. “It was in 2021, during the Covid pandemic,” says Dr Zhiyuan Wang, now of the Max Planck Institute for Quantum Optics, in Germany. “I was super bored at home, so I worked on a weird mathematical problem to entertain myself.”
Wang stumbled on a solution to this problem – known as the Yang-Baxter equation – and wondered whether it had implications for the real world. To his amazement, it did. It appeared to suggest the existence of an entire class of particles – paraparticles – that behave neither like fermions nor bosons.
It’s a discovery that could challenge our understanding of how the world works.
New reality
Paraparticles are a consequence of two quantum properties that have no analogue in the everyday world: spin and indistinguishability.
Spin, despite its name, has nothing to do with the Earth spinning on its axis. In fact, there’s no easy comparison to make, since there is nothing like it in the world we’re familiar with.
Earth’s spin requires a body (the planet) to have an ‘extent’ (the measurable space it occupies) so that the different parts of it are different distances from the axis of rotation. But quantum particles, like electrons, have no extent — their spin is ‘intrinsic’.
Quantum spin only exists in the sub-microscopic world of atoms and their constituents.
If you’re struggling to get your head around this, it can help to think of energy. While Newtonian physics says that a body’s energy only rises above zero if it’s moving, Einstein argued that a body has intrinsic, or ‘rest’, energy even when it’s not moving.
(This intrinsic energy is what gets liberated in a nuclear bomb, represented by the famous E=mc2 equation you may remember being scratched onto the chalkboard in Oppenheimer). Just as rest energy belongs to a body, so does spin.
For many quantities in the quantum world, there is the smallest, indivisible chunk known as the ‘quantum’. For spin, this is ½ times a quantity known as ‘h-bar’: the spin of a particle can be 0, ½ h-bar, 1 h-bar, 3/2 h-bar, and so on (though physicists drop the ‘h-bar’ part and just talk of spin 0, spin ½, spin 1, spin 3/2, and so on).
Particles with integer spin are bosons and ones with half-integer spin are fermions.
In addition to spin, the second quantum property required for us to understand paraparticles is indistinguishability. Like spin, this is unique in the quantum world. After all, no two objects in the everyday world are truly identical.
Two chairs may appear the same at first glance but one may have a tiny scratch on it that another doesn’t or may differ in any of an infinity of ways. But two electrons are truly identical. If they are swapped, you can never tell.
The reason spin combined with indistinguishability has huge significance for the quantum (and everyday) world is because of the ‘wave function’. This is a mathematical entity that encodes everything that is known about a quantum particle.
For example, the square of the height, or ‘amplitude’, of the wave function at any point in space tells you the chance of finding the particle there.
Now, if we have two identical particles, the combined wave function is simply the wave function of the first particle multiplied by the wave function of the second particle.
Because we cannot tell whether two identical particles have been swapped or not, we have no choice but to describe them by their wave function + the wave function swapped. And it is here that spin makes all the difference.
In the case of bosons, adding the two wave functions leads to a bigger wave function. This means that, if there is boson in a particular quantum state, another boson has an enhanced probability of being in the same state. (Remember: bosons are gregarious.)
It is the fact photons, which are bosons, like to be with their mates that enables them to steam along together in vast numbers to make lasers.
In marked contrast, when two identical fermions are swapped, the combined wave function is multiplied by -1. It would take complex maths to explain why, but this means the wave function + their wave function swapped equals 0.
In other words, there is no probability of a fermion (like an electron) being in the same state as another fermion. “It is the fact that electrons cannot get on top of each other that makes tables and everything else solid,” said the Nobel Prize-winning physicist Prof Richard Feynman in The Feynman Lectures on Physics in 1989.
In fact, this antisocial nature of electrons, known as the ‘Pauli exclusion principle’ after Dr Wolfgang Pauli, stops electrons in atoms piling into the same orbit. It’s also responsible for there being 92 different types of atom and not just a single kind.
In short, it creates chemistry and the bewildering variety of the everyday world. You can thank it for being alive!
A simple way to appreciate the profoundly different behaviours of bosons and fermions is to think of two dice.
Imagine the ways, for instance, that you might obtain a total of 10. There are three ways: 4–6, 5–5, 6–4. Now think of the pairings as behaving like identical bosons. 4–6 and 6–4 are indistinguishable, so there are only 2 ways of obtaining 10.
Finally, think of the dice as fermions. Not only are 4–6 and 6–4 indistinguishable but 5–5 is impossible (because fermions can't occupy the same state). So there is only one way 10 can be obtained.
So how is all of this relevant to paraparticles? Well, the same rules don’t apply. “They are fundamentally different from fermions and bosons in that the wave function transforms in a more complicated way when two paraparticles are swapped,” says Wang.
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The missing piece
Wang and Hazzard aren’t the first to consider that fermions and bosons might not be the only possibilities. “Pauli considered it 100 years ago,” says Wang.
The first concrete theory of paraparticles was proposed by physicist Herbert Green in 1953. It appeared to prove that particles matching neither fermions nor bosons might exist. “But, by the 1970s, theorists had shown that [these] paraparticles were merely fermions and bosons in disguise,” says Hazzard.
Physicists also proved theorems that seemed to rule out paraparticles. “The case seemed closed,” says Hazzard. “Until now.”
But the paraparticles envisaged by Wang and Hazzard evade the theorems.
They have internal ‘states’ that can’t be observed directly. Let’s call them ‘yellow’ and ‘purple’ for the sake of illustration. Swapping two identical particles, one yellow and the other purple, would result in the yellow one becoming purple and the purple becoming yellow.
It’s the fact that it’s now possible to know if two ‘identical’ have swapped that enables paraparticles to behave differently from bosons and fermions.
“Many smart people have worked on this area without seeing what Wang and Hazzard have seen,” says Prof Jiannis Pachos, theoretical physicist at the University of Leeds. “It’s pretty amazing.”
Others such as Prof Francesco Toppan at the Brazilian Centre for Research in Physics, in Rio de Janeiro, say the work isn’t so new. “I gave the first formal proof that paraparticles are theoretically detectable in 2021,” he says.
Excitingly, Hazzard and Wang believe their paraparticles could be implemented in the laboratory – and beyond. It’s well known that a large collection of particles can, in some circumstances, behave as one – like a flock of starlings in a murmuration.
In a solid, for example, many atoms can vibrate in synchrony and so act like a single particle known as a phonon. Such ‘excitations’ of matter are called quasiparticles.
Wang and Hazzard envisage several artificial systems in which quasiparticles can be used like paraparticles. For instance, they imagine a one-dimensional string of ultra-cold atoms in which the outer electron is a relatively long way from the central nucleus. These ‘Rydberg atoms’ interact with their neighbours like a long line of tiny magnets.
One magnet budges the next magnet, and so on, and the overall effect is as if a quasiparticle has moved along the line. Such a quasiparticle, says Hazzard, would behave like a paraparticle.
Using these and other systems, Hazzard says it would be possible to implement paraparticles that behave like slightly gregarious bosons or slightly sociable fermions (with two, or three, or more existing in the same state). “There are an infinite number of possibilities,” says Hazzard.
These – currently hypothetical – systems are in one and two dimensions. But they say, in principle, there’s no barrier to creating them in three dimensions. “In practice, however, it’s going to be challenging,” says Hazzard.
Others agree. “Their model requires an enormous amount of fine-tuning, making it impossible to realise in any system now or in the future,” says Prof Paul Fendley of the University of Oxford. “It’s a proof of principle, and principle is unfortunately all it is.”
Optimistically, the work may lead to the creation of new materials that behave like nothing ever imagined before – by definition, it’s impossible to say what they might do.
There’s also the possibility of storing quantum information in the hidden internal states – stable storage has previously been a barrier to the development of powerful ‘quantum computers’.
“This may enable the transmission of information totally secure from eavesdropping,” says Hazzard.
Long ago, physicists realised that, in flat, two-dimensional materials, quasiparticles could arise that didn’t behave like fermions or bosons. These ‘anyons’ have been conclusively observed in the laboratory in the past couple of decades – and put forward as promising candidates for encoding information in quantum computers.
Quantum computers work by exploiting the ability of particles like atoms to be in many states at once to do many calculations at once. But such atoms are notoriously unstable and prone to losing their quantum superpower if not isolated from the disturbance of the surrounding environment.
What makes anyons attractive as candidates for storing quantum information is their stability. They are utterly immune to any outside interference. But what makes them unattractive is that they exist only in two dimensions, which clearly isn’t practical. But Wang and Hazzard’s work could change all of that.
“It had been thought anyons couldn’t exist in three dimensions,” says Pachos. “This work suggests that they might, and they might be more stable.”
Implementing paraparticles as quasiparticles in the laboratory is one thing. But might there exist previously undiscovered fundamental particles that behave like paraparticles? Hazzard thinks this is an exciting question.
“Although we think of quasiparticles as not real, they are excitations of matter just as fundamental particles are excitations of the ‘quantum fields’ physicists imagine fill space,” says Hazzard. “So maybe there are hitherto undiscovered fundamental particles that are paraparticles.”
More speculatively, the existence of an entire class of unsuspected fundamental particles could shake up the Standard Model – the theory that explains how the basic building blocks of matter interact, but which doesn’t include paraparticles.
Currently, we know most of the Universe’s material takes the form of dark matter, outweighing the visible stars and galaxies by a factor of five – and nobody knows what it is. “This is way beyond my area of expertise,” admits Hazzard. “But we’ve been talking to particle physicists and cosmologists.”
As Shakespeare wrote in Hamlet: “There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.” Dark matter being made of paraparticles is certainly nothing anyone has ever dreamt of before.
About our experts
Prof Kaden Hazzard is an associate professor physics and astronomy at Rice University in the US. He is published in various scientific journals, including Nature, Physical Review Letters and Nature Communications.
Dr Zhiyuan Wang is a postdoc researcher at the Max Planck Institute for Quantum Optics in Germany. His work is published in Nature, Physical Review and The European Physical Journal.
Prof Jiannis Pachos is a theoretical physicist professor at the University of Leeds, in the UK. He has been published in the likes of Physical Review Letters, Physical Review and New Journal of Physics.
Prof Francesco Toppan is a professor of physics at the Brazilian Centre for Research in Physics, in Rio de Janeiro, Brazil. His work is published in International Journal of Geometric Methods in Modern Physics, Journal of Physics and Nuclear Physics.
Prof Paul Fendley is a professor of statistical mechanics and mathematical physics at the University of Oxford, in the UK. He is published in the likes of Journal of Statistical Physics, Physical Review B American Physical Society and Journal of Statistical Mechanics: Theory and Experiment IOP Science.
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