Ů��С��Ƶ

Imagine you are out on a walk. Outside the house in the fresh air you may have left the walls behind, but even so there are boundaries that limit where you can wander. In a city, you are constrained by streets and sidewalks. In the countryside, fences bar your way, and if you come upon a hill, you will definitely feel that incline in your legs.

Now, consider the electron, the fundamental particle that carries a charge and lives inside all materials. One of its favourite things to do is to run alongside other electrons, forming electric currents. But just as when you set off on a walk, electrons can’t just do anything they like. In fact, for years, physicists have suspected that electrons must navigate a hidden quantum landscape that constrains their motion.

Could we ever see this landscape? Its shape is set by the laws of quantum physics, while its texture is described by highly complex and abstract mathematics – hopes were never high. But recently researchers published the first full map of this previously unseen realm. “We can now see these hidden textures all of a sudden light up in the experimental data,” says at the Massachusetts Institute of Technology, one of the researchers who created the map.

All this offers a new way to understand and design materials, perhaps leading to, for instance, super-efficient wires that conduct electricity with no resistance. A new view of what actually happens inside materials is bound to lead to new ways to improve them.

Our world is one of “stuff”, whether it be wood for chairs, plastic for toothbrushes or the complex materials that make up magnetic and electronic devices that power modern life. But to understand how stuff behaves, we need to look under its surface. Here there lies a dense tangle of jostling atoms with electrons between them, and how those electrons behave often determines a material’s properties.

Subscriber-only newsletter

Sign up to Lost in Space-Time

Untangle mind-bending physics, maths and the weirdness of reality with our monthly, special-guest-written newsletter.

Sign up to newsletter

Bloch’s energy bands

A notable effort to paint a picture of this internal hustle and bustle came in 1929 from Swiss-American physicist Felix Bloch. He showed that the repeating pattern of atoms within a solid forces electrons to move between them in a periodic way too, similar to how a boat bobs up and down, buoyed by the steady rhythm of waves. He applied this insight to the electrons’ wave functions, the equations that encode all the particles’ quantum properties. This led him to prove that the wave functions repeat in space as well, which gave rise to a whole new picture of the electrons’ world. Based on its “Bloch wave function”, an electron can’t have just any energy as it whizzes through a material. Those energies are constrained to a range or “band”. Thanks to Bloch’s work, we now know that a solid’s electrical character – be it a conductor, semiconductor or insulator – depends on how many electrons are corralled into the same band. For example, if the highest energy band is only partially full of electrons, there is still room for them to move around and carry current, like they do in a conductor.

Bloch’s theory made modern electronics possible. But its framework didn’t always align with reality, a problem that has only grown in the past few decades. In the 1980s and 1990s, physicists began studying materials, such as bismuth telluride, that acted as insulators, but displayed unexpected currents on their surface. And then, in 2018, there was graphene, one-atom thick sheets of carbon, which conducted electricity with virtually no resistance when stacked and twisted – phenomena Bloch’s theory couldn’t explain.

But there were also clues as to what may be hiding within these materials. In the 1980s, British physicist Michael Berry realised that electrons could undergo subtle shifts in their wave function as they moved through quantum systems, especially in loops – one of the first clear hints that they were navigating a richer, more complex quantum landscape than Bloch had imagined.

Mapping an electron’s quantum geometry

Other elements of that topography had already been established. Even before Berry’s work, French physicists Jean-Pierre Provost and Gérard Vallée for mapping it out by offering a recipe for measuring the distance between electrons’ quantum states. Their work, alongside Berry’s, is now summarised by one key mathematical object, which is known as the ‘‘quantum geometric tensor” (QGT). It contains all the keys for charting the secret quantum geometry that might explain the behaviours that Bloch’s model couldn’t. An intrepid explorer of the microscopic world could use it to map the esoteric quantum landscape where electrons reside.

Imagine being dropped into an unfamiliar environment, like a dense rainforest or a desert undulating with sand. Two tools could help you find your bearings. The first is a ruler that determines the shortest path to some destination. The second is a special compass that tells you how moving in a loop reorientates you. It would tell you if you unknowingly turned while walking in a circle back to your starting point and ended up pointing in a different direction. In the quantum world, the QGT provides both (see diagram below).

Mathematically, the QGT is a matrix, or a table of numbers, where each number represents some facet of quantum geometry. You can look at one number to get a reference for how to measure distances, then go to a different part of the table and find a number that describes what happens if you move in a loop.

The entire matrix can be theoretically calculated from electrons’ wave functions, but in practice, the mathematics is often too complicated. A solid contains an enormous number of electrons and their wave functions have many more mathematical dimensions than the three spatial dimensions of any material. Because of this, experimentally measuring the QGT instead is the only way to understand it. Unfortunately, problems abound here as well.

Experiments that directly involve wave functions are devilishly tricky, since a wave function only captures a particle’s probable states, rather than its concrete properties. Measuring the wave function causes these states to collapse, so measurements must be indirect and gentle. For years, this rendered the QGT little more than theory. “The presence of the QGT has been simply an assumption or belief since nobody actually had observed its presence,” says at Seoul National University in South Korea, who collaborated with Comin to create the first quantum map of a solid.

Single atoms captured morphing into quantum waves in startling image

In the 1920s, Erwin Schrödinger wrote an equation that predicts how particles-turned-waves should behave. Now, researchers are perfectly recreating those predictions in the lab

Before Comin and Yang’s work, researchers made progress on filling in some bits of the QGT table, but a full map of quantum geometry within a solid remained elusive. However, in the past decade, physicists have made great strides in engineering and controlling quantum objects, enough to snatch the first glimpses of the entire QGT. The first measurement came in 2020, when at the Kastler Brossel Laboratory in France and his colleagues measured the QGT of , or qubits, embedded in diamond. These were, Goldman says, “probably the most controllable qubits in the world”, and he and his team extracted their QGT by repeatedly nudging them with precisely tuned circularly polarised light and measuring how their wave functions responded.

That same year, at the University of Clermont Auvergne in France and his colleagues did something similar with , or photons, trapped inside a semiconductor cavity. Once again, tight control over the photons made the difference. “You have, really, very direct access to the [photon’s] wave function,” says Malpuech.

However, materials that might prove useful for novel electronic devices are nothing like qubits or carefully controlled photons. They are much more complex. Even Goldman says that in his team’s experiment, adding just one more qubit made the QGT measurement a lot more challenging – and materials, which contain myriad atoms, are immensely more complicated. “There is, a priori, no general recipe for extracting the quantum geometry of those [quantum] states,” he says.

This is the challenge that Comin and his team faced when they started thinking about measuring the QGT for electrons inside a material composed of cobalt and tin nearly five years ago. They turned to angle-resolved photoemission spectroscopy (ARPES), a staple technique in material science labs at many major universities. Here, researchers bombard a material with light, which knocks out electrons that land on a detector. From the detector’s readings, researchers can determine what properties the electrons had while inside the material and map the material’s bands.

Comin’s team tweaked ARPES so the light wouldn’t only dislodge electrons, but also spin them, allowing them to extract the QGT entries that explain what happens to an electron when it moves in loops. Yang’s team then analysed the same data to excavate the parts of the QGT that would provide a ruler for quantum distances. The shape of the quantum world that had been obscured for so long came into focus. “We did it together,” says Comin. “I was personally extremely excited.” In November 2024, they had their topographical map, the of a solid material’s internal quantum landscape.

More successes followed. In June this year, Yang and a different team of collaborators repeated the experiment with black phosphorus, this time with even greater precision.

Hunting for a better superconductor

Just as Bloch’s picture of where electrons live started the path towards the invention of transistors, the map revealed by the QGT may herald a breakthrough in creating other new materials. One exciting possibility is materials that conduct electricity with no resistance. These “superconductors” could replace traditional wires and help create electronics that are thousands of times more energy efficient, something especially important with the expansion of digital technology and AI. “In superconductors, we have huge scientific and technological potential, and it has been, in my opinion, a little bit underappreciated how big the potential is,” says  at Aalto University in Finland.

In 2022, Törmä and her colleagues were the to explain the puzzling observation that stacked, twisted layers of graphene could superconduct. According to Bloch’s theory, these materials have “flat” bands, which means that their electrons have the same energy no matter how fast they move or what direction they are moving in. An electron in a flat band is like one that exists in a perfectly flat landscape – there are no hills it could roll down and it has no incentive to ever really change its motion. Because of this, researchers expect electrons in flat bands to do next to nothing. Certainly, they don’t expect them to form perfectly efficient supercurrents.

Törmä and her colleagues explained how they form supercurrents anyway by considering the material’s quantum geometry. They found that when the stacked graphene layers are twisted just right, electrons’ wave functions overlap enough to reshape their world. A bridge may suddenly appear in their quantum landscape, connecting electrons that were previously separated by a large distance, allowing once estranged charges to couple up and superconduct. This quantum geometry is richer than Bloch’s theory alone can capture, and it potentially unlocked the secrets to the material’s behaviour.

“This was very influential to the community. It gave us a hint that there was a solution,” says at Harvard University. Since then, the idea that quantum geometry could be a key ingredient in future superconductors has been a major feature of Törmä’s work.

She thinks that experiments like Comin’s and Yang’s could strengthen the case that values in the QGT and superconductivity are deeply connected. “In experiments, you’d like to measure both the physical response and the quantum geometric tensor to really establish this connection,” she says. She currently leads the , which aims to achieve a superconductor breakthrough by 2033.

But they have their work cut out for them. To form lossless currents, electrons need to form pairs, yet they naturally repel each other. More than a century after the first superconductor was discovered, the only materials of this type we know of still require either ultra-low temperatures or extremely high pressures to overcome this difficulty. If electrons could be nudged into pairing by the intrinsic geometry of their quantum world, that could lead to more practical superconductors.

To do that, what researchers need, says Törmä, is a checklist of key “ingredients” for a room-temperature and ambient-pressure superconductor – and its accurate QGT may be an important entry on that list. “Most superconductors that exist now have been found by experimentalists’ intuition,” says Törmä. “If quantum geometry affects superconductivity positively, then we can use it as a design tool.”

Banerjee is all for this idea. He and his colleagues are specifically experimenting with stacked graphene, the material that Törmä’s team tackled in 2022. Earlier this year, Banerjee’s team with microwaves and use its response to learn more about the behaviour of electrons in it when it superconducts. They quantified how much a supercurrent resists change, like a river of electrons being steered or sped up, a number that Banerjee expects to match one of the entries in the QGT table.

If he is right, then his team would have strong evidence for Törmä’s theory that quantum geometry is behind its strange superconductivity. Ů��С��Ƶs could then design the superconductor of their dreams by twisting and stacking graphene sheets, or some similarly thin material, in a way that maximises quantum properties linked to the QGT, such as stronger superconductivity. But for now, no one has managed to measure the full QGT in stacked graphene, and the samples are too small and thin to submit to techniques that work for chunky solids like the ones that Comin and Yang studied. Comin is also on his own quest to find a superconductor, but he is searching in bulky three-dimensional materials that are conducive to his ARPES method.

Remarkably, the list of electronic effects that stem from quantum geometry doesn’t stop with superconductivity. A variety of exotic effects – like currents spontaneously forming in materials – have recently been linked with some parts of the QGT. One example is the anomalous Hall effect, where electrons veer to the side as if nudged by an invisible magnetic force. These effects could emerge from the underlying geometry of quantum states, rather than classical forces, and may be useful in designing devices where directional control of current is key. Transistors – the building block at the heart of all existing electronics – perform exactly this current control function. Instead of needing multiple components to manipulate the flow of charge, materials shaped by quantum geometry may do this by default.

The same geometry could also govern how some materials respond to light, causing them to fill up with currents when illuminated. This could open the door for new kinds of solar cells or light sensors.

The quantum world: A concise guide to the particles that make reality

Get to grips with the deepest layer of reality we know of with our inventory of the subatomic realm, from known particles like quarks and the Higgs boson to hypotheticals including the fifth force and strings in 11 dimensions

at Boston University in Massachusetts says that studying the QGT could even benefit a broader swathe of science that deals with materials. He first came across it while studying how systems change from one phase to another, the more complex quantum analogues of how liquid water changes into solid ice. In these systems, phase changes mark sudden shifts in big collectives of particles, like when a magnet flips its alignment. He found that the distance between quantum states, measured by the ruler in the QGT, can stretch or even diverge near this critical transition point. “I started seeing [quantum] geometry everywhere. It just appears in all aspects of physics,” he says.

These days, Polkovnikov is interested in whether the quantum geometry of differs from those that never become chaotic. And he is convinced that quantum geometry could become an important concept in chemistry, where it helps explain what some electrons are doing during fast and abrupt chemical reactions.

We are only just beginning to explore the hidden topography of the quantum world inside materials – the ink is still drying on those first maps. Even so, the interest is really growing, says Törmä. “In the beginning, I was kind of following every paper,” she says. “Now, I have given up. There’s so much.”