An Experiment That Could Rewrite Textbooks
Imagine two children spinning merry-go-rounds in the same direction at a playground. You'd expect the two rotations, combined, to produce an even stronger rotation in the same direction.
In May 2026, scientists in Dresden, Germany, saw the opposite happen inside a crystal.
They used an intense terahertz laser to drive atoms in a crystal along circular orbits. Then a second ultrafast laser captured, frame by frame like a strobe light, what the atoms did next.
Two same-direction rotations met and combined inside the crystal — producing a rotation in the opposite direction.
Not slowing to a stop. Not pausing. It turned around directly, spinning backward at double the frequency.
The researchers call this phenomenon "1+1=−1."
1915: Einstein's Spinning Iron Cylinder
To understand why this matters, we need to go back 110 years.
In 1915, during World War I, Einstein and the Dutch physicist Wander Johannes de Haas performed a seemingly simple experiment in Berlin. They suspended an iron cylinder by a thin filament inside a coil, magnetized it with current, then suddenly reversed the current — flipping the magnetization.
The iron cylinder began to spin by itself.
This wasn't electromagnetic induction, or torque from the current. It was conservation of angular momentum.
Iron's magnetism comes from electron spin. Each electron is like a tiny gyroscope with spin angular momentum. When you reverse the magnetization, every electron's spin flips — their angular momentum goes from +L to −L. But angular momentum can't vanish into nothing. It has to go somewhere. So the entire iron cylinder starts rotating to compensate for the angular momentum lost by the flipped spins.
Einstein and de Haas's experiment proved for the first time: magnetism and mechanical rotation are two sides of the same coin. The angular momentum of electron spin ultimately transfers to the macroscopic rotation of the entire lattice.
This effect was later named the Einstein–de Haas effect, and it remains one of the oldest experiments in condensed matter physics.
A 110-Year Black Box
But here is a question that troubled physicists for over a century.
Einstein and de Haas showed that angular momentum eventually transfers from electron spins to the lattice's macroscopic rotation. But what happens in between?
How does angular momentum move, step by step, from a single electron's spin to the whole crystal?
After a spin flips, angular momentum first passes into lattice vibrations — the collective oscillations of atoms in the crystal. But lattice vibrations have many modes, like the many strings of a guitar. What rules govern how angular momentum is passed between these modes?
For 110 years, no one had directly seen this process.
You knew angular momentum was conserved. You knew it must flow through the lattice. But you couldn't see how it flows — like knowing water runs from mountain to sea while fog hides the riverbed.
"This is a huge gap," said Sebastian Maehrlein, who led the study at Helmholtz-Zentrum Dresden-Rossendorf (HZDR) and is a professor at TU Dresden. "Tracing the continuous flow of angular momentum from the initially excited phonon to other lattice modes has been an unsolved mystery since the pioneering experiments of Einstein, de Haas, and Barnett."
Terahertz Lasers: Finally Seeing It
On May 24, 2026, Maehrlein's team published a paper in *Nature Physics* reporting the first direct observation of angular momentum transfer between lattice modes.
The material was bismuth selenide (Bi₂Se₃), a topological insulator. It was chosen not for fashion but for its exceptionally clean structure: inversion symmetry keeps its infrared-active and Raman-active phonons cleanly separated, like two independent racetracks.
The core idea:
A circularly polarized terahertz laser, tuned exactly to a 2.0 THz infrared-active phonon mode (the E_u mode) in Bi₂Se₃. Circularly polarized light means the electric field vector traces a circle in space. That circle drives atoms in the crystal into circular motion — like tracing circles along the rim of a bowl to set a marble spinning inside.
This injects angular momentum into the lattice.
Then, a second ultrashort laser pulse — with femtosecond precision (one femtosecond is one quadrillionth of a second) — captured the atomic motion like a strobe light.
What did they see?
After the 2 THz E_u phonon was excited, angular momentum began transferring to another mode: a 4.0 THz Raman-active phonon (the E_g mode). Note the frequency relation — 4.0 is exactly twice 2.0. Two E_u phonons combine into one E_g phonon. Energy conservation: 2+2=4. Frequency doubling makes perfect sense.
But what about angular momentum?
Both E_u phonons are right-handed, each carrying +1 unit of angular momentum. The resulting E_g phonon should be right-handed too, +2, right?
No. It is left-handed.
"1+1=−1": Arithmetic Written by Symmetry
This is the most counterintuitive part of the whole experiment. Two phonons rotating in the same direction combine into a phonon rotating the opposite way.
Olga Minakova, the paper's first author and a PhD student at the Fritz Haber Institute, said: "I find it very elegant how the laws of physics are directly dictated by nature's symmetry."
The secret lies in the crystal's symmetry.
In free space, angular momentum is continuously conserved. It can take any value — +1, +2, +3.7, −π... anything. Space is rotationally symmetric at every angle.
But a crystal is not free space. A crystal has discrete symmetries.
Bi₂Se₃ has threefold rotational symmetry — written C₃. The crystal looks identical after every 120° rotation. Not at every angle — only at 120°, 240°, and 360°.
This changes the rules of angular momentum conservation.
Under C₃ symmetry, angular momentum is no longer continuously conserved — it is conserved modulo 3. It can only take values of 0, +1, or −1 (equivalently 0, 1, 2, since 2 ≡ −1 mod 3). +3 is equivalent to 0, and +4 is equivalent to +1.
Now look at the combination: two E_u phonons, each carrying +1 angular momentum. Total = +1 + +1 = +2.
But in the C₃ world, +2 ≡ −1 (because 2 − (−1) = 3, a multiple of 3).
So the resulting E_g phonon must have angular momentum −1. Left-handed.
1 + 1 = −1.
The math isn't broken, and energy isn't violated. This is arithmetic written by the crystal's symmetry. In the continuous world, 1+1=2. In the C₃ world, 2 and −1 are the same thing.
Umklapp: The Crystal's "Bounce"
Physicists have a name for this: an Umklapp process — German for "flip-over."
Umklapp processes aren't new in condensed matter physics. Back in 1929, Peierls used linear-momentum Umklapp processes to explain crystal thermal conductivity. When two phonons' linear momenta add up beyond what the lattice can hold (a reciprocal lattice vector), the excess is "refunded" to the lattice and the momentum direction flips.
Think of it like running on a circular 400-meter track. You run 300 meters; another person also runs 300 meters. Together that's 600 meters. But on a 400-meter track, the 600-meter mark is the same point as the 200-meter mark. Your "combined position" isn't at 600 meters — it's at 200 meters, as if you "bounced" back.
Linear-momentum Umklapp processes have been experimentally verified for nearly a century. But an angular-momentum Umklapp process — a "bounce" of rotation — had never been directly observed.
"Although Umklapp processes are known in other areas of condensed matter physics," Maehrlein said, "this is the first experimental demonstration of an Umklapp process involving lattice angular momentum."
Why It Matters
You might ask: what does a phonon flipping its rotation direction in a crystal have to do with me?
Three levels of relevance.
First: fundamental physics. Angular momentum conservation is one of physics' most fundamental laws. But in crystals it holds in a "modulo n" form — an idea long proposed but never directly verified. This experiment fills a century-old gap between the Einstein–de Haas effect and modern spintronics. Maehrlein said: "We have found something fundamentally new that is expected to make it into textbooks."
Second: magnetism and spintronics. Angular momentum transfer through the lattice is a key step in ultrafast demagnetization. Understanding it is the precondition for controlling it. Faster magnetic storage and spintronic devices could benefit. The paper establishes "axial nonlinear phononics" as a new tool for ultrafast control of material properties.
Third: symmetry as arithmetic. This is the most fascinating level.
Symmetry IS Arithmetic
We're used to one arithmetic: 1+1=2. That's arithmetic over the integers, corresponding to physics in continuous space.
But nature is full of discrete symmetries. Crystals have C₃, C₄, C₆ symmetries. Molecules have various point-group symmetries. In these discrete worlds, arithmetic works differently.
In the C₃ world, 1+1=−1. In the C₄ world, 1+1=2, but 2+2=0. In the C₆ world, 3+3=0, and 2+2+2=0.
This isn't a math game. These "mod n" rules determine how phonons scatter, how light propagates, how magnetism relaxes, how heat conducts. They are the physical laws of the crystalline world.
Symmetry isn't just a geometric property that "makes things look nice." Symmetry defines the rules by which information flows in a system. Under continuous symmetry, information (angular momentum, momentum) flows continuously. Under discrete symmetry, information flows "modulo n" — it can bounce, flip, or seem to disappear (actually converted into the rigid motion of the lattice as a whole).
Which brings up a thought.
"Umklapp" in AI
In machine learning, we increasingly exploit symmetry too.
Equivariant neural networks are a class of networks that respect physical symmetries. E(3)-equivariant networks, for example, guarantee outputs transform along with rotations and translations of inputs. They've proven powerful in molecular modeling and protein structure prediction.
But equivariant networks also face discrete-symmetry situations. Crystal graph neural networks (crystal GNNs), for instance, must handle a crystal's space-group symmetries. In such processing, the "angular momentum" of features — if expanded in spherical harmonics — follows similar "mod n" rules.
Taking a tensor product of two l=1 features decomposes into l=0, l=1, l=2 under continuous SO(3). But under C₃ symmetry, the decomposition is completely different — certain channels are forbidden, and some channels flip sign.
That's the AI counterpart of "1+1=−1": under symmetry constraints, the rules for combining information aren't universal addition, but the "modular arithmetic" defined by the symmetry group.
Going deeper: attention mechanisms in large language models are essentially information combination in a high-dimensional space. Transformers have no explicit symmetry constraints, so their "arithmetic" is continuous and free. But if we impose symmetry constraints on a model — say, when modeling physical systems — the combination rules change. Two "right-handed" features under C₃ constraint might combine into a "left-handed" one.
That's not a bug; it's a feature. Symmetry constraints make models respect physical laws, but they also introduce "bounces" and "flips" in information flow. Understanding these bounces is essential to designing AI that truly respects physical symmetry.
Epilogue: The Elegance of Symmetry
I keep returning to what Minakova said: "I find it very elegant how the laws of physics are directly dictated by nature's symmetry."
It's worth savoring.
We usually think of physical laws as "discovered" — we run experiments, observe phenomena, summarize regularities. But this experiment demonstrates a different logic: symmetry precedes phenomena. C₃ symmetry wasn't discovered by experiment; it's dictated by the crystal structure. And once you know the symmetry, you can predict 1+1=−1 — no experiment needed, the math tells you.
The experiment merely confirmed the prediction.
Physics has shown this logic before. Dirac predicted the positron as a mathematical requirement of (Lorentz) symmetry. Yang and Mills' gauge field theory was a mathematical consequence of (gauge) symmetry. The entire Standard Model was nearly "written" by symmetry.
But "1+1=−1" has a particular simplicity. No abstract high-energy mathematics — just two rotations combining into a reverse rotation. You can draw it on a blackboard, gesture it with your hands. It compresses the profound principle that "symmetry dictates physics" into an arithmetic statement a middle schooler can grasp.
One hundred and ten years ago, Einstein and de Haas used a suspended iron cylinder to show that magnetism and rotation are the same thing. One hundred and ten years later, Maehrlein and Minakova used a terahertz laser to show that rotation in crystals follows an arithmetic we had never directly witnessed.
Two experiments, spanning a century, telling the same story: symmetry is not decoration — it is the rule itself.
Next time you see a merry-go-round, consider: if it were built on a threefold-symmetric crystal, two children spinning the same direction might launch a third one flying backward.
That's not magic. That's the arithmetic of C₃.
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Paper: Olga Minakova et al., "Observation of angular momentum transfer among crystal lattice modes," *Nature Physics*, 2026. DOI: 10.1038/s41567-026-03274-8
Institutions: Helmholtz-Zentrum Dresden-Rossendorf (HZDR), Fritz Haber Institute of the Max Planck Society, TU Dresden, Forschungszentrum Jülich, Eindhoven University of Technology