An Experiment That Could Rewrite Textbooks
Imagine two children spinning carousels in the same direction. You'd expect the combined result to be an even stronger rotation in that same direction.
In May 2026, scientists in Dresden, Germany saw the opposite in a crystal.
They used an intense terahertz laser to drive atoms in a crystal along circular orbits. Then a second ultrafast laser, acting like a stroboscopic flash, filmed frame by frame what the atoms did next.
Two same-direction rotations met and combined inside the crystal — and produced a counter-rotating one.
Not slowing to zero, not stopping. It reversed outright, rotating in the opposite direction at twice the frequency.
The researchers call this phenomenon "1+1=−1".
1915: Einstein's Spinning Iron Cylinder
To understand why this matters, go back 110 years.
In 1915, amid World War I, Einstein and Dutch physicist Wander Johannes de Haas performed a deceptively simple experiment in Berlin. They hung an iron cylinder by a thin filament inside a coil, magnetized it with current, and suddenly reversed the current — flipping the magnetization.
The iron cylinder started spinning on its own.
This was not electromagnetic induction, nor torque from the current. It was conservation of angular momentum.
Iron's magnetism comes from electron spin. Each electron is like a tiny top with spin angular momentum. When you flip the magnetization, all electron spins flip — their angular momentum changes from +L to −L. But angular momentum cannot vanish into nothing. It must go somewhere. So the entire cylinder begins to rotate, compensating for the angular momentum lost by the flipping spins.
The Einstein–de Haas experiment first proved that magnetism and mechanical rotation are two sides of the same coin. Electron spin angular momentum is ultimately transferred to the macroscopic rotation of the whole lattice.
110 Years in a Black Box
But one question troubled physicists for a century: what happens in between? How does angular momentum travel, step by step, from individual electron spins to the entire crystal?
After a spin flip, angular momentum first passes into lattice vibrations — the collective oscillations of atoms. But there are many vibrational modes, like strings on a guitar. What rules govern how angular momentum flows among them?
For 110 years, no one had seen this process directly.
"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. "Since the pioneering experiments of Einstein, de Haas, and Barnett, tracking the continuous flow of angular momentum from initially excited phonons to other lattice modes has remained an open mystery."
Terahertz Lasers: Finally Seeing It
On May 24, 2026, Maehrlein's team published in *Nature Physics* the first direct observation of angular momentum transfer between lattice modes.
The material was bismuth selenide (Bi₂Se₃), a topological insulator — chosen not for fashion, but for its clean structure: inversion symmetry keeps its infrared-active and Raman-active phonons neatly separated, like two independent race tracks.
The core idea:
A circularly polarized terahertz laser, tuned to a 2.0 THz infrared-active phonon mode (the E_u mode), drives atoms along circular orbits — like circling a finger along a bowl's wall to make marbles rotate. This injects angular momentum into the lattice.
Then a second ultrashort laser pulse — femtosecond precision — stroboscopically records the atomic trajectories.
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 frequencies — 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 sense.
But angular momentum?
Both E_u phonons rotate clockwise, each carrying +1 unit of angular momentum. The combined E_g phonon should be clockwise, +2 — right?
No. It is counterclockwise.
"1+1=−1": Arithmetic Written by Symmetry
This is the most counterintuitive part: two co-rotating phonons combine into a counter-rotating one.
First author Olga Minakova, a doctoral researcher at the Fritz Haber Institute, said: "I find it very elegant how the laws of physics are directly dictated by nature's symmetries."
The secret lies in crystal symmetry.
In free space, angular momentum is continuously conserved — it can take any value. But a crystal has discrete symmetries.
Bi₂Se₃ has threefold rotational symmetry, denoted C₃: the crystal looks identical after every 120° rotation.
This changes the rules. Under C₃ symmetry, angular momentum is conserved only modulo 3: it effectively takes values 0, +1, −1 (since +3 ≡ 0, +4 ≡ +1).
Now the combination: two E_u phonons, each +1. Total = +2. But in the C₃ world, +2 ≡ −1 (since 2 − (−1) = 3).
So the resulting E_g phonon must carry −1. Counterclockwise.
1 + 1 = −1.
This isn't broken math or violated energy conservation. It is arithmetic written by crystal 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 call this an Umklapp process — German for "flip-over".
Umklapp processes aren't new in condensed matter physics. In 1929, Peierls used linear-momentum Umklapp processes to explain thermal conductivity: when two phonons' momenta sum exceeds what the lattice can hold (a reciprocal lattice vector), the excess is "returned" to the lattice and the momentum direction flips.
Like running on a 400 m circular track: 300 m plus 300 m is 600 m — but on a 400 m track, the 600 m mark is the 200 m mark. Your "combined position" looks like a bounce backwards.
Linear-momentum Umklapp has been experimentally verified for nearly a century. But angular-momentum Umklapp — a "bounce" of rotation — had never been directly observed.
"Although Umklapp processes are known in other areas of condensed matter physics," said Maehrlein, "this is the first experimental demonstration of an Umklapp process involving lattice angular momentum."
Why It Matters
Fundamental physics. Angular momentum conservation is among the most fundamental laws of physics. In crystals it is conserved "modulo n" — long proposed, but never directly verified. This experiment fills a century-old gap between the Einstein–de Haas effect and modern spintronics. Maehrlein: "We have found something fundamentally new that is expected to make it into textbooks."
Magnetism and spintronics. Angular momentum transport in the lattice is a key step in ultrafast demagnetization. Understanding it is a prerequisite for controlling it — relevant for faster magnetic storage and spintronic devices. The paper establishes "axial nonlinear phononics" as a new means of ultrafast material control.
Symmetry as arithmetic. In a C₃ world, 1+1=−1. In a C₄ world, 1+1=2 but 2+2=0. In a C₆ world, 3+3=0. These "mod n" arithmetic rules govern how phonons scatter, how light propagates, how magnetism relaxes, how heat conducts. Symmetry is not decoration — it defines the rules by which information flows in a system.
"Umklapp" in AI
Machine learning increasingly exploits symmetry too. Equivariant neural networks respect physical symmetries and excel in molecular modeling and protein structure prediction. Crystal graph neural networks must handle space-group symmetries, where feature "angular momenta" obey analogous mod-n rules. A tensor product of two l=1 features decomposes differently under C₃ than under continuous SO(3) — some channels are forbidden, some flip sign.
This is "1+1=−1" in AI: under symmetry constraints, the composition rule for information is not universal addition but a modular arithmetic defined by the symmetry group. Symmetry constraints make models respect physical laws — but they also introduce "bounces" and "flips" in information flow. Understanding those bounces is essential for designing truly symmetry-respecting AI.
Epilogue: The Elegance of Symmetry
Minakova's words are worth savoring: "I find it very elegant how the laws of physics are directly dictated by nature's symmetries."
We usually think of physical laws as *discovered*. But this experiment shows another logic: symmetry precedes phenomenon. C₃ symmetry isn't found by experiment — it's dictated by crystal structure. Once you know the symmetry, you can predict 1+1=−1 without any experiment. The experiment merely confirmed the prediction.
Physics has shown this logic before: Dirac predicting the positron from Lorentz symmetry; gauge field theory from gauge symmetry; the whole Standard Model written out of symmetry. But "1+1=−1" has a special plainness — two rotations combining into a reverse rotation, something you can draw on a blackboard.
110 years ago, Einstein and de Haas showed that magnetism and rotation are the same thing. Today, Maehrlein and Minakova show that rotation in crystals obeys an arithmetic we had never directly seen.
Two experiments, a century apart, telling one story: symmetry is not decoration — it is the rule itself.
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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