Bethe Strings: A 1931 Prediction Captured in an Ultracold Gas

Thomas Lee Abshier, ND · October 13, 2026

A Bound State Without a Bond

Ordinary matter is held together by attraction that pulls particles into a low-energy arrangement and keeps them there. A molecule is two or more atoms sharing electrons. A nucleus is made of protons and neutrons held together by the strong force. In each case, a force acts, a well forms, and the particles sit in it.

Bethe strings are a different kind of bound state. They are clusters of particles that move together along a line as a single object, held together only by short-range contact interactions between the particles themselves. They exist as exact solutions of certain one-dimensional quantum models, and their stability comes from a special mathematical property of those models called integrability. Hans Bethe found the first examples in 1931. In August 2026, a group at the University of Innsbruck led by Hanns-Christoph Nägerl reported in Nature Communications that they had produced a dense mixture of such strings, many of them large, in a gas of cesium atoms cooled to a few billionths of a degree above absolute zero.

This essay explains what the strings are, why they appear only in one dimension, how the Innsbruck team made them and detected them, and what the result does and does not establish.

Bethe’s 1931 Paper

The problem Bethe solved

In 1931 Bethe published “Zur Theorie der Metalle” in Zeitschrift für Physik. He studied a chain of magnetic atoms, each with a spin that can point up or down, with each spin coupled to its neighbors. This is the Heisenberg spin chain. Bethe found a way to write down its exact quantum states and energies, not as an approximation but as an exact solution. His method is now called the Bethe ansatz (“ansatz” is German for an educated starting guess about the form of the solution).

The key idea is that each state of the chain can be described by a set of numbers, now called rapidities, that act like the momenta of independent particles moving along the chain. When two of these particles meet, they exchange momenta and pick up a phase, but they do not scatter into new momenta. Many-particle collisions reduce to a sequence of two-particle collisions. That property is the signature of an integrable system.

Where the strings come from

When the rapidities are real numbers, they describe particles that travel freely along the chain. Bethe noticed that the equations also allow complex rapidities. A complex momentum means a wavefunction that decays with distance, and a decaying wavefunction describes particles that stay close together. In the spin chain, these solutions are bound groups of flipped spins that move as a unit. Later workers showed that bound groups of any size n exist, and that their rapidities line up in evenly spaced columns in the complex plane. Plotted that way, they look like a string of beads, which is where the name comes from.

From spin chains to gases of atoms

The same mathematics applies to a gas of bosons confined to a line and interacting only when two particles touch. Elliott Lieb and Werner Liniger solved the repulsive version of this gas exactly in 1963. In 1964, J. B. McGuire solved the attractive version and showed that its eigenstates include bound clusters of n particles, with a binding energy that grows as n(n² − 1). C. N. Yang extended the method in 1967 to particles with internal spin states. McGuire’s clusters are the Bethe strings of the Bose gas, and they are what the Innsbruck group set out to make.

In the limit of many particles, a McGuire cluster becomes the bright soliton of the nonlinear Schrödinger equation, a localized wave packet that keeps its shape as it moves. The Innsbruck authors describe Bethe strings as the quantum counterpart of bright solitons. Soliton trains had been made in attractive Bose condensates as early as 2002 at Rice University, but only at weak interaction, where the classical wave picture applies. The Innsbruck experiment works in the strongly quantum regime, where each cluster has a definite small number of atoms and the wave picture fails.

Why One Dimension Matters

Attraction usually means collapse

Bosons have no Pauli exclusion principle to keep them apart. If they attract, nothing stops many of them from piling into the same small region. In three dimensions, an attractive Bose condensate collapses once it exceeds a critical size, and the high density drives three-body collisions that form deeply bound molecules and eject atoms from the trap. Experiments at Rice in 2000 and at JILA in 2001 watched exactly this kind of collapse and explosion.

Integrability blocks the cascade

In one dimension with a contact interaction, the gas is integrable. Collisions cannot redistribute momenta, so a string cannot break into smaller pieces or absorb another string by colliding with it. Two strings that meet pass through each other and emerge intact, shifted only in position. This is why the strings can exist as stable objects, and it is also why they cannot exist in three dimensions, where the extra directions allow ordinary scattering and the integrable structure is lost.

Real experiments are never perfectly integrable. The tubes are not infinitely narrow, the atoms sit in a weak trap along the tube, and three atoms can still meet and form a molecule. The Innsbruck system is “nearly integrable,” which is enough for the strings to live for many milliseconds, long enough to measure.

The Innsbruck Experiment

Making many one-dimensional gases at once

The experiment begins with a Bose–Einstein condensate of cesium-133 atoms, between 20,000 and 60,000 of them, held in crossed laser beams and levitated against gravity by a magnetic field gradient. Two additional laser beams crossing at right angles in the horizontal plane form a two-dimensional optical lattice. Each node of that lattice is a narrow tube about 83 nanometers across, in which atoms can move only along the tube’s axis. The condensate splits into 3,000 to 5,000 populated tubes, each holding on average between 7 and 30 atoms. The transverse confinement is so tight (about 10.5 kHz) that no atom has the energy to move sideways. Each tube is, for practical purposes, a one-dimensional world.

Tuning attraction with a magnetic field

Cesium is a favored atom in Innsbruck because its interactions can be tuned by a magnetic field through a Feshbach resonance. Near such a resonance, the scattering length that describes how two atoms interact can be swept from positive (repulsive) through zero to negative (attractive) by small changes in the field. The team starts with strong repulsion, at which the atoms in each tube avoid one another so effectively that the bosons behave much like fermions (the Tonks–Girardeau regime). They then ramp the interaction steadily through zero and into attraction over tens of milliseconds.

The ramp rate matters. If the interaction were changed infinitely slowly, each tube would follow its ground state, and the ground state of an attractive Bose gas is a single giant cluster containing every atom. If the interaction were switched suddenly, the gas would be thrown into an excited state in which the atoms stay apart even though they attract (the “super Tonks–Girardeau” gas, which the same Innsbruck group first produced in 2009). A ramp at an intermediate rate leaves the gas in a far-from-equilibrium mixture of strings of many sizes. That intermediate case is what the experiment targets, following a theoretical proposal by Koch, Bastianello and Caux in 2021.

Seeing a binding energy

A camera sees atoms, not bonds. The group detected the strings by comparing two kinds of expansion.

In the first, they turned off the weak trap along the tubes and let the atoms spread along the tube axis while keeping the tight transverse confinement. A string spreads as a unit, so the measured release energy reflects only the center-of-mass motion of the strings. In the second, they also released the transverse confinement, letting the atoms expand into three dimensions. Strings cannot survive in three dimensions, so each one comes apart and its binding energy turns into extra kinetic energy of its atoms.

On the repulsive side of the ramp, the two measurements agree, as they must for a gas with no bound states. On the attractive side they separate sharply. At an interaction strength of −1 per micrometer, the one-dimensional release energy levels off near h × 35 Hz per atom, while the three-dimensional release energy climbs to about h × 110 Hz. The difference, about h × 75 Hz per atom, is the average binding energy that the strings released when they broke.

Two further measurements support the same picture. The momentum distribution after one-dimensional expansion narrows as the interaction approaches zero and then broadens again on the attractive side, as expected for atoms confined in tight clusters (a tighter cluster has a broader momentum spread, by the uncertainty principle). And Tan’s contact, a measure of how often two atoms are found at the same place, rises to nearly six times its repulsive-side value at equal interaction strength, consistent with atoms crowded together in strings.

All three measurements agree with simulations based on generalized hydrodynamics, a theory developed in 2016 that describes how integrable systems flow on large scales by tracking the population of each type of quasiparticle, including strings of every size.

Proportion Check: How Big Are the Strings?

The binding energy of an isolated n-atom string in the Lieb–Liniger model is

Eᵦ(n) = ħ² c² n(n² − 1) / (24 m),

where c is the interaction strength and m is the mass of a cesium atom. Evaluating this at c = 1 per micrometer gives the following values.

Atoms in string (n)	Total binding energy (h × Hz)	Binding per atom (h × Hz)	Binding (nanokelvin)	Approximate size (µm)
2	19	9.5	0.9	2.0
3	76	25	3.7	1.0
4	190	48	9.1	0.67
5	380	76	18	0.50
6	665	111	32	0.40
8	1,597	200	77	0.29
10	3,137	314	151	0.22

The four-atom row reproduces the paper’s own worked example (190 Hz, about 9 nK, bond length 0.67 µm), which confirms the formula and constants.

The table shows how a lower bound on string size follows from a single number. Binding energy per atom rises steeply with n. If every atom were in a pair, the average binding per atom could be no more than 9.5 Hz. If every atom were in a four-atom string, it could be no more than 48 Hz. A measured average of about 75 Hz per atom at this interaction strength therefore cannot be produced by strings of four or fewer; some atoms must sit in strings of at least five. The paper applies the same reasoning across its full data set and reports a lower bound of six on the largest string present. Its generalized-hydrodynamics model indicates that larger strings, up to and beyond ten atoms, are also populated.

Two features of the table explain the experiment’s limits. First, the strings are small compared with the tube length but large compared with the tube width (83 nm). The larger strings approach the scale at which three atoms can meet closely enough to form a deeply bound molecule and leave the trap. The paper reports that roughly 20 to 40 percent of the atoms are lost during the ramp at the strongest attraction, and that the loss is well described by three-body recombination within strings. Second, because binding grows as the cube of n, the largest strings are the most tightly bound and the most vulnerable to that loss. The experiment can show that large strings are present but cannot yet count them one size at a time.

What the Result Establishes

The experiment establishes three things.

First, Bethe strings of the attractive Lieb–Liniger gas, which until now were a theoretical object for continuous atomic gases, have been produced and identified by their binding energy, their momentum distribution and their short-range correlations.

Second, the gas contains a dense mixture of strings of many sizes at once, including strings of six or more atoms. Earlier observations of Bethe-type bound states involved one small string at a time, of two to five constituents.

Third, generalized hydrodynamics, previously tested mainly in repulsive gases, gives a quantitative description of this strongly attractive, far-from-equilibrium regime.

What it leaves open

The detection is indirect. No image shows an individual six-atom cluster; the strings are inferred from energy and correlation measurements averaged over thousands of tubes and compared with theory. The statement that strings “collide without breaking apart” is a property of integrable dynamics that the one-dimensional expansion data are consistent with (strings scatter during that expansion and still release their binding energy afterward), rather than a direct observation of a single collision. The size distribution comes from the model, not from direct counting. The authors name several next steps: following the strings through the crossover from one to three dimensions, connecting them to Efimov three-body states, and studying their role in anomalous transport, where large strings in spin chains are already known to drive a scaling behavior called Kardar–Parisi–Zhang universality.

Why It Matters

Exactly solvable models are rare in physics. Most many-body problems are approximated, and the approximations are tested against experiments that are themselves hard to control. The one-dimensional Bose gas is one of the few systems in which an exact theory, written down decades ago, can be compared directly with a laboratory system whose interaction strength, density and temperature can be set by hand. The Innsbruck experiment extends that comparison into the attractive regime, where the exact theory predicts its most unusual objects. The agreement is a demonstration that the mathematics Bethe introduced to describe a magnetic chain also describes real atoms moving in a line, nearly a century later and in a physical system Bethe could not have imagined.

Notes on the Source

This essay was prompted by a Morning Overview article (an AI-assisted news summary) describing the Innsbruck result. The account above was checked against the Nature Communications paper itself and against the earlier literature. Several points in the source article needed correction or qualification.

1. Earlier observations were not limited to solid-state magnets. The source states that Bethe strings had previously been detected only in solid-state magnetic systems. Bound strings were observed in the crystal SrCo₂V₂O₈ by terahertz spectroscopy (Wang et al., 2018) and neutron scattering (Bera et al., 2020). But two-magnon bound states, the smallest Bethe strings, were also observed in a chain of ultracold atoms in an optical lattice at the Max Planck Institute of Quantum Optics in 2013 (Fukuhara et al.), and bound states of up to five microwave photons were observed on a ring of superconducting qubits at Google in 2022 (Morvan et al.). The Innsbruck paper’s own abstract cites both quantum magnets and superconducting qubits. What is new in 2026 is a dense mixture of many strings, including large ones, in a continuous gas.
2. “Six” is a lower bound on the largest string, not a confirmed size or an upper limit. The source says in its opening that the larger clusters “held six or more particles each,” and in its closing that the largest confirmed string is “bounded at six.” The paper’s statement is that a model-independent analysis of the binding energy requires the largest strings to contain at least six atoms, and that the generalized-hydrodynamics model indicates still larger strings are present. The arXiv preprint (May 2025) phrased this as “more than six”; the published version says “at least six.”
3. The attractive solution is McGuire’s, not Lieb and Liniger’s. The source credits Lieb and Liniger (1963) with the exact solution of the one-dimensional Bose gas and says strings appear “once the interaction turns attractive.” Lieb and Liniger solved the repulsive case. The attractive case and its bound clusters were worked out by McGuire in 1964. Yang’s 1967 extension concerned particles with internal spin.
4. Bethe’s 1931 model was a spin chain. The source says Bethe derived the strings “for particles confined to a single line.” His paper treated a chain of interacting spins. The strings there are bound groups of flipped spins. Their extension to atoms moving freely along a line came three decades later.
5. The strings were inferred, not watched. The source says the group “watched them behave as predicted” and describes collisions without breakup as an observed behavior. The evidence is the difference between one- and three-dimensional release energies, the momentum distribution and Tan’s contact, all compared with theory. Collision stability is a known property of integrable systems that the data are consistent with.
6. Title. The published title is “Probing Bethe strings in an attractive one-dimensional Bose gas.” The arXiv preprint was titled “Observing Bethe strings in an attractive Bose gas far from equilibrium.”

Funding details and quotations attributed to the authors in the source article were not independently verified and are not repeated here.

References

1. Horvath, M., Bastianello, A., Dhar, S., Koch, R., Guo, Y., Caux, J.-S., Landini, M. & Nägerl, H.-C. Probing Bethe strings in an attractive one-dimensional Bose gas. Nature Communications (published 29 August 2026). doi:10.1038/s41467-026-76018-0. Preprint: arXiv:2505.10550 (May 2025).
2. Bethe, H. Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette. Zeitschrift für Physik 71, 205 (1931).
3. Lieb, E. H. & Liniger, W. Exact analysis of an interacting Bose gas. I. The general solution and the ground state. Physical Review 130, 1605 (1963).
4. McGuire, J. B. Study of exactly soluble one-dimensional N-body problems. Journal of Mathematical Physics 5, 622 (1964).
5. Yang, C. N. Some exact results for the many-body problem in one dimension with repulsive delta-function interaction. Physical Review Letters 19, 1312 (1967).
6. Takahashi, M. Thermodynamics of One-Dimensional Solvable Models. Cambridge University Press (1999).
7. Fukuhara, T. et al. Microscopic observation of magnon bound states and their dynamics. Nature 502, 76 (2013).
8. Wang, Z. et al. Experimental observation of Bethe strings. Nature 554, 219 (2018).
9. Bera, A. K. et al. Dispersions of many-body Bethe strings. Nature Physics 16, 625 (2020).
10. Morvan, A. et al. Formation of robust bound states of interacting microwave photons. Nature 612, 240 (2022).
11. Strecker, K. E., Partridge, G. B., Truscott, A. G. & Hulet, R. G. Formation and propagation of matter-wave soliton trains. Nature 417, 150 (2002).
12. Haller, E. et al. Realization of an excited, strongly correlated quantum gas phase. Science 325, 1224 (2009).
13. Koch, R., Bastianello, A. & Caux, J.-S. Adiabatic formation of bound states in the one-dimensional Bose gas. Physical Review B 103, 165121 (2021).
14. Castro-Alvaredo, O. A., Doyon, B. & Yoshimura, T. Emergent hydrodynamics in integrable quantum systems out of equilibrium. Physical Review X 6, 041065 (2016).
15. Bertini, B., Collura, M., De Nardis, J. & Fagotti, M. Transport in out-of-equilibrium XXZ chains: exact profiles of charges and currents. Physical Review Letters 117, 207201 (2016).
16. Tan, S. Energetics of a strongly correlated Fermi gas. Annals of Physics 323, 2952 (2008).