Physicists finally saw ‘Bethe strings,’ atom clusters predicted in 1931, in ultracold cesium

A cloud of cesium atoms cooled to within a few billionths of a degree above absolute zero has produced bound clusters that physicists have sought since 1931. The clusters, called Bethe strings, formed inside several thousand parallel one-dimensional tubes in an Innsbruck laboratory, and the larger ones held six or more particles each.

Hans Bethe, the German-born physicist who later won the Nobel Prize for explaining how stars produce energy, derived the strings as exact solutions for particles confined to a single line. For roughly a century they existed on paper and, in indirect form, in the magnetic behavior of certain solids. The Innsbruck group, led by Hanns-Christoph Nägerl, reports that it has now made them in a gas of atoms and watched them behave as predicted.

Bethe’s 1931 ansatz and the strings hidden in its solutions

In 1931 Bethe introduced a method, now known as the Bethe ansatz, for finding the exact energy levels of a chain of interacting magnetic particles. It was later extended to one-dimensional gases of bosons, and the solutions it produces include collective bound states in which several particles lock together through their mutual interactions rather than through chemical bonds. Those are the strings. They can exist only in one dimension, and before this work they had been detected only in solid-state magnetic systems, according to the University of Innsbruck’s release of 14 September 2026.

Bethe himself was recognized by the 1967 Nobel Prize in Physics for his discoveries concerning energy production in stars, work unrelated to the strings.

Cesium atoms tuned from repulsive to attractive

Nägerl’s team cooled cesium atoms to within a few billionths of a degree of absolute zero and split the cloud into several thousand extremely narrow tubes, so that each atom could move along only one direction. The experimenters then tuned the interactions between atoms from repulsive to attractive. Once the force turned attractive, the atoms bound into states of different sizes, including clusters of six or more particles.

The result appears in Nature Communications headed “Probing Bethe strings in an attractive one-dimensional Bose gas,” published on 29 August 2026. Its authors are Milena Horvath, Alvise Bastianello, Sudipta Dhar, Rebekka Koch, Yanliang Guo, Jean-Sébastien Caux, Manuele Landini and Nägerl. The paper reports that a binding-energy analysis sets a lower bound of at least six on the largest string present, and that generalized hydrodynamics modeling predicts even larger strings are also excited. An earlier arXiv version of the work, posted in May 2025, already described bound states of more than six particles.

Expansion tests that separate strings from loose atoms

Binding energy is invisible to a camera, so the group used expansion as the signature. In one run the clusters expanded while still confined to the tubes. In another the confinement was removed so they could spread in three dimensions. Strings fell apart in three dimensions, and the energy released by the break-up showed up as faster motion, while unbound atoms showed essentially the same energy in both cases. Horvath, a lead author, described the simplest experiment as letting the strings expand.

The one-dimensional runs produced the most striking behavior. “This is a remarkable feature of the strings: they can collide without breaking apart,” Horvath said in the Innsbruck release. Dhar, the other lead author, said the setup now lets the group create the strings in the laboratory, manipulate them, make them collide and probe that collisional stability directly.

One-dimensional gases as a testing ground for exact theory

The experiment sits in a long line of work on exactly solvable models. In 1963 Elliott Lieb and Werner Liniger gave the exact solution of a one-dimensional gas of bosons with point-like interactions, a model that the same Bethe-style machinery handles, and C. N. Yang generalized the approach in 1967, per the reference history of the method. Strings are among the states that such solutions contain once the interaction turns attractive. In the cesium cloud, the Innsbruck group swept from repulsion to attraction and compared the measured binding energy, momentum distribution and a quantity called Tan’s contact with generalized hydrodynamics predictions, which the Nature Communications abstract says agree with the data.

Alvise Bastianello served as the lead theorist, and theory groups at the University of Amsterdam and the Technical University of Munich contributed to interpreting the data, as ScienceDaily’s summary of the Innsbruck release notes. Bastianello said the result “opens new possibilities for studying how these collective quantum objects form and interact.”

The work was funded by the Austrian Science Fund (FWF) through a Wittgenstein Prize grant, by a European Union ERC grant and by the UK’s Engineering and Physical Sciences Research Council. Horvath belongs to the FWF doctoral program Atoms, Light and Molecules. The largest string size the experiment confirmed is bounded at six particles in the paper’s own analysis, so the larger strings the models predict remain a target for later runs.

This article was produced with the assistance of AI and reviewed by Morning Overview editors prior to publication.

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