Unconventional superconductors, which don’t obey the Bardeen-Cooper-Schrieffer theory, are tricky to model in two or more dimensions. Tractable models, however, do exist for one-dimensional systems. In the lab, on the other hand, many unconventional superconductors are 2D and 3D, and 1D systems are hard to find or create. Without a shared theoretical and experimental system, testing theoretical models is hard to do.

About 25 years ago, researchers managed to design 1D chains of cuprates, a family of unconventional superconductors (see the article by Alex Malozemoff, Jochen Mannhart, and Doug Scalapino, Physics Today, April 2005, page 41). But in the intervening years, they have struggled to dope them. Doping over a range of concentrations provides information about the charge-carrier interactions that are essential for verifying theoretical models.

Now Zhi-Xun Shen of Stanford University, his postdoc Zhuoyu Chen, and their colleagues have successfully doped 1D cuprate chains over a range of hole-doping values. They found that the Hubbard model, which describes particles interacting in a lattice with just two terms, needs an important addition to match experimental results.

Shen and his colleagues grew their Ba_2 − xSr_xCuO_3 + δ chain by molecular-beam epitaxy. The growth process required high temperatures (635 °C) and a continuous supply of ozone, which delivered oxygen to the compound more efficiently than O₂. The researchers adjusted the material’s charge doping by shutting off the ozone supply at different stages of the sample cooling process. Oxygen would work its way into the material and pluck an electron out of the cuprate chain, so the longer the ozone ran, the higher the hole doping was.

The researchers prepared cuprate chains with hole doping from 9% to 40% and performed angle-resolved photoemission spectroscopy. In that technique’s measurements, photons eject electrons from the material, and the measured kinetic energies and emission angles of those electrons reveal the material’s band structure.

The strong electron correlations in a 1D system are predicted to cause spin–charge separation, in which excitations divide into two kinds of quasiparticles- ones with spin but no charge, called spinons, and ones with charge but no spin, called holons. The lab members saw bands for spinons and holons, as expected. But they also saw another pair of bands that didn’t match the Hubbard model.

John Hubbard devised his model in 1963 to be as simple as possible while still capturing the essential physics of exotic many-body phenomena. In it, charges hop between the sites of a discrete lattice and interact only with other particles on the same site.

Shen and his colleagues found that the other spectral features they observed could be modeled by the addition of a strong near-neighbor Coulomb attraction between holons at different sites. Ordinarily the Coulomb interaction is repulsive, so the coupling must be mediated by bosonic excitations—most likely phonons. With that additional term in the Hamiltonian, theory matched experiment.

The team’s augmented Hubbard model could qualitatively describe 2D high-temperature cuprate superconductors—including d-wave superconductors, which have been particularly difficult to model. (Z. Chen et al., Science 373, 1235, 2021.)