Spinons may help electrons pair along stripes in some superconductors

15/09/2026
Artistic representation of a quantum colored string (the blue "river") running through an antiferromagnetic lattice (the checkerboard pattern of alternating spins). The red rectangles highlight regions where spinons—topological magnetic defects—pair into singlets, acting as the elementary pairing units that generate the d-wave superconducting pattern. The image evokes the game of Go (Weiqi), where strategic formations along a boundary can determine the outcome of the game—much as spinon singlets along a stripe determine the local pairing physics in high-temperature superconductors. Credit: Xue-Feng Zhang / Chongqing University.


Superconductors are materials that carry electricity with zero resistance below specific temperatures. Many of these materials become superconducting at very low temperatures, yet some enter superconducting phases at higher temperatures.

While high-temperature superconductivity has been widely studied over the past few decades, its underlying physical processes have not yet been clearly elucidated. In some high-temperature superconductors, holes (i.e., electron vacancies) can organize themselves into stripes.

Researchers at the Beijing Computational Science Research Center, Beijing Normal University and Chongqing University recently investigated the possible contribution of this stripe organization to d-wave pairing, in which paired electrons have a quantum wave pattern that changes sign between perpendicular directions and is often represented as a four-lobed cloverleaf shape.

Their paper, published in Physical Review Letters, offers a possible new explanation for how electron pairing could emerge around stripes in models of strongly correlated, hole-doped materials.

"In 2005, Science Magazine listed the mechanism of high-temperature superconductivity as one of the 100 most important unanswered questions in science," Xue-Feng Zhang, senior author of the paper, told Phys.org. "It appeared again in their 2021 '125 Questions' collection. That tells you how stubborn this puzzle is. My collaborators and I have been obsessed with it since 2023."

The unsolved mystery of electron pairing along stripes

Earlier experiments and numerical simulations suggested a close relationship between superconductivity and stripes in cuprates, copper-oxide materials containing layers of copper and oxygen atoms. Calculations indicated that in models of these materials, electron pairs tend to accumulate along hole-rich stripes. Physicist Jan Zaanen described these stripes as "rivers of charge" separated by insulating regions.

"Looking at a stripe and seeing pairing is like hearing an orchestra and knowing the melody is beautiful, without knowing which instrument is playing it," said Zhang. "Our objective was simple but terrifying: look inside the stripe and identify the musician."

An emerging technique called quantum gas microscopy opened new possibilities for studying strongly interacting quantum systems and their underlying physics. This technique essentially allows experimental physicists to take actual photographs of individual atoms in optical lattices.

"We asked ourselves: can we do the same thing with our numerical wavefunctions?" said Zhang. "If we could take a high-resolution snapshot of a stripe, maybe we could finally catch the pairing mechanism in the act."

As part of their study, Zhang and his colleagues decided to study two specific models describing how strongly interacting electrons behave in lattice materials. These two models, an extended version of the Fermi-Hubbard model and the t-J model, have been widely used to study copper-based high-temperature superconductors.

The Fermi-Hubbard model describes electrons that move between sites on a grid while repelling other electrons occupying the same site. The t-J model, on the other hand, describes how electrons move through a grid without sharing a site and how the spins of neighboring electrons in a lattice interact.

By studying these models, the team tried to identify a possible microscopic mechanism underlying the d-wave electron pairing observed in some high-temperature superconductors.

"Think of the two models as a Go board where alternating sites hold black and white stones, representing electrons with opposite spins," explained Zhang. "Like Go stones, they can hop to neighboring empty intersections, but they strongly repel each other if forced to share the same position. These are the simplest equations that still capture the essential physics of copper-oxide superconductors."

Reconstructing quantum behavior from computational snapshots

Zhang and his colleagues tried to solve equations capturing the basic physics of copper-oxide superconductors using a technique called density-matrix renormalization group (DMRG). DMRG compresses the mathematical description of a quantum system while retaining the information considered most important for accurately describing its state.

"The real innovation of our study, however, was combining this approach with 'perfect sampling,' a numerical technique inspired by quantum gas microscopy experiments," said Zhang. "This technique lets us take snapshots of the quantum state, just as an experimentalist would photograph atoms in a lab."

To make sense of the snapshots they collected, the researchers used an updated version of a theory they had previously developed, called the Quantum Colored String Model. This theory describes a stripe linked with superconductivity as a fluctuating rope made of three distinct types of beads. These include spinons (magnetic defects), holons (charge defects), and dual-holes (hypothetical quasiparticles representing two adjacent empty electron sites in a stripe).

"By tracking how these beads move and interact, we could reconstruct the wavefunction and see exactly how the pairing emerges," said Zhang.

"Our most striking finding is that the pairing inside a stripe is carried by objects called spinon singlets—pairs of magnetic defects with opposite chirality that fit together like the two interlocking halves of a yin-yang symbol: opposite in nature, yet forming a stable, complete whole. These are not ordinary Cooper pairs; they are topological defects that live inside the stripe itself. When they pair up, they naturally create the alternating plus-minus pattern characteristic of d-wave superconductivity."

To test their theoretical predictions, the team performed a numerical test. In this calculation, they turned off the magnetic interaction that binds spinons.

They found that this prompted singlets to dissolve and the d-wave pattern to vanish. When they turned the magnetic interaction back on, however, the pairing emerged again.

"That is strong evidence of causality—not just a beautiful coincidence," said Zhang. "The broader implication is that stripes are not just passive backgrounds for superconductivity; they are active factories that produce the elementary pairing units.

"This challenges older scenarios where pairing is assumed to happen uniformly across the material. It also gives experimentalists a concrete prediction: both quantum simulators and scanning tunneling microscopy experiments can observe the signatures of these spinon pairs."

A new possible route to d-wave pairing

While this study does not offer a conclusive explanation of high-temperature superconductivity, it outlines a microscopic mechanism that could explain the formation of electron pairs in a single fluctuating stripe and how their quantum correlations could produce the sign-changing pattern characteristic of d-wave pairing.

"Turning that local dance into a global superconducting state—across many stripes and at finite temperature—is still the next mountain to climb," said Zhang. "Yet I believe we now know what the dancer looks like."

So far, the team's analysis primarily explored what happens inside a single stripe in hole-doped materials. The next step will be to investigate what happens when there are multiple stripes.

"In our paper, we already see hints that spinon singlets can tunnel from one stripe to another, which could be the bridge that turns local pairing into global superconductivity," explained Zhang. "We plan to map out this inter-stripe coherence in detail."

Zhang and his colleagues are now also working with experimental physicists to test their predictions. In particular, they would like to test their theoretical framework's predictions using cold-atom quantum simulators, laboratory systems in which ultracold atoms are arranged in optical lattices to emulate the behavior of interacting electrons, including their formation into stripes.

"If an experimentalist can photograph a spinon singlet in an optical lattice, it would be the smoking gun that validates the entire scenario," added Zhang.

"We are optimistic that this conversation between theory and experiment will accelerate rapidly in the next few years. Furthermore, in our recent preprint, we make a prediction of a subtle 2-lattice-constant shift between positive and negative energy patterns in the scanning tunneling microscopy experiment. We have received preliminary positive feedback from experimental groups regarding this prediction."

Source: https://tinyurl.com/3wmss9ws via Phys.org