A research team at the University of Illinois Urbana-Champaign and other institutions has found that periodic electronic patterns remain on the surface of uranium ditelluride (UTe₂) even when the material is warmed past the temperature at which it loses superconductivity. Examining how the patterns respond to temperature and magnetic fields, the researchers found they may be explained by a "pair density wave," a state in which electron pairs form a spatially periodic structure distinct from uniform superconductivity.
The researchers compare this state to the Cheshire Cat, which vanishes while its grin remains. However, what was directly observed is a charge modulation on the surface. The work does not show that a zero-resistance superconducting state continues to exist.
Behind the "ghost of superconductivity" metaphor is an experiment that revisits the relationship between electrons forming pairs and the whole crystal entering a uniform superconducting state.
The peer-reviewed paper, by Zhen Zhu, Professor Vidya Madhavan, Professor Eduardo Fradkin and colleagues, was published in PNAS on July 15, 2026. The university presented the research on September 9, and on October 2 it was distributed on EurekAlert! as a university release. Measurement details, including supplementary material, are in the authors' preprint.
Patterns that remain after superconductivity is gone
UTe₂ is a heavy-fermion compound in which superconductivity was reported in 2019. Strong interactions between electrons make them behave as though they have a very large effective mass. It is also a leading candidate for superconductivity based on spin-triplet electron pairs, but the nature of the pairing and the overall picture of the superconducting state are still debated.
The Cooper pairs responsible for superconductivity consist of two correlated electrons. Conceptually, electrons forming pairs and many pairs cooperating to create a uniform superconducting state can be thought of as separate phenomena.
The question this study asked was whether, even after that uniform superconducting state is lost, a different kind of order, in which electron pairs form a spatial pattern, could remain on the surface.
A pair density wave (PDW) is a state in which the superconducting order parameter varies periodically in space. The order parameter is a quantity that represents the strength and quantum-mechanical phase of pairing. It is easy to picture as "a wave of light and dark in the distribution of electron pairs," though it is not defined solely by the distribution of the number of pairs.
A charge density wave (CDW), meanwhile, is an electronic state in which the distribution of charge varies periodically in space. A PDW and a CDW can be coupled, so observing charge patterns can be a way to probe the order of electron pairs.
However, finding a CDW does not by itself prove that a PDW exists. The team therefore examined not only the period of each pattern but also the temperatures at which each disappears and differences in how each responds to the direction of the magnetic field.
The "Cheshire Cat's grin" metaphor used by Professor Fradkin in the university's announcement describes a state in which only order derived from electron pairs remains after uniform superconductivity has disappeared. Whether that "grin" is truly connected to the vanished superconductivity is what the team examined by combining multiple measurements.
The two kinds of patterns vanished at different temperatures
The samples were high-quality UTe₂ single crystals grown by the molten-flux method, with a superconducting transition temperature of about 2.1 K.
The researchers cleaved the crystals in ultra-high vacuum to expose fresh surfaces and measured them with a scanning tunneling microscope (STM). An STM uses the tunneling current flowing between a sharp tip and the sample surface to probe the arrangement of atoms and the distribution of electronic states available at each position.
Many of the low-temperature measurements were made at 300 mK. When the resulting images are Fourier transformed, spatially repeating patterns appear as peaks at specific wavevectors.
In addition to the previously known q₁, q₂ and q₃ charge modulations, the team identified components with smaller wavevectors, such as p₃. p₂ appears when a magnetic field is applied.
Using high-quality crystals was important for reducing defect-derived patterns that can hide weak signals. The supplementary material shows that measuring defect-rich regions makes signals such as p₃ hard to see, but that they reappear when image processing removes the effect of defects. Improved crystal quality made it possible to examine components that had previously been buried.
In these measurements, the p₃ charge modulation and the p₂ modulation induced by the magnetic field disappeared near the superconducting transition at about 2.1 K, while the q components remained even at 4.2 K.
| Charge modulation observed on the surface | Main measurement conditions | Change as temperature rises |
|---|---|---|
| p₃ | Observed at zero field | Drops to near-background intensity above the transition temperature of about 2.1 K |
| p₂ | Appears when a magnetic field is applied | Observed at low temperature, but gone at 2.5 K, above the transition temperature |
| q₁, q₂, q₃ | Temperature sweep at zero field; field measurements at 4.2 K | Remain above the transition temperature; strongly attenuated at 4.8 K |
This table organizes the temperatures and measurement conditions in Figures 2–4 of the authors' preprint for comparison. Because conditions differ from figure to figure, it cannot be assumed that all were measured on the same crystal. Nor is it a table comparing the absolute intensities of the components.
p₁ has also been observed under a magnetic field, but its temperature dependence was not tracked in this study. Symmetry leads researchers to expect it to behave like p₂ and p₃.
The wavevectors of the p components are about half those of the corresponding q components, so their spatial period is about twice as long. Looking at the periods alone, one could regard them as a fundamental and its harmonic of the same charge wave.
However, the shorter-period q components remain after the longer-period p components vanish. A simple fundamental-and-harmonic relationship therefore has difficulty explaining the temperature dependence.
Quasiparticle interference caused by electron scattering can also produce periodic patterns in STM images. The authors confirmed that the wavevectors of the new peaks barely change when the energy of the measured electrons is varied, and that the peaks are also observed in energy regions outside the superconducting gap.
This result is one of the grounds for interpreting the observed patterns as charge density waves rather than quasiparticle interference.
Magnetic-field direction preserved a clue to superconductivity
4.2 K is well above the superconducting transition temperature of these samples. At this temperature, the team examined how the q components change with magnetic field.
They used a vector-field STM, which can adjust not only the strength of the magnetic field but also its direction.
Along the a axis, the q components remained at 0.4 T, but at 0.8 T q₁ and q₂ disappeared and q₃ was only barely visible. For a field near the c axis, peaks that remained at 1.5 T were suppressed at 3 T, and for a field near the b axis they were suppressed when the field was raised from 4 T to 8 T. The authors compared the field-induced changes while keeping the same field of view and measurement conditions.
These values are not precise determinations of the critical fields at which each peak disappears. They are measurement points confirming states in which the peaks remain and states in which they are suppressed.
The directions near the b and c axes are written b* and c* in the paper, and do not strictly mean directions parallel to the crystal axes.
Still, the order was clear: the field needed to suppress the pattern is largest along b*, followed by c*, then a.
This order corresponds to the directional dependence of the upper critical field that destroys UTe₂'s uniform superconductivity at low temperature. In other words, even at temperatures where uniform superconductivity does not exist, the surface charge pattern retained a dependence on field direction similar to that of the superconducting state.
This correspondence is an important clue for considering that the q components that survive at high temperature are also linked to the order of electron pairs.
Of course, a CDW that responds to a magnetic field does not allow a single cause to be pinned down. In this study, the match in field-direction dependence, in addition to the relationship between periods and the difference in vanishing temperatures, strengthens the grounds for considering explanations that include a PDW.
How the electron-pair wave was inferred from the charge waves
In the theoretical model the team presented, multiple components of the PDW combine to produce a charge modulation at wavevector q = 2p. Producing this modulation does not require a uniform superconducting component.
When the PDW couples to a uniform superconducting component, on the other hand, a charge modulation at wavevector p appears.
In this model, when the temperature is raised and the uniform superconducting component is lost, the p charge modulation disappears. But if the surface PDW itself remains, the q charge modulation can persist afterward.
A single mechanism can thus explain both the relationship between the p and q wavevectors and the fact that they vanish at different temperatures.
The model also explains how a magnetic field changing the orientation of the spin structure of the triplet electron pairs could make charge components such as p₂, which were not visible at zero field, appear.
This is, however, a theoretical model built on the observations. The authors used Landau free energy and assumed that the PDW forms at a temperature higher than the transition temperature of uniform superconductivity.
The model takes about 5 K, where the q components appear, as the PDW transition temperature, but independent measurements have not directly confirmed that electron pairs form at 5 K.
The PNAS abstract also words this as "suggesting" the existence of a PDW at high temperature, and names a surface PDW as the most natural and simple candidate for explaining the observed state.
Distinguishing between the directly observed charge modulations and the electron-pair order inferred from them is important for understanding this result. By tracking multiple patterns in both temperature and magnetic field, the team narrowed the conditions a theory involving a PDW must satisfy.
What is seen on the surface must be considered separately from the crystal interior
This is not the first study to search for a PDW in UTe₂. In a paper published in Nature in 2023, Gu and colleagues used a superconducting tip to visualize the pairing gap varying spatially at low temperature. The magnitude of the modulation was about 10 μeV.
The significance of the present study is not that it detected a PDW itself for the first time. It lies in explaining a state that persists at temperatures higher than the transition temperature of uniform superconductivity, using the temperature dependence and magnetic-field response of multiple charge modulations.
For the crystal interior, on the other hand, a different measurement places an important constraint. In a paper published in Nature Communications in 2024, Kengle and colleagues used resonant elastic X-ray scattering to examine charge order inside the bulk.
That study used a different single crystal with a superconducting transition temperature of 1.8 K and measured at 2.2 K. It examined the vicinity of the wavevectors corresponding to the previously reported CDW, but within the detection limit it could not confirm any signal above background.
This result does not negate the present surface measurements, nor does it prove that a PDW does not exist. The two probe different depths and different physical quantities.
At the same time, there is no basis for generalizing the charge modulations observed on the surface to an order that also exists inside the crystal.
The authors' preprint also acknowledges that the high-temperature PDW interpretation involves strong assumptions and, in the absence of bulk evidence, must be considered a state localized at the surface.
On a surface, the energy states and interaction conditions of electrons differ from those in the interior. It is therefore possible that electron-pair order persists to higher temperatures on the surface alone.
However, a more microscopic theory is needed to establish whether this mechanism is really what forms the surface PDW in UTe₂.
Also, the persistence of a charge pattern at 4.2 K does not mean that a zero-resistance superconducting state continues up to 4.2 K.
Going forward, measurements that more directly confirm electron-pair order above the superconducting transition temperature, and experiments that connect the states of the surface and the crystal interior, will be important.
If these can be confirmed, it would become possible to examine in detail, using the concrete material UTe₂, how the order of electron pairs and charge is rearranged before and after uniform superconductivity emerges.