The Long Read: All in a spin

It is one of the most famous experiments in physics. Light illuminates a pair of slits in a wall, generating an array of bright and dark patches on a screen. The British physicist Thomas Young first performed the experiment at the turn of the 19th century to demonstrate that light can interfere with itself, behaving as a wave. Much later, quantum versions of the experiment would demonstrate something far more mysterious: that photons, electrons and other particles can exhibit wave-like interference patterns, but apparently only when no-one is watching. The experiment “has in it the heart of quantum mechanics”, wrote the American physicist and Nobel laureate Richard Feynman. “In reality, it contains the only mystery.”

Today, few scientists doubt the merits of quantum mechanics. It has proved itself through mind-boggling predictive power, not to mention a host of practical applications: semiconductor electronics, lasers, superconducting magnets, quantum cryptography and quantum computing, to name but a few. Yet it is still a subject ripe with puzzles, both in its basic interpretation and in its role in condensed matter, where each material can serve as a quantum playground.

One puzzle is the existence of peculiar types of magnetism, as studied by ESRF users such as Markus Grüninger from the University of Cologne in Germany. Unravelling these phenomena has led Grüninger and his colleagues to shift the boundaries of an X-ray technique – amazingly, in such a way as to recall Young’s famous experiment once again. “Our experiments rely on the excellent beam quality at the ESRF, the outstanding performance of the set-up at beamline ID20, and the fruitful collaboration with the beamline staff,” says Grüninger.

The technique in question is resonant inelastic X-ray scattering (RIXS). This begins with an X-ray photon knocking a tightly bound electron up to a higher atomic energy level. Almost instantaneously an electron from another high energy level relaxes into the resultant hole, releasing a new photon. By measuring the difference in energy between the incoming and outgoing photons, users can learn how the process has changed the solid in collective excitations of electron charge and spin – the latter being the basis of magnetism. The ESRF has helped develop RIXS since the 1990s, and currently offers it at two dedicated, world-leading beamlines: ID32 with soft X-rays, and ID20 with hard X-rays.

Hard X-ray photons can transfer a lot of momentum to a sample. In 2019, an international team led by Grüninger wanted to push ID20’s capabilities, and record an even greater range of momentum transfer than usual. Drawing on theory by Jeroen van den Brink at IFW Dresden in Germany, and making use of new ID20 instrumentation developed by beamline scientists Giulio Monaco (now at the University of Padova in Italy) and Marco Moretti (now at the Polytechnic University of Milan, also in Italy), the team studied the effect of large changes in momentum transfer on the intensity of the outgoing X-rays. Their sample was a crystal of an iridium oxide containing pairs or “dimers” of iridium ions. To their delight, the researchers found an interference pattern, demonstrating that the X-ray photons were exciting electrons at both iridium sites in the dimers at once – similar to light passing through Young’s double-slit, although in this case putting the dimer in an excited state (Figure 1) [1].
 

Click figure to enlarge

Fig. 1: a) RIXS spectra of the iridium oxide Ba₃CeIr₂O9 measured at different momentum transfers q, corresponding to maxima (5Q, 7Q) and minima (4Q, 6Q) of the intensity. As q changes, the peak energy remains the same, whereas the intensity varies periodically, the hallmark of interference. This is the basis of RIXS interferometry, which can provide insights into the spatial delocalization and entanglement of electrons in novel magnetic materials. b) An integration of the RIXS intensity (over the energy range highlighted by the blue box) yields the periodic interference pattern as a function of q in units related to the lattice constant c or the intradimer Ir-Ir distance d = π/Q.

The experiment marked the beginning of RIXS interferometry, a technique that was predicted as far back as the mid 1990s. By demonstrating that the electrons in the iridium dimers experience a quantum, wave-like delocalization over a quasi-molecular dimer orbital, RIXS interferometry opened the door to the study of materials with novel magnetic properties, which physicists have been trying to understand for decades.

The most familiar type of magnetism – the sort that exists in a common fridge magnet – is ferromagnetism. In metals such as iron, it results from conduction electrons that are delocalized over an entire crystal, with spins able to align parallel to one another, producing a net magnetic moment. This is very different to one type of material with novel magnetism, the Mott insulator. Conduction in this type of material is forbidden due to strong electron repulsion, but it still has magnetism because its spins, while localized on individual ions, can interact with each other. Even more intriguing is the cluster Mott insulator, an emerging new class of material that exhibits what could be called a “local delocalization”. Here, electrons are fully delocalized over a dimer (or another small collection of ions), but they cannot propagate from one dimer to another. This results in local magnetic moments, residing not on individual ions but on quasi-molecular clusters. “In contrast to the usual electron spin, these cluster moments are something that we can tailor, by choosing the ionic species, cluster geometry, electron count, pressure and so on,” says Grüninger.

In 2022, Grüninger and colleagues used their new RIXS interferometry to unambiguously identify a cluster Mott insulator for the first time. The ID20 data could directly reveal the presence of three electron spins delocalized over an iridium dimer, creating a cluster magnetic moment [2] in a compound that is a candidate for a quantum spin liquid. The data also paved the way for a systematic exploration of more complex compounds, for example with trimers [3] or tetramers, rather than dimers. “Our results show that the trimers reside in an unexpected parameter regime that promises non-trivial magnetic moments,” says Grüninger. “They challenge previous views on trimer physics, highlighting the strength of RIXS interferometry.”

Cluster Mott insulators are exciting because of their potential as microscopic, fine-tuned magnets, as well as for their still-unexplored quantum properties. They also have potential to realize quantum “spin liquids”. First predicted by the US physicist and Nobel laureate Philip Anderson back in the 1970s, though experimentally elusive, spin liquids excel by the quantum-driven absence of magnetic order – even at temperatures close to absolute zero – that defines more conventional magnets. They are characterized by a quantum-entangled network of strongly fluctuating spins, driven by competing interactions that cannot be satisfied simultaneously. A simplified example of the situation is three spins on the vertices of a triangle: they may all want to align antiparallel to each other, but this is possible only for a pair of them, not all three simultaneously.   
 

Christoph Sahle (left), PhD student Lara Pätzold and Markus Grüninger set up another experiment on the ID20 beamline.

According to Grüninger, one can picture the electron spins as musicians in a band. In a ferromagnet, the musicians are playing the same note. In a disordered conventional magnet, on the other hand, the musicians are busy warming up their instruments, producing a cacophony of sounds. But the spin liquid is something far more avant-garde – more like a free jazz band, in which the individual musicians are in the groove, closely interacting, producing music that somehow keeps changing without falling apart. “The unfamiliar ear may find it difficult to distinguish their music from noise, but a connoisseur can spot the close interactions between all of them,” says Grüninger. One might add: with help from RIXS interferometry.

In practice, physicists try to realize spin liquids by tuning the network of interactions between spins, and cluster Mott insulators provide one possibility to achieve this, through adjustment of their magnetic moments. The payoff could be big. In the late 1980s, Anderson predicted that high-temperature superconductivity could result from a spin-liquid state. Moreover, spin liquids host novel particles with fractionalized quantum numbers – Majorana fermions, for instance, which are their own anti-particles. Such exotic excitations are desired as the basis of fault-tolerant qubits for quantum computing. 

One particular theoretical model known to harbour Majorana quasiparticles is named after Alexei Kitaev at the California Institute of Technology in the US. Although pure Kitaev behaviour has proved hard to realize experimentally, in 2020 Grüninger and colleagues used RIXS interferometry to “sharpen their ears” and spot exotic Kitaev-type interactions in the iridium oxide Na₂IrO₃ with a honeycomb lattice of spins [4]. RIXS interferometry was established on dimers, but it can be extended – as here – to an infinite lattice, because Kitaev-type interactions between spins dynamically “cut” the lattice into a fluctuating soup of dimers. 

One of the defining features of the Kitaev model is that, unusually, the spin entanglement depends on the honeycomb’s three bond directions. In other words, along one bond the entanglement exists only for the x component of the spins, while along another bond it exists for the y component, and along the third, for the z component. Or in free-jazz terms: the musicians only listen to each other in pairs, and so long as they are staring into each other’s eyes, turning and changing partners all the while. Two years ago, Grüninger’s team managed to observe this bond-directional behaviour of spin excitations for the first time – a crucial step towards identifying more authentic Kitaev spin liquids and Majorana quasiparticles (Figure 2) [5].

Click figure to enlarge

Fig. 2: The integrated RIXS intensity (blue to red) of magnetic excitations in Na₂IrO₃ is plotted as a function of momentum transfer along different crystallographic directions and at different temperatures. (The q-space equivalent of the honey-comb lattice is overlaid in black lines.) In a normal magnet, spin excitations would be the same along the three equivalent bond directions. Here, the intensity varies depending on the bond and its direction – one of the hallmarks of the Kitaev spin-liquid model. Kitaev quantum spin liquids are desirable for their potential to host Majorana fermions for quantum computing, among other applications.

According to Christoph Sahle, the scientist in charge of ID20, these challenging experiments have become possible thanks to a host of improvements: permanent experimental installations, a frequent X-ray injection scheme, a dedicated hutch with upgraded optics, a huge spectrometer, and of course the Extremely Brilliant Source itself. “It’s great to collaborate with users who make full use of the unique set of properties that this kind of machine provides,” he says.

Traditionally, neutron scattering has been the go-to technique for exploring magnetism. But Grüninger thinks RIXS interferometry is emerging as a complementary technique, and in certain cases is even setting a new standard. “We’re currently working on applying RIXS interferometry to a much larger class of materials, making it even more powerful,” he says. 

See this article in the March edition of the ESRFnews magazine
 

Text by Jon Cartwright

References
[1] A. Revelli et al., Sci. Adv. 5 eaav4020 (2019).
[2]  A. Revelli et al., Phys. Rev. B 106 155107 (2022).
[3] M. Magnaterra et al., Phys. Rev. B 111 085122 (2025).
[4] A. Revelli et al., Phys. Rev. Research 2 043094 (2020).
[5] arXiv:2301.08340, https://doi.org/10.48550/arXiv.2301.08340