The first-ever “space–time crystal” has been created in the US by Hanqing Zhao and Ivan Smalyukh at the University of Colorado Boulder. The system is patterned in both space and time and comprises a rigid lattice of topological solitons that are sustained by steady oscillations in the orientations of liquid crystal molecules.

In an ordinary crystal atomic or molecular structures repeat at periodic intervals in space. In 2012, however, Frank Wilczek suggested that systems might also exist with quantum states that repeat at perfectly periodic intervals in time – even as they remain in their lowest-energy state.

First observed experimentally in 2017, these time crystals are puzzling to physicists because they spontaneously break time–translation symmetry, which states that the laws of physics are the same no matter when you observe them. In contrast, a time crystal continuously oscillates over time, without consuming energy.

A space–time crystal is even more bizarre. In addition to breaking time–translation symmetry, such a system would also break spatial symmetry, just like the repeating molecular patterns of an ordinary crystal. Until now, however, a space–time crystal had not been observed directly.

Rod-like molecules

In their study, Zhao and Smalyukh created a space–time crystal in the nematic phase of a liquid crystal. In this phase the crystal’s rod-like molecules align parallel to each other and also flow like a liquid. Building on computer simulations, they confined the liquid crystal between two glass plates coated with a light-sensitive dye.

“We exploited strong light–matter interactions between dye-coated, light-reconfigurable surfaces, and the optical properties of the liquid crystal,” Smalyukh explains.

When the researchers illuminated the top plate with linearly polarized light at constant intensity, the dye molecules rotate to align perpendicular to the direction of polarization. This reorients nearby liquid crystal molecules, and the effect propagates deeper into the bulk. However, the influence weakens with depth, so that molecules farther from the top plate are progressively less aligned.

As light travels through this gradually twisting structure, its linear polarization is transformed, becoming elliptically polarized by the time it reaches the bottom plate. The dye molecules there become aligned with this new polarization, altering the liquid crystal alignment near the bottom plate. These changes propagate back upward, influencing molecules near the top plate again.

Feedback loop

This is a feedback loop, with the top and bottom plates continuously influencing each other via the polarized light passing through the liquid crystal.

“These light-powered dynamics in confined liquid crystals leads to the emergence of particle-like topological solitons and the space–time crystallinity,” Smalyukh says.

In this environment, particle-like topological solitons emerge as stable, localized twists in the liquid crystal’s orientation that do not decay over time. Like particles, the solitons move and interact with each other while remaining intact.

Once the feedback loop is established, these solitons emerge in a repeating lattice-like pattern. This arrangement not only persisted as the feedback loop continued, but is sustained by it. This is a clear sign that the system exhibits crystalline order in time and space simultaneously.

Accessible system

Having confirmed their conclusions with simulations, Zhao and Smalyukh are confident this is the first experimental demonstration of a space–time crystal. The discovery that such an exotic state can exist in a classical, room-temperature system may have important implications.

“This is the first time that such a phenomenon is observed emerging in a liquid crystalline soft matter system,” says Smalyukh. “Our study calls for a re-examining of various time-periodic phenomena to check if they meet the criteria of time-crystalline behaviour.”

Building on these results, the duo hope to broaden the scope of time crystal research beyond a purely theoretical and experimental curiosity. “This may help expand technological utility of liquid crystals, as well as expand the currently mostly fundamental focus of studies of time crystals to more applied aspects,” Smalyukh adds.

The research is described in Nature Materials.

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Researchers in the US have directly imaged a class of extremely low-energy atomic vibrations called moiré phasons for the first time. In doing so, they proved that these vibrations are not just a theoretical concept, but are in fact the main way that atoms vibrate in certain twisted two-dimensional materials. Such vibrations may play a critical role in heat and charge transport and how quantum phases behave in these materials.

“Phasons had only been predicted by theory until now, and no one had ever directly observed them, or even thought that this was possible,” explains Yichao Zhang of the University of Maryland, who co-led the effort with Pinshane Huang of the University of Illinois at Urbana-Champaign. “Our work opens up an entirely new way of understanding lattice vibrations in 2D quantum materials.”

A second class of moiré phonons

When two sheets of a 2D materials are placed on top of each other and slightly twisted, their atoms form a moiré pattern, or superlattice. This superlattice contains quasi-periodic regions of rotationally aligned regions (denoted AA or AB) separated by a network of stacking faults called solitons.

Materials of this type are also known to possess distinctive vibrational modes known as moiré phonons, which arise from vibrations of the material’s crystal lattice. These modes vary with the twist angle between layers and can change the physical properties of the materials.

In addition to moiré phonons, two-dimensional moiré materials are also predicted to host a second class of vibrational mode known as phasons. However, these phasons had never been directly observed experimentally until now.

Imaging phasons at the picometre scale

In the new work, which is published in Science, the researchers used a powerful microscopy technique called electron ptychography that enabled them to image samples with spatial resolutions as fine as 15 picometres (1 pm = 10^-12 m). At this level of precision, explains Zhang, subtle changes in thermally driven atomic vibrations can be detected by analysing the shape and size of individual atoms. “This meant we could map how atoms vibrate across different stacking regions of the moiré superlattice,” she says. “What we found was striking: the vibrations weren’t uniform – atoms showed larger amplitudes in AA-stacked regions and highly anisotropic behaviour at soliton boundaries. These patterns align precisely with theoretical predictions for moiré phasons.”

Zhang has been studying phonons using electron microscopy for years, but limitations on imaging resolutions had largely restricted her previous studies to nanometre (10^-9 m) scales. She recently realized that electron ptychography would resolve atomic vibrations with much higher precision, and therefore detect moiré phasons varying across picometre scales.

She and her colleagues chose to study twisted 2D materials because they can support many exotic electronic phenomena, including superconductivity and correlated insulated states. However, the role of lattice dynamics, including the behaviour of phasons in these structures, remains poorly understood. “The problem,” she explains, “is that phasons are both extremely low in energy and spatially non-uniform, making them undetectable by most experimental techniques. To overcome this, we had to push electron ptychography to its limits and validate our observations through careful modelling and simulations.”

This work opens new possibilities for understanding (and eventually controlling) how vibrations behave in complex 2D systems, she tells Physics World. “Phasons can affect how heat flows, how electrons move, and even how new phases of matter emerge. If we can harness these vibrations, we could design materials with programmable thermal and electronic properties, which would be important for future low-power electronics, quantum computing and nanoscale sensors.”

More broadly, electron ptychography provides a powerful new tool for exploring lattice dynamics in a wide range of advanced materials. The team is now using electron ptychography to study how defects, strain and interfaces affect phason behaviour. These imperfections are common in many real-world materials and devices and can cause their performance to deteriorate significantly. “Ultimately, we hope to capture how phasons respond to external stimuli, like how they evolve with change in temperature or applied fields,” Zhang reveals. “That could give us an even deeper understanding of how they interact with electrons, excitons or other collective excitations in quantum materials.”

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Electron–phonon interactions in a material have been modelled by combining billions of Feynman diagrams. Using a modified form of the Monte Carlo method, Marco Bernardi and colleagues at the California Institute of Technology predicted the behaviour of polarons in certain materials without racking up significant computational costs.

Phonons are quantized collective vibrations of the atoms or molecules in a lattice. When an electron moves through certain solids, it can interact with phonons. This electromagnetic interaction creates a particle-like excitation that comprises a propagating electron surrounded by a cloud of phonons. This quasiparticle excitation is called a polaron.

By lowering the electron’s mobility, while increasing its effective mass, polarons can have a substantial impact on the electronic properties of a variety of materials – including semiconductors and high-temperature superconductors.

However, physicists have struggled to model polarons and it would be extremely helpful for them to represent polarons using Feynman diagrams. These are a mainstay of particle physics, which are used to calculate the probabilities of certain particle interactions taking place. This has been challenging because polarons emerge from a superposition of infinitely many higher-order interactions between electrons and phonons. With each successive order, the complexity of these interactions steadily increases – along with the computational power required to represent them with Feynman diagrams.

Higher-order trouble

Unlike some other interactions, each higher order becomes more and more important in representing the polaron as accurately as possible. As a result, calculations cannot be simplified using standard perturbation theory – where only the first few orders of interaction are required to closely approximate the overall process.

“If you can calculate the lowest order, it’s very likely that you cannot do the second order, and the third order will just be impossible,” Bernardi explains. “The computational cost typically scales prohibitively with interaction order. There are too many diagrams to compute, and the higher-order diagrams are too computationally expensive. It’s basically a nightmare in terms of scaling.”

Bernardi’s team – which also included Yao Luo and Jinsoo Park  – approached the problem with the Monte Carlo method. This involves taking repeated random samples within a space of all possible events contributing to a process, then adding them together. It allows researchers to build up a close approximation of the process, without accounting for every possibility.

The team generated a series of Feynman diagrams spanning the full range of possible electron–phonon interactions. Then, they combined the diagrams to gain precise descriptions of the dynamic and ground-state properties of polarons in real materials.

Statistical noise

One issue with a fully-random Monte Carlo approach is the sign problem, which arises from statistical noise that can emerge as electrons scatter between different energy bands during electron–phonon interactions. Since different bands can contribute positively or negatively to the interaction probabilities represented by Feynman diagrams, these contributions can cancel each other out when added together.

To avoid this, Bernardi’s team adapted the Monte Carlo method to evaluate each band contribution in a structured, non-random way – preventing sign cancellations. In addition, the researchers applied a matrix compression approach. This vastly reduced the size and complexity of the electron–phonon interaction data, without sacrificing accuracy. Altogether, this enabled them to generate billions of diagrams without significant computational costs.

“The clever diagram sampling, sign problem removal, and electron–phonon matrix compression are the three key pieces of the puzzle that have enabled this paradigm shift in the polaron problem,” Bernardi explains.

The trio hopes that its technique will help us understand polaron behaviours. “The method we developed could also help study strong interactions between light and matter, or even provide the blueprint to efficiently add up Feynman diagrams in entirely different physical theories,” Bernardi says. In turn, it could help to provide deeper insights into a variety of effects where polarons contribute – including electrical transport, spectroscopy, and superconductivity.

The research is described in Nature Physics.

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