January 02, 2025 -- In a new study, researchers from the Max Planck Institute of Quantum Optics under the lead of Timon Hilker demonstrated evidence of stripe formation, i.e. extended structures in the density pattern, in a cold-atom Fermi-Hubbard system. By using a quantum gas microscope and a special mixed-dimensional geometry, they were able to observe unique higher-order correlations in spin and charge densities related to those seen in some high-temperature superconducting materials. These findings, which shed light on a key phenomenon in condensed matter physics, suggest that individual stripe structures could form at higher temperatures than the much-debated stripe phase. This experiment represents a major step forward in using quantum simulators to explore the most fundamental properties of materials. The work is published this week in Nature.

Quantum phases, such as those apparent in exotic quantum matters like superconductors, superfluids or Bose-Einstein condensates arise from interactions between particles forming complex, often non-traditional arrangements. At the microscopic level, these phases are governed by competing forces, like charge and spin interactions, that cause emergent behaviors unexplained by classical physics. Since the discovery of these phases in the early 1980s, their microscopics have been under debate. It is, for example, unclear how high-temperature superconductivity arises in Cuprates, where the pairing mechanism enables a resistance-free electron flow. In this paper, the team investigated a question related to this phenomenon : How does the strong interaction of dopants with a magnetic background lead to effective interactions between the dopants? These fundamentals are highly relevant for advancements in condensed matter physics and ultimately the development of novel future materials.

A game of chess of Holes and Magnetism

In a new study using a Fermi-Hubbard system – a certain kind of cold-atom quantum simulator – researchers at the Max Planck Institute of Quantum Optics were now able to present a novel method to understand the movement and interaction of particles in a material, specifically focusing on how they behave at certain temperatures. In their study, the researchers investigated exotic phases of doped antiferromagnetic two-dimensional materials. Therefore, they utilized a mixed-dimensional quantum simulator where dopant particles, or “holes,” are confined to one-dimensional motion, while spins remain two-dimensional. This mixed setup fosters conditions conducive to observing stripe formations – alternating patterns of charge and spin density often associated with superconductivity.

What the researchers observed are long-range correlations between holes, indicating an increased probability of hole “bunching”: a behavior that could signal the formation of “stripes”. One can think of a “stripe” as a string that lies loosely on the table. It is a one-dimensional object with some wiggles. In this case, the string is made up of holes in a square array of spins. Another way to see it could be like the lines of pawns on a board of chess. The squares of the chess board symbolize the lattice places and the pawns are the holes. Some of them have made their move but the player ensured according to the rules of the game that they stay linked either directly or diagonally next to each other.

The board in this analogy fits the antiferromagnetic background in which the holes align. But contrary to an invariable chess board, the magnetism itself is affected by the arrangement of holes. Here, the field of the game changes, so to speak, with the behaviour of its figures.  It is this backaction between the charge motion of the holes and the local antiferromagnetism, which drives the intricate physics leading ultimately to striped phases and superconductivity.

‘Sweet spot’ for first signs of fluctuating stripes

Using a lithium quantum gas microscope, the researchers found that the holes form patterns that indicate that they tend to form these lines – directly or diagonally.

But the structure is not perfect as it doesn’t stretch through the entire system. “We mainly see fragments of two to five aligned holes. They are clearly not randomly distributed but cluster in a one-dimensional way,” says Dominik Bourgund, PhD candidate and first author of the paper. At lower temperatures, approaching absolute zero, the holes form perfect lines equivalent to the pawns right at the beginning of a chess game. This has been confirmed in numerical calculations and experimental observations in real materials. At much higher temperatures, where previous quantum simulations took place, the holes were randomly distributed, comparable to the image of a chess board towards the end of the game. “What we investigated is somewhat like the sweet spot, at which the materials show first signs of short, strongly fluctuating stripes. Our measurements suggest that under certain doping levels, these structures are present long before they order into a charge-density wave”.

With this novel approach, it was possible to observe these fluctuating structures of the quantum states and perform actual measurements unlike in solid-state experiments. The team’s results highlight that even in a many-body system with only repulsive interaction structures of bound particles emerge robustly at an intermediate temperature regime. These findings are particularly relevant for fundamental research on strongly correlated quantum materials and show compelling evidence that cold-atom quantum simulators can be used to explore these complex quantum phases, advancing research into the mechanisms behind high-temperature superconductivity and potentially guiding the development of future materials.

“We are thrilled that this technically very demanding experiment had worked out and we see several exciting research directions leading from here. Our next project steps involve testing for the first signs of superconductivity in the system, expanding our system to two layers instead of one, and cooling our setups to even lower temperatures,” says research group leader Timon Hilker with a prospect to the near future.