A quintessential quantum simulator takes a 10 000-fold leap

The many-electron wavefunction of a solid is extremely complicated. Electrons move continuously in 3D space, influenced by the potential-energy landscape of the atomic nuclei (which themselves are also moving) and the long-range Coulomb repulsion of all the other electrons.

In contrast, the FHM is admirably simple. Its fermionic particles occupy only the discrete nodes of a lattice, and they interact only with particles on the same node. (Typically, the fermions are taken to have spin ½, and each node can accommodate at most two particles: one with spin up and one with spin down.) The particles can hop to neighboring nodes, but they can’t change their spin states. The system is characterized by only a handful of tunable parameters: the interaction energy of particles on the same node, the energy required to hop nodes, the temperature, and the average density of particles per node.

Given that simplicity, it’s perhaps surprising that the FHM captures so many of the real effects of solid-state physics, such as antiferromagnetism. When the same-node interaction is repulsive, the temperature is low enough, and the particle density is near the so-called half-filled level of one particle on average per node, particles settle into a state in which exactly one sits at each node. Even though they’re not sharing nodes and therefore not interacting with one another, the subtleties of quantum mechanics and Fermi–Dirac statistics drive them toward a pattern of alternating spins.

The antiferromagnetic phase is also observed in the cuprates and other superconductors near zero doping—that is, when the material composition provides neither extra electrons nor holes to carry charge through the otherwise superconducting layers. The FHM’s antiferromagnetism is a tantalizing hint that a superconducting phase may be lurking nearby. But to get there, researchers need to move to still lower temperatures and away from half filling, and that’s where the understanding breaks down.

The half-filled FHM is one of the few cases that theoretical studies can grapple with reasonably well. Away from half filling, theorists run up against the sign problem: The integrals involved are dominated by large positive and negative contributions that almost, but don’t quite, cancel out, so they’re extremely difficult to calculate accurately. Meanwhile, experimenters have been stalled in their quest for lower temperatures.

Fermionic atoms in optical traps are a reasonable approximation of the FHM’s particles on discrete lattice nodes. And arrays of equally spaced optical traps easily emerge—in 1D, 2D, or 3D lattices—from the interference patterns of pairs of counterpropagating laser beams.

But that setup requires exceptionally low temperatures. To mimic the physics that arises in real materials at tens to hundreds of kelvin, a trapped-atom FHM experiment must be cooled to tens of nanokelvin—near the limit of what cold-atom physicists can currently achieve.

Another big limitation is the system uniformity. Laser beams as typically generated have Gaussian profiles: They’re brightest in the center and fade away around the edges. As a result, in a 2D or 3D lattice of traps made from Gaussian beams, the traps in the middle are deeper than those around the periphery. In an experiment on more than a few dozen of those traps, it’s likely that different parts of the system would be in completely different phases.

The USTC researchers took on both those challenges. For the latter, they built custom-designed diffractive optical elements to convert their Gaussian beams into flat-top beams with uniform intensity over almost the entire beam profile. With three pairs of flat-top beams, they formed a uniform lattice nearly 100 sites wide in each dimension, for 800 000 sites total.

But the benefits of homogeneity don’t stop there. In a typical FHM experiment, researchers hold the atomic gas in a single large Gaussian trap before loading it into the lattice of smaller traps. The trap is deepest in the center, so the gas is densest there—and the inhomogeneity of the gas density is a source of entropy in the lattice.

What Pan, Chen, Yao, and colleagues did instead was hold the gas in a box trap: a hollow cylinder made of light, whose walls repel the atoms and keep them inside. By allowing the gas to equilibrate to a uniform density over the volume of the trap, they could load it into the lattice much more uniformly. “In retrospect, that’s obvious, but they were the first to realize it,” says Hulet. The more uniform loading leads to significantly lower entropy—by at least a factor of two—and therefore lower temperature.

With a 3D lattice that’s large, cold, and uniform, the researchers were uniquely positioned to observe something that had never been seen before in the FHM: the phase transition to antiferromagnetic order. Importantly, although Greiner and colleagues had seen antiferromagnetic correlations in their 2D experiment, they didn’t see an actual antiferromagnetic phase, which doesn’t even exist in 2D. Rather, the antiferromagnetic correlations start small and gradually spread across the 2D system at lower temperatures. When Greiner and colleagues saw a checkerboard pattern spanning their 80-site lattice, it was because the model’s correlation length had grown larger than the system they were looking at.

On the other hand, whereas Greiner and colleagues used a quantum gas microscope to see the checkerboard pattern directly, that option wasn’t available to the USTC researchers. Instead, they used Bragg scattering to measure the spin ordering in their 3D lattice, similar to how x-ray scattering probes the ordering of atoms in a real crystal.

Figure 2 shows one of their experiments that studied the antiferromagnetic phase transition. Panel a is a sketch of the system’s phase diagram in terms of entropy (related to temperature) and the particle density n; the antiferromagnetic phase forms a symmetric dome on either side of the half-filled state n = 1. The series of blue dots shows how the researchers tuned n to probe a slice of phase space that cuts through the antiferromagnetic dome.

Panel b shows the USTC researchers’ measurements of the spin structure factor S, which quantifies how well-ordered the spins are. Near n = 1, S is large, as expected of an antiferromagnetic phase. But outside of the phase boundaries, which the researchers estimate to lie somewhere in the gray bands, S doesn’t abruptly drop to zero. Rather, it tails off with a power-law dependence.

The power law is defined by a critical exponent, and there are only a few values the exponent could plausibly take. A wide variety of seemingly disparate physical systems fall into a small number of universality classes, each with its own characteristic scaling behavior (see Physics Today, July 2023, page 14). The FHM is thought to belong to the same universality class as the 3D Heisenberg model, which would give it a critical exponent of 1.396. But that’s never been confirmed, because the FHM phase transition had never been observed before.

When the researchers drew a line with slope −1.396, they found that it agreed reasonably well with their data in the log–log plot in figure 2b. Importantly, though, the experiment doesn’t constitute a measurement of the critical exponent. “Accurately determining the critical exponent of a power-law function requires making measurements over several orders of magnitude,” explains Yao. “In our current work, we did not fulfill that condition. But in the future, we hope to determine the value precisely.”

Pan, Chen, Yao, and colleagues have performed the most quantitative and informative FHM experiment to date, but there’s much more to be done. The superconducting phase, if it exists, lies at temperatures even lower than the researchers have achieved, and they’ll need further experimental improvements to access it.

If and when researchers do reach the superconducting phase, the next step will be to perform detailed experiments to try to uncover the mechanism by which the fermionic particles combine into bosonic pairs that condense into a superfluid. Part of the reason that cuprate superconductivity has been so enigmatic is that there’s no way to tune individual properties in isolation. Just to change the charge-carrier density, for example, it’s necessary to make a new sample with a different chemical composition, which changes other properties in tandem.

In the FHM, on the other hand, changing the particle density is as straightforward as reloading the lattice with more or fewer atoms. Other parameters can be tuned too, including those that take the model beyond the classic FHM to simulate effects such as phonons or spin fluctuations. By testing how each parameter does or doesn’t contribute to superconductivity, researchers could finally uncover the mysterious electron-pairing mechanism.

But understanding superconductivity isn’t the only goal. Strongly correlated electron systems give rise to many other physical phenomena, some of which show up in the FHM at the temperatures researchers can achieve already. “Due to the difficulty in numerical calculations, little is currently known about the 3D FHM at low temperatures and away from half filling,” says Yao. “Mapping out its phase diagram is important in its own right.”

And the USTC group won’t be the only one working on the FHM. Box traps, the key to lowering the quantum gas’s entropy and temperature, are an established technology, so now that their importance for creating low-entropy gases is known, other groups can start using them too. The diffractive optical elements used to create the flat-top beams were custom designed, but similar products are available commercially. “It will absolutely be possible for other groups to replicate these results,” says Hulet. “Pan’s group is ahead of everybody else, but only by a few months.”