Mysteries of the bizarre ‘pseudo-cap’ in quantum physics finally solved

By cleverly applying a computational technique, scientists have made a breakthrough in understanding “pseudo-cap,” a long-standing enigma in quantum physics closely related to superconductivity. The discovery, presented in Sciencewill help scientists in their quest for room-temperature superconductivity, a holy grail of condensed matter physics that would enable lossless energy transmission, faster MRI machines and ultra-fast levitating trains.

Some materials made from copper and oxygen exhibit superconductivity (electricity flows without resistance) at relatively high, but still frigid, temperatures below minus 140 degrees Celsius. At higher temperatures, these materials fall into what is called the pseudogap state, where they sometimes behave like a normal metal and sometimes more like a semiconductor.

Scientists have discovered that the pseudogap appears in all so-called high-temperature superconducting materials. But they don’t understand why or how it appears, or whether it persists when the temperature drops to absolute zero (minus 273.15 degrees Celsius), the unattainable lower limit of temperature at which molecular motion stops.

By better understanding how the pseudogap arises and how it relates to the theoretical properties of superconducting materials at absolute zero, scientists are getting a clearer picture of these materials, says study co-author Antoine Georges, director of the Flatiron Institute’s Center for Computational Quantum Physics.

“It’s like you have a landscape with a lot of fog, whereas before you could only see a few valleys and a few peaks,” he says. “Now the fog is clearing and we can see more of the landscape as a whole. It’s really an exciting time.”

Quantum physicists can study states such as the pseudogap using computational methods that model the behavior of electrons in a material. But these calculations are extremely difficult to perform because of quantum entanglement, in which electrons connect and cannot be treated individually even after they are separated. For more than a few dozen electrons, it is impossible to directly calculate the behavior of all the particles.

“Calculating the properties of these materials is a real challenge: it is impossible to simulate them accurately, even on the most powerful computer,” explains Georges. “You have to use clever algorithms and simplified models.”

The Hubbard model is a famous model: researchers view matter as a chessboard on which electrons can jump from one adjacent space to another, like a tower. Electrons can have an up-spin or a down-spin. Two electrons can share a space on the chessboard only if they have opposite spins and pay an energy cost. With this model, which emerged in the 1960s, scientists can deploy different calculation methods, each with strengths and weaknesses in different situations.

“There is a class of methods that work very well at zero temperature, and another class of methods that work very well at finite temperature,” says Fedor Šimkovic IV, lead author of the new study, who was a postdoc with co-author Michel Ferrero at the École Polytechnique and Collège de France in Paris and is now a team leader at IQM Quantum Computers in Munich, Germany. “These two worlds don’t usually communicate with each other, because in between, at very low but finite temperatures, is actually the most computationally difficult regime.”

It is precisely in this intermediate state that the pseudogap lies. To address this problem, the team applied an algorithm called schematic Monte Carlo, first described in 1998 and improved in 2017 by Riccardo Rossi, a co-author of the new paper. Unlike quantum Monte Carlo, a well-known and successful algorithm that uses randomness to examine small areas of the model at a time and aggregates these examinations to reach conclusions, schematic Monte Carlo considers interactions across the entire board at once.

“The approach of the schematic Monte Carlo method is very different,” explains Rossi, a researcher at the CNRS and Sorbonne University. “We can simulate, in principle, an infinite number of particles.”

Armed with the schematic Monte Carlo method, the team figured out what happens to pseudogap materials as they cool toward absolute zero. From previous research, they knew that the materials could become superconductors, or that they could develop “bands,” in which electrons arrange themselves into rows of matching spins separated by rows of empty squares.

The state in which the Hubbard model enters absolute zero depends on the number of electrons. When the model has exactly as many electrons as there are squares on the chessboard, the entire board becomes a stable checkerboard pattern of up and down rotations, making the material an electrical insulator (profoundly uninteresting for superconductivity research, since insulators are the opposite of conductors). Adding or removing electrons can cause superconductivity and/or striping.

At higher temperatures, at which electrons are still moving, the researchers knew that electron removal would cause the pseudogap, but they didn’t know what would happen when the material cooled.

“There was a question about whether the pseudogap always evolved into a band state,” Georges says. “Our paper answers that important question in the field and closes that window.” The study found that as pseudogap materials cool toward absolute zero, they do indeed develop bands. Interestingly, Georges adds, by modifying the Hubbard model to allow for diagonal motions, like that of a bishop, the pseudogap evolves into a superconductor as it cools.

The study also answered the question of what causes the pseudogap, in which the arrangement of electrons is no longer uniform as it was at absolute zero, but instead includes striped areas, squares with two electrons, holes, and checkerboard pattern areas. The researchers found that as soon as these checkerboard areas appeared in the electron arrangement, the materials fell into the pseudogap. These two big answers about the pseudogap help further unravel Hubbard’s model.

“On a broader scale, this is all part of a collective effort by the scientific community to combine computational approaches to solve these difficult problems,” Georges says. “We are living in a time where these problems are finally starting to be clarified.”

These results will also benefit applications beyond numerical calculations, including quantum gas simulation, a 20-year-old field at the intersection of quantum optics and condensed matter physics. In these experiments, atoms are cooled to ultracold temperatures and then trapped by lasers in a grid similar to Hubbard’s model. With new developments in quantum optics, researchers can now lower these temperatures almost to the point where the pseudogap forms, thus uniting theory and experiment.

“Our paper is directly relevant to these ultracold quantum gas simulators,” Georges says. “These quantum simulators are now on the verge of being able to observe this pseudogap phenomenon, so I expect some really interesting developments in the next year or two.”