Generalization of the Euler Maclaurin formula shows new superconductors Spektrum.de

Mathematical Physics: Two previously unknown types of superconductors predicted

Superconductors have enormous technological potential, but how they work remains a mystery to experts to this day. By generalizing a 300-year-old formula, four researchers have come closer to understanding the materials.

Andreas A. Buchheit, Torsten Keßler, Peter K. Schuhmacher, Benedikt Fauseweh

© koto_feja / Getty Images / iStock (detail)

Although they have been studied for many decades, many aspects of superconductors (here's an illustration) remain mysterious.

The quantum world seems counterintuitive and strange even to most experts. This is because quantum physical phenomena are mostly hidden from us in everyday life. But there are also exceptions: in some materials, quantum effects occur even at a macroscopic level. Phenomena impossible in the world of classical physics can then be observed with the naked eye. An example of this are superconductors, which, due to a principle of quantum mechanics, have no electrical resistance and can therefore transport electricity over any distance without any loss.

Since their discovery, these materials have made physicists dream: imagine energy networks that carry electricity generated in wind turbines in the North Sea across the country without loss. Another quantum effect causes superconductors to float above magnets – so they can serve as the basis for “Maglev trains” that travel as fast as a plane flies. These exotic substances could also be used in medical imaging to detect individual quanta of light. And superconductors already serve as the basic building block for modern quantum computers.

Although they are already being used in some areas, these miracle materials have not yet started a technological revolution. The reason for this is a major limitation that superconductors bring with them: they only work at extremely low temperatures. Even high-temperature superconductors generally need to be cooled to the temperature of liquid nitrogen, which is less than -196 degrees Celsius.

A superconductor that works at room temperature is therefore widely considered to be the Holy Grail of solid-state physics. Initial announcements of such room-temperature superconductors (like the LK-99 material, which made headlines in the summer of 2023) have so far turned out to be failures. A big problem here is that superconductors, although discovered in 1903, are still not fully understood in theory.

We are now closer to understanding these exotic materials, as we report in an article published in October 2023 in “Physical Review Research”. In it, we were able to expand the phase diagram of superconductors with the help of the generalization of a 300-year-old mathematical formula. In other words: Using new mathematical and physical methods, we predicted two previously unknown phases of superconductors – one of which looks extremely promising for the development of quantum computers.

Electrons shape the different states of superconductors

Not all superconductors are created equal. Similar to how water can take on completely different states depending on temperature and pressure (from solid to liquid to gas), superconductors also have different phases. The state it is in depends not only on pressure and temperature, but above all on the strength and range of electron interactions within the material.

While electrons in a normal material avoid each other due to their repulsive charge, in a superconductor an attractive force arises between the negatively charged particles. The cause of this force can be, for example, lattice vibrations or magnetic excitations in the material. This force means that two electrons can combine to form so-called Cooper pairs. They no longer all move differently, but exhibit collective behavior due to a quantum mechanical effect. If you generate electricity in a commercially available conductor, the electrons collide with imperfections or defects in the material, slow down, and the electricity is lost as heat. In a superconductor, due to collective behavior, electron pairs no longer perceive local defects and move without resistance in the material.

The bonded pair of electrons is stable. This means that electrons must overcome an energy barrier to exit the superconducting state. This barrier is called the superconducting energy gap and is a fundamental characteristic of a superconductor. To simulate the superconductor, the energy gap must therefore be calculated.

Mathematically, the energy gap can be viewed as a solution to an integral equation. Until now, experts have focused on short-range interactions, in which electrons first have to get very close to each other to sense each other. It has become clear to many that this does not necessarily correspond to reality – but describing interactions at large distances is extremely complex and was previously beyond the scope of computational possibilities. Using a new mathematical method, our team was able to include long-range interactions between electrons in the description of superconductors. This led to surprising results.

If you change the strength and range of electron interaction, different superconducting phases are created – similar to varying the pressure and temperature of water. We found a surprising result: superconductors are typically found in two phases; one is the “S-wave superconductor”, the other the “D-wave nodal superconductor”. The letters denote the different symmetries of the energy range, for example s, p or d, as they also occur in electron orbitals.

The S-wave superconductor has the simplest form: the energy gap is similar throughout space. In the d-wave superconductor, however, the energy gap changes significantly depending on the speed of the electron pairs. In d-wave nodal superconductors, there are even knot-like structures in which the energy gap disappears at certain electron speeds. There the superconductor behaves similarly to a normal material. Copper oxide-based high-temperature superconductors are a famous example of this.

Two new superconducting states give hope

As we now discover, two new phases in long-range interactions emerge. Just as ice and gas could hardly be more different, these superconducting states also have properties completely different from those of conventional superconductors. The first is the s+d wave, which combines superconducting properties of the s and d waves. Here, the S-wave component removes the nodes from the D-wave conductor and makes the material superconducting everywhere. However, the more complex structure of the d-wave superconductor remains intact.

The newly discovered second phase occurs with strong and very long-range interactions. We call this the p+ip+d wave phase because it combines three different symmetries. It has a wave of a p-wave component. P-wave superconductors, similar to d-wave, have a complex spatial structure. But that's not all: the p-wave portion of the new phase is intertwined with a rotated portion of the p-wave, which we call p+ip. This creates an extremely complex and stable structure, as if two ropes were tied together. The resulting phase presents interesting properties. Similar to a knot in a thread, the state is very stable, what theorists call topological: it can be stretched and deformed, but the knot remains – and external fluctuations or impurities in the material cannot harm this superconducting state.

Exotic phases | In addition to the familiar S-wave (blue) and D-wave phase (green), two new exotic phases (red and yellow) appear in long-range interactions (left). The diagrams on the right show the structure of the superconducting energy gap (z-axis), depending on the speed and direction of electron movement (xy-axis).

This stability could be relevant for technological applications, especially quantum computers. Their quantum processors are made up of individual qubits, which are very sensitive and therefore need to be very well isolated and cooled from the outside world. The stable excitations arising at the edges of p+ip superconductors could be used to develop more robust and scalable quantum computers.

The long-range interactions between electrons required for this could occur naturally in high-temperature superconductors, or the interactions could be created artificially in the laboratory – this could, for example, make a common material a superconductor. This works by bombarding a material with laser light, which is amplified by highly reflective mirrors in a so-called cavity resonator.

The complex mathematics behind long-range interactions

The two previously unknown phases remained hidden for so long because calculating them correctly was extremely time-consuming. The reason for this lies in the enormous number of electrons in a sample of material; typically we're talking about about 1,023 particles. While objects that interact at short range are influenced only by their immediate surroundings, in the long-range case the influence of all electrons in a single particle must be taken into account. For the calculation, this means you must calculate a sum of 1,023 particles. Even modern mainframes cannot perform this task.

In an article published in the “Journal of Scientific Computing” in December 2021 and a follow-up article in the journal “Nonlinearity”, we present an exact formula that can be used to calculate this enormous sum on a standard laptop. Our method is a generalization of the 300-year-old Euler-Maclaurin molecular formula for long-range interactions in three-dimensional space.

The idea here is to divide the sum of long-range interactions into two contributions. The first can be written as an integral, which ignores the microscopic structure of the material and only takes into account what happens on large scales. However, the microscopic crystal structure of the material also plays an important role in correctly predicting physical effects. This is addressed in the second term, the network contribution. It is based on a generalization of the Riemann zeta function, which was studied at the beginning of the 20th century by mathematician Paul Epstein and which bears his name. While the zeta function known in number theory sums natural numbers, Epstein's zeta function forms a sum over a multidimensional crystal lattice. Using well-known techniques from number theory, we can now efficiently evaluate the Epstein zeta function.

Our new equation for calculating the band gap energy of superconductors contains the Epstein zeta function, which takes into account both the long-range interaction and the microscopic structure of the material. This allowed us to expand the phase diagram of superconductors and investigate how exotic materials behave when electrons interact with each other over long distances. Our new discoveries, especially the discovery of the particularly stable topological phase, could give a huge boost to other fields of research, such as the development of quantum computers.

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Andreas A. Buchheit, Torsten Keßler, Peter K. Schuhmacher, Benedikt Fauseweh

Andreas A. Buchheit is a mathematician and physicist at Saarland University. Torsten Keßler is a mathematician at the Eindhoven University of Technology. Peter K. Schuhmacher is a physicist at the German Aerospace Center Cologne (DLR). Benedikt Fauseweh is a physicist at the Technical University of Dortmund and the DLR.