The new theory combines quantum geometry with electron-phonon coupling

A new study published in Natural physics introduces the theory of electron-phonon coupling, which is influenced by the quantum geometry of electronic wave functions.

The movement of electrons in the lattice and their interaction with lattice vibrations (or phonons) play a key role in phenomena such as superconductivity (resistanceless conduction).

Electron-phonon coupling (EPC) is the interaction between free electrons and phonons, which are quasi-particles representing the vibrations of a crystal lattice. EPC leads to the formation of Cooper pairs (pairs of electrons), responsible for superconductivity in certain materials.

A new study explores the field of quantum geometry in materials and how these can contribute to the power of EPCs.

Phys.org spoke with the study’s first author, Dr. Jiabin Yu, Moore Postdoctoral Fellow at Princeton University.

Speaking about the motivation behind the study, Dr. Yu said, “My motivation is to go beyond conventional wisdom and find out how the geometric and topological properties of wave functions affect interactions in quantum materials. In this work, we focus on EPC, one of the most important interactions in quantum materials.”

Electronic wave functions and EPC

A quantum state is described by a wave function, a mathematical equation containing all information about the state. The electronic wave function is basically a way to measure the probability of where an electron is in the lattice (the arrangement of atoms in a material).

“In condensed matter physics, people have long used energies to study the behavior of materials. In the last few decades, a paradigm shift has led us to understand that the geometric and topological properties of wave functions are fundamental to understanding and classifying realistic quantum materials,” explained Dr. Yu.

In the context of EPC, the interaction between the two depends on the location of the electron in the crystal lattice. This means that the electronic wave function controls to some extent which electrons can couple with phonons and affect the conduction properties of that material.

The researchers in this study wanted to investigate the effect of quantum geometry on EPC in materials.

Quantum geometry

A wave function, as already mentioned, describes the state of a quantum particle or system.

These wave functions are not always static and their shape, structure and distribution can evolve in space and time, just as a wave in the ocean changes. But unlike waves in the ocean, quantum mechanical wave functions follow the laws of quantum mechanics.

Quantum geometry investigates this variation in the spatial and temporal characteristics of wave functions.

“The geometric properties of single-particle wave functions are called band geometry or quantum geometry,” explained Dr. Yu.

In condensed matter physics, the band structure of materials describes the energy levels available to electrons in the crystal lattice. Think of them as rungs on a ladder, with energy increasing the higher you go.

Quantum geometry affects the band structure by influencing the spatial extent and shape of the electron wave functions in the lattice. Simply put, electron distribution affects the energy structure or distribution of electrons in a crystal lattice.

The energy levels in the lattice are crucial because they determine important properties such as conductivity. Additionally, the band structure will vary from material to material.

Gaussian approximation and skipping

The researchers built their model using a Gaussian approximation. This method simplifies complex interactions (such as those between electrons and phonons) by approximating the distribution of variables such as energies as a Gaussian (or normal) distribution.

This facilitates mathematical manipulation and inference about the effect of quantum geometry on EPC.

“The Gaussian approximation is essentially a way to relate the electron jump in real space to the quantum geometry of momentum and space,” said Dr. Yu.

Electron hopping is a phenomenon in crystal lattices where electrons move from one place to another. For hopping to occur efficiently, the wavefunctions of electrons at adjacent sites must overlap, allowing electrons to tunnel across potential barriers between sites.

The researchers found that the overlap was affected by the quantum geometry of the electronic wave function, which affected the hopping.

“EPC often originates from a hopping change with respect to lattice vibrations. Thus, EPC should naturally be enhanced by strong quantum geometry,” explained Dr. Yu.

They quantified this by measuring the EPC constant, which is indicative of the strength of the bond or interaction, using a Gaussian approximation.

To test their theory, they applied it to two materials, graphene and magnesium diboride (MgB₂).

Superconductors and applications

The researchers decided to test their theory on graphene and MgB₂ as both materials have superconducting properties controlled by EPC.

They found that for both materials the EPC was strongly influenced by geometrical contributions. Specifically, geometric contributions of 50% and 90% were measured for graphene and MgB₂respectively

They also discovered the existence of a lower bound or limit to contributions due to quantum geometry. Simply put, there is a minimal contribution to the EPC constant due to the quantum geometry and the rest of the contribution is from the electron energy.

Their work suggests that raising the superconducting critical temperature, which is the temperature below which superconductivity is observed, can be done by increasing the EPC.

Some superconductors like MgB₂ are phonon-mediated, meaning that the EPC directly affects their superconducting properties. According to the research, the strong quantum geometry implies a strong EPC, which opens a new way to search for relatively high-temperature superconductors.

“Although EPCs cannot mediate superconductivity alone, they can help cancel some of the repulsive interaction and help generate superconductivity,” added Dr. Yu.

Future work

The theory developed by the researchers has only been tested for certain materials, which means it is not universal. Dr. Yu believes the next step is to generalize this theory so that it is applicable to all materials.

This is particularly important for the development and understanding of various quantum materials (such as topological insulators) that could be influenced by quantum geometry.

“Quantum geometry is ubiquitous in quantum materials. Scientists know it should influence many quantum phenomena, but often lack theories that clearly capture this effect. Our work is one step toward such a general theory, but we are still a long way from fully understanding it. far because it’s basically unrealistic.” ” concluded Dr. Yu.

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