The new theory links quantum geometry to electron-phonon coupling

A new study published in Nature Physics presents a theory of electron-phonon coupling that is influenced by the quantum geometry of electronic wave functions.

The movement of electrons in a lattice and their interactions with lattice vibrations (or phonons) play a major role in phenomena such as superconductivity (conduction without resistance).

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

The new study explores the realm of quantum geometry in materials and how these can contribute to the strength of EPC.

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 common wisdom and discover 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 that holds all the information about the state. An electronic wave function is essentially a way to measure the probability of where the 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 made us realize that the geometric and topological properties of wave functions are crucial to understanding and classifying materials quantum realistic.” explained Dr. Yu.

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

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

Quantum geometry

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

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

Quantum geometry explores this variation of 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 a crystal lattice. Think of them as rungs on a ladder, with energy increasing the higher you go.

Quantum geometry affects the band structure by affecting the spatial extent and shape of the electron wavefunctions within the lattice. In simple terms, electron distribution affects the energy structure or arrangement of electrons in a crystal lattice.

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

Gaussian approximation and hopping

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

This makes it easier to mathematically treat and draw conclusions about the influence of quantum geometry on EPC.

“The Gaussian approximation is essentially a way to relate the hopping real-space electron to the quantum momentum-space geometry,” said Dr. Yu.

Electron shedding is a phenomenon in crystal lattices where electrons move from one place to another. For hopping to occur effectively, the wavefunctions of electrons in neighboring 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, thereby affecting the hopping.

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

They estimated this by measuring the EPC constant, which indicates the strength of the coupling or interaction, using the Gaussian approximation.

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

Superconductors and applications

The researchers chose to test their theory on graphene and MgB₂ because both materials have EPC-induced superconducting properties.

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

They also found the existence of a limit or lower limit for contributions due to quantum geometry. In simple words, there is a minimal contribution to the EPC constant due to the quantum geometry, and the rest of the contribution is from the energy of the electrons.

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

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

“Even if EPC cannot mediate superconductivity alone, it can help cancel out 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 to make it applicable to all materials.

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

“Quantum geometry is ubiquitous in quantum materials. Researchers know that it must influence many quantum phenomena, but they often lack theories that clearly capture this effect. Our work is a step toward such a general theory, but we we are still a long way from fully understanding it,” concluded Dr. Yu.

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