New theory links quantum geometry to electron-phonon coupling

A new study published in Nature Physics introduces a theory of electron-phonon coupling that is affected by the quantum geometry of the electronic wavefunctions.

The movement of electrons in a lattice and their interactions with the lattice vibrations (or phonons) play a pivotal role in phenomena like superconductivity (resistance-free conductivity).

Electron-phonon coupling (EPC) is the interaction between free electrons and phonons, which are quasiparticles representing 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 to the first author of the study, Dr. Jiabin Yu, Moore Postdoctoral Fellow at Princeton University.

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

Electronic wavefunctions and EPC

A quantum state is described by a wavefunction, a mathematical equation holding all the information about the state. An electronic wavefunction is basically a way to measure the probability of where the electron is located in the lattice (arrangement of atoms in a material).

"In condensed matter physics, people have long used energies to study the behavior of materials. In the last several decades, a paradigm shift led us to understand that the geometric and topological properties of wavefunctions are crucial in 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 within the crystal lattice. This means that the electronic wavefunction, to some extent, governs which electrons can couple with phonons and impact the conductivity properties of that material.

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

Quantum geometry

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

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

Quantum geometry explores this variation of spatial and temporal characteristics of wavefunctions.

"The geometric properties of single-particle wavefunctions 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 steps of a ladder, with the energy increasing the higher you go.

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

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

Gaussian approximation and hopping

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

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

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

Electron hopping is a phenomenon in crystal lattices where electrons move from one site to another. For hopping to occur effectively, the wavefunctions of electrons at neighboring sites must overlap, allowing electrons to tunnel through the potential barriers between sites.

The researchers found that the overlapping was affected by the quantum geometry of the electronic wavefunction, thus affecting hopping.

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

They quantified this by measuring the EPC constant, which tells 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).