New theory links quantum geometry to electron-phonon coupling

A new study published in Natural physics introduces a theory of electron-phonon coupling that is affected 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 central 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. The EPC leads to the formation of Cooper pairs (electron pairs), responsible for the superconductivity of certain materials.

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

Phys.org spoke with the study’s first author, Dr. Jiabin Yu, a Moore postdoctoral scholar 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 the 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 the information about the state. An electron wavefunction is essentially 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 recent decades, a paradigm shift has led us to understand that the geometric and topological properties of wave functions are crucial to understand and classify realistic quantum materials.” Dr. Yu explained.

In the context of the EPC, the interaction between the two depends on the location of the electron in the crystal lattice. This means that the electronic wave function determines, to some extent, which electrons can couple to phonons and impact 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 earlier, describes the state of a particle or quantum system.

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

Quantum geometry explores 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 a crystal lattice. Think of them as steps on a ladder, with the energy increasing as you climb.

Quantum geometry influences band structure by affecting the spatial extent and shape of electronic wave functions within the lattice. Simply put, the distribution of electrons affects the energy structure or arrangement of electrons in a crystal lattice.

Energy levels in a network are crucial because they determine important properties like conductivity. Additionally, the structure of the strip varies from material to material.

Gaussian approximation and jumps

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 facilitates mathematical manipulation and allows conclusions to be drawn about the influence of quantum geometry on the EPC.

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

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

The researchers found that the overlap was affected by the quantum geometry of the electronic wave function, thereby affecting the jump.

“EPC often arises from the change of jumps relative to lattice vibrations. Naturally, EPC should be enhanced by strong quantum geometry,” explained Dr. Yu.

They quantified 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 as both materials have EPC-driven superconducting properties.

They found that for both materials, the EPC was strongly influenced by geometric contributions. Specifically, the geometric contributions were measured at 50% and 90% for graphene and MgB.₂respectively.

They also discovered the existence of a lower limit for contributions due to quantum geometry. Simply put, there is a minimum contribution to the EPC constant due to quantum geometry, and the rest of the contribution comes from the energy of the electrons.

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

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

“Even though EPC alone cannot ensure superconductivity, it can help cancel part of the repulsive interaction and generate superconductivity,” Dr. Yu added.

Future work

The theory developed by the researchers has only been tested for certain materials, meaning 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 developing and understanding different quantum materials (like topological insulators) that could be affected by quantum geometry.

“Quantum geometry is ubiquitous in quantum materials. Researchers know that it is expected to affect many quantum phenomena, but theories that clearly capture this effect are often lacking. Our work is a step toward such a general theory, but we are still far from fully understanding it,” Dr. Yu concluded.

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