The properties of most of materials around us are dictated by their charge carriers, e.g. the electrons. Consequently, **one of the most sought-after problems in condensed matter physics is to understand how electrons behave in a quantum material**, that is a material that display characteristics that could only be explained using quantum physics. In a wide class of materials, electrons interact weakly with each other which allows to describe their behavior as small perturbations of the otherwise well-known properties, or in other words assuming that the interacting system properties will not differ by much once weak interactions are included. But that is not always the case.

In the **regime of strongly correlated quantum matter** instead, electron interactions are very significant, and it is not possible to use intuition or knowledge of the weakly interacting cases to describe their properties. In certain cases, one finds that **new and vastly different properties emerge** as a result of collective interaction between the electrons, that can no longer be traced back to the limit of non-interacting electrons. Moreover, **the complexity of the strongly correlated electron problem scales exponentially with system sizes**. This means that it becomes harder to approach the problem numerically as one considers larger and larger systems. In a recent article published as Editors’ suggestion in Physical Review Letters, however, researchers from the Department of Physics at Stockholm University, tackle an example of a strongly correlated problem that shows in graphene Moiré systems — and uncover the existence of novel emergent phases of matter with unusual and remarkable properties.

## Moiré patterns in quantum materials

Moiré patterns are formed in everyday life. They occur as a result of interference between similar but slightly shifted or rotated patterns. For example, here is the famous mathematician Roger Penrose demonstrating such pattern using two sheets of paper. Recently, there has been an increased interest in Moiré patterns formed in quantum materials like graphene due to the ability to manufacture them in experiments. A famous example is twisted bilayer graphene which shows such patterns. When **two layers of graphene** are put on each other but with a relative tiny twist, an enlarged Moiré superlattice emerges with a lattice constant that is much larger than the original lattice constant of a single layer of graphene.

## Properties of Moiré superlattices

By controlling the relevant parameters such as the twist angle, Moiré superlattices show very interesting properties.

One property is that they could host a number of **flat bands**. Bands in a material tell us about the energy of a single electron moving with a given momentum. The flatness of a band means that the kinetic energy of the occupying electrons is quenched and the interaction between them is enhanced which opens a new world of possibilities for interaction driven phases of matter such as **superconductivity** and the **fractional quantum Hall effect**. The quantum Hall effects** **happens when two dimensional materials are placed in a strong magnetic field. This results in the quantization of a quantity known as the Hall conductance, that is the ratio of the electric current in one direction and the voltage in the transverse direction. The value of this Hall conductance could be quantized to either integers or fractions, hence the terminology “fractional quantum Hall effect”.

## New properties found

In the article “Particle-Hole Duality, Emergent Fermi Liquids and Fractional Chern Insulators” by Ahmed Abouelkomsan, Zhao Liu and Emil J. Bergholtz, two Moiré systems, namely **trilayer graphene and twisted bilayer graphene aligned with hexagonal boron nitride**, have been theoretically investigated by considering the core problem of electrons interacting through Coulomb interaction in a fractionally filled flat band — in shape contrast to previous studies that focused on integer band fillings.

The authors show that by looking at the problem from the **perspective of holes** instead of electrons, through a particle-hole transformation, useful insights could be gained. A hole can be thought of as an absence of an electron. A hole occupying a certain site in a lattice corresponds to an electron not occupying that site. **From the perspective of holes, the problem becomes weakly interacting at various fillings, a state of matter known as a Fermi liquid.** Such state is amenable to the standard techniques used when dealing with weakly interacting systems. In addition, the authors identify a new emergent parameter that controls the physics of that Fermi liquid, that is a single-hole dispersion driven solely from the interactions.

Moreover, the authors show that in twisted bilayer graphene aligned with boron nitride, fractional Chern insulator states could be stabilized at certain band fillings. **Fractional Chern insulators** (FCI) are states of matter that are lattice analogues of fractional quantum Hall states —i.e., by possessing a fractional quantized Hall conductivity and the same universal properties.

However, a crucial difference is **the unnecessity of an applied magnetic field**. A strong magnetic field imposes a limitation on the usual fractional quantum Hall experimental setup. Another limitation is the small gap that is less than 1 Kelvin making those states very difficult to survive at higher temperatures. A preliminary analysis has showed that **the fractional Chern insulator states predicted by the authors have a significantly larger gap that allows the FCI states to survive up to temperatures of at least 10 Kelvin**. These findings thus open new routes for experiments on manufacturing high temperature quantum Hall states with no magnetic field, and ultimately making them available for applications.

This work is part of the PhD thesis work by Ahmed Abouelkomsan supervised by Emil Bergholtz, and in collaboration with Emil’s former postdoc Zhao Liu (now assistant professor at Zhejiang University).

## Journal publication:

Ahmed Abouelkomsan, Zhao Liu, and Emil J. Bergholtz, Phys. Rev. Lett. 124, 106803 (2020). Available online at https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.124.106803

Contact

Ahmed Abouelkomsan, ahmed.abouelkomsan@fysik.su.se

Emil Bergholtz, emil.bergholtz@fysik.su.se