Graphene Multilayers Achieve Diverse Electronic Crystal Phases, Revealing Isospin Cascade Sequences

The fascinating behaviour of electrons in stacked layers of graphene continues to reveal surprising new phases of matter, and recent work by Wangqian Miao of The Pennsylvania State University and The Hong Kong University of Science and Technology, along with Chu Li, details the discovery of multiple distinct electronic crystal phases within rhombohedral multilayer graphene. This research systematically explores how increasing the number of electrons causes a cascade of ordering, resulting in unique states possessing non-zero Chern numbers, a property linked to robust electronic transport. The team demonstrates that these topological electronic crystals exhibit nearly identical energy levels, potentially leading to an extended anomalous Hall effect, and importantly, connects these theoretical predictions to observations made in recent experiments, offering a pathway to understanding and harnessing the exotic properties of this material.

Multilayer graphene exhibits complex electronic behaviour, investigated using comprehensive self-consistent Hartree Fock calculations combined with an ab initio tight binding model. As carrier density increases, the research uncovers a cascade of isospin phase transitions, leading to a rich variety of ordered states, including electronic crystals with non-zero Chern numbers. The study further demonstrates the near-degeneracy of these topological electronic crystals, which host an extended quantum anomalous Hall effect, and characterizes pressure driven phase transitions between these states. Finally, the work discusses thermodynamic signatures, particularly the behaviour of the inverse compressibility, in relation to recent experimental observations.

Density Functional Theory Calculations of Moiré Materials

Researchers employed Density Functional Theory (DFT) calculations to determine the structural properties of pentalayer graphene under pressure, crucial because the electronic properties of these materials depend on their atomic structure. They utilized the VASP software package and the RPBE exchange-correlation functional within DFT, employing the PAW method to represent electron-core interactions and incorporating DFT-D3 corrections for van der Waals interactions. Calculations were performed on a supercell with ABC stacking, ensuring convergence through established thresholds, to map changes in lattice constant and interlayer spacing under pressure, results used to construct a phase diagram. Hartree-Fock (HF) calculations were then used to study the electronic structure and phases of the material, a more sophisticated method than DFT for correlated electron systems. These calculations involved concepts such as the mini Brillouin Zone, Green’s Function, and Density Matrix, to investigate the formation of anomalous Hall crystal states. The team addressed limitations of finite-sized systems by employing finite size scaling, extrapolating results to the thermodynamic limit to improve accuracy and reliability of condensation energies for different electronic crystal states.

Isospin Transitions and Topological Phases in Graphene

Scientists have uncovered a cascade of isospin phase transitions in rhombohedral multilayer graphene, revealing a rich variety of ordered electronic states, including electronic crystals exhibiting non-zero Chern numbers. This work systematically investigates the emergence of these electronic crystals using comprehensive self-consistent Hartree Fock calculations combined with an ab initio tight binding model, demonstrating a pathway to novel topological phases of matter. Measurements of the perpendicular displacement field revealed a value of 5 meV/Å, with an interlayer spacing of approximately 3.33 Å.

Experiments revealed that as carrier density increases, the system undergoes first-order phase transitions, as depicted in a detailed phase diagram for tetralayer, pentalayer, and hexalayer graphene. The data shows the emergence of quarter-metal, half-metal, three-quarter-metal, and full-metal phases, with the number of isolated flat bands at the Fermi level varying depending on the specific phase. Researchers identified that electronic crystal phases preferentially appear near isospin transitions, particularly at an electron density of approximately 3x 10 12cm -2, indicating a strong interplay between isospin polarization and symmetry breaking. The breakthrough delivers a microscopic foundation for understanding these phenomena through a Stoner-type instability, where the density of states at the Fermi level, multiplied by the effective Hubbard interaction, exceeds one, establishing a clear connection between tunable interlayer asymmetry, gap opening under external fields, and the emergence of these novel electronic crystal phases in rhombohedral multilayer graphene.

Topological States and Anomalous Hall Effect

This research presents a detailed investigation into the behaviour of electrons within multilayer graphene, specifically exploring the emergence of an “electronic crystal” structure. Through advanced computational modelling, scientists have mapped a sequence of transitions occurring as electron density increases, revealing a variety of ordered states, some possessing non-zero Chern numbers, a property linked to unusual electronic behaviour. The team demonstrates that these topological electronic crystals exhibit near-degeneracy and host an extended anomalous Hall effect, a phenomenon where electrons are deflected at right angles to an applied electric field, without the need for a magnetic field. The study further characterises how external pressure influences these states, identifying a phase diagram dependent on both pressure and electron density. Importantly, the results connect to experimental observations of inverse compressibility, a measure of how electron density responds to changes in energy, showing a correlation between the predicted isospin phase transitions and the negative inverse compressibility observed in experiments on similar materials. The authors acknowledge that their calculations employ a simplified model, utilising Hartree-Fock theory, and that incorporating electron correlation effects and the influence of realistic pressure-enhanced moiré potentials represents an important avenue for future work.