Topological Insulators and Quantised Hall Effect in Depleted Square Lattices.

The behaviour of electrons in strong magnetic fields and periodically structured lattices continues to reveal unexpected topological states of matter, with potential applications in robust quantum technologies. Researchers are now exploring how deliberately disrupting the perfect order of these lattices, through a process termed ‘depletion’, impacts the emergence of these states. A recent investigation, published in a leading peer-reviewed journal, details how the interplay between lattice depletion and electron hopping influences the quantum Hall effect and the formation of Chern phases, characterised by their unique topological properties. Sara Aghtouman from the University of Zanjan, Godfrey Gumbs from Hunter College, City University of New York, and Mir Vahid Hosseini from the University of Zanjan, present their findings in the article, “Quantum Hall Effect and Chern Phases in the 1/5-Depleted Square Lattice”, where they demonstrate a tunable pathway to engineer robust topological insulators within square lattice systems. Their work utilises a tight-binding model, a method approximating electron behaviour in solids, to analyse the resulting electronic structure and Hall conductivity, quantified by calculating Chern numbers, which describe the topological character of the electronic bands.

Topological phases represent a developing area within condensed matter physics, offering potential advancements in materials science and electronics. Researchers actively investigate novel materials and mechanisms to realise and control these exotic states of matter, focusing on systems exhibiting non-trivial topological properties. This study details a computational investigation into the emergence of topological phases within a two-dimensional square lattice, demonstrating how lattice geometry and electron hopping parameters dictate the material’s electronic structure and topological characteristics. The research establishes a clear pathway for engineering robust Chern insulators, materials exhibiting insulating behaviour in the bulk but possessing conducting states on their surface, protected by topology.

The investigation begins with a detailed analysis of a square lattice subjected to a 1/5 depletion – the removal of a fraction of lattice sites – and a perpendicular magnetic field. Researchers employ a tight-binding model, a computational method approximating electron behaviour in solids by describing electrons as being bound to atoms, incorporating both nearest and next-nearest neighbour hopping terms to accurately describe electron movement between atoms. This modelling allows them to map the Hofstadter butterfly, a fractal diagram representing the energy spectrum of electrons in a periodic potential under a magnetic field, and calculate Chern numbers, topological invariants that characterise the quantum Hall effect. The computational framework provides a robust platform for exploring the interplay between geometric frustration, electron correlations, and topology in creating and controlling exotic quantum states of matter.

In the absence of diagonal hopping, a parameter describing electron movement to non-adjacent sites, the system exhibits particle-hole and flux-inversion symmetries, resulting in a zero total Chern number across all energy bands. However, the introduction of diagonal hopping breaks these symmetries, deforming the Hofstadter butterfly and creating energy gaps within the electronic band structure. Remarkably, this symmetry breaking also leads to a non-zero total Chern number, indicating the emergence of unconventional topological phases of matter. Topological phases are characterised by robust electronic states protected by the topology of the material’s band structure, offering potential advantages for developing fault-tolerant quantum devices.

Researchers systematically vary the hopping parameters, meticulously identifying regimes where individual Chern indices are large, signifying strong topological character and enhanced robustness against perturbations. They also pinpoint parameter windows that optimise gap stability and the formation of Hall plateaus, flat regions in the Hall conductivity indicating quantized conductance. The Hall effect describes the voltage generated across a conductor when a magnetic field is applied perpendicular to the current flow, providing a direct measure of the material’s topological properties.

The results demonstrate that combining lattice depletion with diagonal hopping provides a tunable mechanism for engineering robust Chern insulators, offering a pathway for creating materials with tailored electronic properties. Chern insulators exhibit insulating behaviour in the bulk but possess conducting states on their surface, protected by topology.

The investigation confirms that the interplay between lattice depletion and diagonal hopping within a two-dimensional square lattice significantly alters the electronic band structure and topological properties, establishing a clear pathway for engineering materials with tailored functionalities. Researchers demonstrate that manipulating the hopping parameters allows for the tuning of individual Chern indices and the optimisation of gap stability, essential for observing robust quantum Hall plateaus. These plateaus represent quantized Hall conductivities, a hallmark of topological insulators, offering potential advantages for developing low-power electronic devices.

The computational methodology employed provides a powerful tool for predicting and understanding the behaviour of complex materials, accelerating the discovery of new topological phases and their potential technological applications. Researchers plan to extend these calculations to incorporate electron-electron interactions, which are known to play a crucial role in determining the properties of correlated materials. They also intend to investigate the effects of disorder and imperfections on the topological properties of the system, aiming to develop materials that are robust against real-world conditions. The ultimate goal is to design and synthesise materials that exhibit robust topological properties and can be used to create revolutionary electronic devices.

Future research will focus on exploring the effects of different types of defects and impurities on the topological properties of the system, aiming to develop materials that are robust against real-world conditions. Researchers also plan to investigate the possibility of realising these topological phases in other materials systems, such as two-dimensional materials and heterostructures. The study represents a significant step towards realising the full potential of topological materials and harnessing their unique properties for technological applications.