Flat Bands in Novel Lattice Confine Electrons and Resist Disorder

Researchers are increasingly focused on harnessing topological phenomena in materials to create novel electronic devices. Joydeep Majhi from the AbCMS Lab, Department of Metallurgical Engineering and Materials Science at the Indian Institute of Technology Bombay, and Biplab Pal from the Department of Physics, School of Sciences, Nagaland University, demonstrate a new two-dimensional model based on a diamond-dodecagon lattice that exhibits both compact localised states and tunable topological phase transitions. This collaborative work reveals that the unique geometry of this lattice supports multiple flat bands, robust against disorder, and allows for control of topological properties via applied magnetic flux. The resulting interplay between localisation, topology, and transport characteristics establishes the diamond-dodecagon lattice as a promising platform for exploring flat-band physics and potentially realising advanced photonic or ultracold atom-based technologies.

This innovative material promises a new platform for exploring fundamental physics and potentially enabling advanced quantum technologies, demonstrating how manipulating magnetic flux through the lattice can precisely control the behaviour of electrons, leading to the emergence of states with unique characteristics.

Specifically, the lattice hosts three completely flat bands, where electrons exhibit zero velocity and infinite mass, arising from destructive interference in electron hopping pathways. These flat bands give rise to compact localized states, where electrons are confined to specific regions of the lattice due to its inherent geometrical frustration. Crucially, these localized states remain stable even with minor imperfections in the material, enhancing its practical viability.

By threading a uniform magnetic flux through the diamond-shaped components of the lattice, researchers have shown they can tune the electronic band structure and induce topological phase transitions, with certain bands developing nontrivial topological features quantified by integer values known as Chern numbers, indicating the presence of robust edge states. Multi-terminal transport calculations reveal flux-tunable resonances and suppression of transmission, directly linked to the flat bands and their topological characteristics. This interplay between localization, topology, and transport establishes the diamond-dodecagon lattice as a robust and tunable system for studying flat-band physics and magnetic flux-controlled phenomena, suggesting potential applications in photonic crystals and systems utilising ultracold atoms, opening avenues for realising and manipulating exotic quantum states in engineered materials.

Calculating electron transmission using the scattering matrix method

A scattering matrix formalism, implemented within the Kwant package, underpinned the multi-terminal transport calculations performed on this two-dimensional lattice system. This approach efficiently computes transmission probabilities for arbitrary tight-binding geometries connected to external leads, offering a robust method for analysing electron flow.

The scattering matrix, denoted as S, mathematically relates incoming and outgoing electron modes in the leads, allowing extraction of the total transmission probability between terminals. This computational technique is formally equivalent to the non-equilibrium Green’s function method, validated by the Fisher-Lee relation, which accurately accounts for the influence of the leads on the central scattering region via plane-wave superpositions.

Specifically, the total transmission probability, T(E), from a source lead ‘p’ to a drain lead ‘q’ at energy E is calculated by summing over all modes ‘a’ in ‘p’ and ‘b’ in ‘q’, using the scattering matrix element S(ba)pq. To determine the scattering matrix, the time-independent Schrödinger equation, Hfullψi = Eψi, was solved, where Hfull represents the full Hamiltonian of the system.

This Hamiltonian incorporates the scattering region, the lead unit cells, and the coupling matrices connecting them. The resulting wave function, ψi, describes the system’s state within the scattering region, seamlessly matched to propagating and evanescent modes in the leads. Calculations were performed on a finite-size 10×10 diamond-dodecagon lattice, with semi-infinite leads attached along the ±x directions. The transmission characteristics were then mapped as a function of magnetic flux, Φ, and electron energy, E, revealing a complex phase diagram exhibiting regions of complete transmission suppression alongside ballistic transmission, and re-entrant transitions between these states.

Flat bands and tunable topological states in a diamond-dodecagon lattice

The diamond-dodecagon lattice exhibits three completely flat bands at energies of -2, 0, and 2, arising from destructive interference in electron hopping processes. These flat bands are characterised by a complete lack of dispersion, meaning electrons possess zero group velocity and are effectively localised. Analytical construction of compact localised states confirms real-space confinement of electrons solely due to the lattice’s geometrical frustration.

The density of states analysis reveals a significant accumulation of states within these flat bands, indicating a high degree of electronic localisation. Introduction of a uniform magnetic flux through the diamond plaquettes demonstrably tunes the band structure, inducing topological features in certain bands. Calculations of the Chern number reveal non-zero integer values, specifically 1 and -1, confirming the topological nature of these bands.

This tunability allows for precise control over the electronic properties of the lattice through external magnetic fields. The system’s robustness against weak random onsite disorder, with flat bands remaining largely unaffected, highlights its potential for practical applications. Transmission probability calculations reveal distinct signatures of flat band transport, including sharp resonant features at energies corresponding to the flat bands.

Flux-dependent transmission suppression was also observed, with minima occurring at specific magnetic flux values, demonstrating a clear interplay between localisation, topology, and transport. These resonant features and suppression effects establish the potential of this model as a flux-tunable quantum transport device.

The Bigger Picture

Scientists are increasingly focused on harnessing the unusual properties of electrons confined to two dimensions, and this work presents a particularly compelling new geometry for doing so. The challenge has long been to engineer materials where electrons behave in highly predictable, even ‘flat’, ways, suppressing unwanted dispersion and unlocking exotic quantum phenomena.

This diamond-dodecagon lattice offers a route to achieving just that, exhibiting multiple flat bands that are remarkably resilient to imperfections. What distinguishes this approach is the inherent geometrical frustration within the lattice itself, creating localized electron states without needing complex external controls. This is a significant step forward because it simplifies the path towards realising these effects in real materials.

The ability to then tune the system’s properties via magnetic flux adds another layer of control, opening possibilities for designing novel electronic devices. The demonstrated interplay between localization, topology, and transport suggests potential applications in areas like low-power electronics and quantum information processing. Translating these theoretical findings into practical devices is not straightforward.

While the model demonstrates robustness against some disorder, the impact of more realistic imperfections remains an open question. Furthermore, fabricating such a precise lattice structure presents a substantial materials science hurdle. Future research will likely focus on identifying or creating physical systems, perhaps photonic lattices or ultracold atomic gases, that can accurately mimic this diamond-dodecagon geometry, and on exploring the limits of its tunability and stability. The broader effort to control and exploit flat bands is gaining momentum, and this work provides a valuable new design principle for the field.