Square-octagon Lattices Exhibit Quantum Hall Crossovers with Chern Numbers 1 and 2, Enabling Topological Insulators

The pursuit of materials exhibiting both strong electron interactions and unusual quantum properties has led researchers to investigate lattices with uniquely flat energy bands, which can host exotic states of matter. Amrita Mukherjee, Rahul Verma, and Pritesh Srivastava, alongside Bahadur Singh, all from the Tata Institute of Fundamental Research, now demonstrate how the square-octagon lattice responds to the influence of spin-orbit coupling and magnetic fields. Their work reveals a surprising interplay between different quantum Hall effects and the emergence of topologically protected states, including those carrying quantized charges at the material’s corners. Importantly, the team identifies several real-world materials, such as octagraphene and specific metal dichalcogenides, that could potentially exhibit these properties, paving the way for new devices based on correlated topological matter and offering a pathway towards realising fractional Chern insulator states.

Coexistence of nontrivial topology and flat electronic bands within low-energy lattices creates fertile ground for exploring correlated quantum states. The square-octagon lattice, possessing both Dirac nodes and flat bands at half-filling, provides an ideal setting for investigating novel quantum phenomena. Researchers thoroughly examine the electronic structure of this lattice, focusing on how topological properties and flat bands interact to influence electron behaviour. The study details how specific geometric arrangements within the lattice give rise to these flat bands, localizing electrons and enhancing their interactions. Furthermore, the work explores how electrons respond to strong magnetic fields, revealing transitions to quantum Hall states and the potential for observing exotic topological phases of matter.

Topological Phases in Square-Octagon Lattices

This study investigates topological and flat-band physics within the square-octagon lattice, employing computational models to explore the influence of spin-orbit coupling and staggered magnetic flux. Researchers systematically varied parameters within these models to map a rich landscape of topological phases, including quantum spin Hall, anomalous Hall, and higher-order topological insulator states. To characterize these phases, the team calculated band structures and identified key topological invariants, such as spin Chern numbers and the Z2 topological index. They demonstrated the emergence of a quantum spin Hall phase with a spin Chern number of one, alongside anomalous Hall phases exhibiting Chern numbers of one and two.

Introducing staggered magnetic flux lifted the degeneracy of electron states and induced these anomalous Hall phases. They also investigated higher-order topological insulator phases, identifying quantized quadrupolar corner charges as a defining characteristic. The team meticulously tuned the intercell hopping parameter to drive transitions between these phases, observing the evolution of chiral edge modes and floating edge bands. Calculations revealed that manipulating these parameters could yield quasi-flat bands with remarkably high flatness ratios, reaching approximately 22 for specific configurations, creating conditions conducive to fractional Chern insulator states.

Beyond theoretical modelling, the study examined several material candidates for realizing these phenomena. Researchers analyzed the electronic structure of octagraphene, revealing metallic behaviour dominated by carbon pz orbitals. They also investigated monolayer MoS2, finding coexisting Dirac-like bands and a quantum spin Hall state near the Fermi level. Synthetic MoSi2N4 monolayers were designed and studied, demonstrating tunable topological phases through chemical substitution of nitrogen and arsenic. Finally, the team considered magnetic compounds with square-octagonal frameworks, such as hollandites, identifying partially flat bands and potential for spin-liquid-like behaviour. These material investigations, combined with the theoretical modelling, establish a pathway toward realizing and manipulating correlated topological matter.

Spin-Orbit Coupling Drives Topological Insulator Transition

This research addresses the largely unexplored influence of intrinsic spin-orbit coupling (SOC) and staggered magnetic flux on the topological and flat-band properties of this lattice. Researchers examine this lattice using computational models and reciprocal space analysis to reveal a rich phase diagram governed by the interplay between SOC, magnetic flux, and filling fraction. The results demonstrate that the system transitions from a conventional insulator to a topological insulator as the strength of spin-orbit coupling increases, with a band inversion occurring at the Dirac point. Specifically, the band gap closes and reopens, leading to the formation of topologically protected edge states. Furthermore, the introduction of staggered magnetic flux induces quantum Hall-like behaviour even in the absence of an external magnetic field, with quantized Hall conductivity values observed at fractional filling fractions. The analysis indicates that these fractional quantum Hall states are associated with the formation of composite fermions, arising from the strong correlations between electrons in the flat bands.

Lattice Tuning Reveals Quantum Hall Phases

This research demonstrates a rich variety of topological phases within the two-dimensional square-octagon lattice, a structure notable for hosting both Dirac states and nontrivial flat bands. The team established that incorporating spin-orbit coupling induces a quantum spin Hall phase, characterized by a spin Chern number of one. Further manipulation through staggered magnetic flux introduces quantum anomalous Hall phases with Chern numbers of one and two, breaking time-reversal symmetry and lifting the degeneracy of electronic states. Systematic adjustment of the lattice’s intercell hopping parameter drives transitions between these quantum anomalous Hall phases, exhibiting chiral edge modes, and higher-order topological insulator phases, distinguished by floating edge bands and quantized quadrupolar charge.

Importantly, the inclusion of these factors renders the flat bands quasi-flat and topologically nontrivial, achieving remarkably high flatness ratios conducive to the formation of fractional Chern insulator states. The researchers also identified several realistic material candidates, including octagraphene and molybdenum dichalcogenides, as potential platforms for realizing these diverse topological phases. Unlike similar lattices, the flat bands in the square-octagon system remain pinned at the Dirac point, enhancing their robustness and suitability for interaction-driven topological phenomena.