Electrons Bind to Form Stable Insulator Building Blocks

Researchers are advancing the understanding of fractional Chern insulators with a novel theoretical framework centred on composite bosons. Guangyu Yu from the Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, and Zheng Zhu, also of the Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, present a real-space approach defining FCIs through locally bound electron-orbital composites. This work establishes a crucial link between two-body interactions and stable FCI states, validated through numerical evidence using the Haldane model. Significantly, this research bridges the gap between continuum and lattice descriptions of the fractional Hall effect, offering a unified and intuitive interpretation of correlated phases and a new method for identifying and designing both Abelian and non-Abelian topologically ordered phases in diverse material systems.

Scientists have unlocked a deeper understanding of exotic materials that could revolutionise future electronics. Their work offers a new way to visualise and control fractional Chern insulators, substances with the potential for lossless energy transfer. This breakthrough bridges fundamental physics with the practical design of next-generation devices.

Scientists have developed a new theoretical framework for understanding fractional Chern insulators, exotic materials that mimic the behaviour of the fractional quantum Hall effect but within a solid lattice structure. This work introduces a real-space approach, interpreting these insulators not through traditional band topology but as assemblies of “composite bosons”, effectively, electrons bound to their surrounding energetically excluded orbitals.

The central innovation lies in constructing a radially ordered set of maximally localised orbitals, a basis that doesn’t require the continuous rotational symmetry often assumed in previous models. Within this framework, a stable fractional Chern insulator emerges when the orbitals excluded around central electrons maximise two-body interaction energy, providing a surprisingly simple organising principle for complex many-body physics.

This research validates the composite boson concept with direct numerical evidence obtained from the Haldane model, a well-established system for studying topological phases. Demonstrating that the proposed criterion accurately characterises these insulators, the analysis reveals a connection between the fractional Hall effect observed in continuum systems and its lattice-based counterpart.

This unification offers an intuitive real-space interpretation for distinct correlated phases, establishing a foundation for both diagnosing and designing topologically ordered phases, including both Abelian and non-Abelian varieties, across diverse material platforms. The conceptual workflow centres on constructing a radially ordered set of maximally localised orbitals within a topological flat band, followed by evaluating the energetic stability of composite bosons formed by electrons and their excluded orbitals.

The study moves beyond conventional approaches that prioritise band properties and often assume specific interaction forms, such as Coulomb interactions. By focusing on the intrinsic properties of the composite bosons, the researchers circumvent limitations inherent in embedding-dependent quantities, revealing the essential origin of fractional Chern insulators.

Specifically, the framework postulates that a stable FCI state arises from the Bose-Einstein condensation of these composite bosons, governed by the energetic favourability of individual boson configurations. This is determined by assessing the two-body interaction energy between electrons in central and surrounding orbitals, offering a transparent criterion for predicting FCI emergence under generic interactions. The Haldane model calculations confirm this prediction, providing direct evidence of composite boson formation and FCI emergence, and highlighting a unified organising principle applicable across different Landau levels.

Radially localised orbitals reveal electron exclusion in fractional Chern insulators

Within the Haldane model, a lattice system known to host fractional Chern insulators, a radially ordered, maximally localised basis was constructed to reveal the internal structure of these exotic phases. Diagonalisation of the band projection operator, followed by spatial localisation and subsequent diagonalisation of the squared position operator, yielded orbitals exhibiting the desired radial ordering.

This process effectively sorted orbitals by their distance from a chosen origin, establishing a clear hierarchy within the topologically flat band. Analysis of the occupation correlation, ⟨Ψ|PnP0|Ψ⟩, directly revealed the exclusion patterns surrounding central electrons. Specifically, calculations demonstrated that for a filling fraction of one-third, the occupation correlation between the central orbital and the first two surrounding orbitals exhibited a marked suppression.

This indicates that these orbitals remain largely unoccupied when a central orbital is filled, consistent with the formation of composite bosons. The flatness ratio of the band exceeded 50, confirming the dominance of interaction effects crucial for FCI behaviour. This high flatness ratio was achieved using parameters t1 = −1, t2 = −0.6 exp(i0.4π), and t3 = 0.58 within the Haldane Hamiltonian.

Further refinement of the basis involved restricting the Hilbert space to a finite circular region, primarily affecting orbitals near the boundary while preserving the accuracy of wavefunctions near the centre. Diagonalisation of the matrix Smn = ⟨m| P (n)|n⟩, and subsequent recombination into the maximally localised basis, ensured spatial resolution of the orbitals.

The resulting eigenvectors, denoted |ψλ⟩= 1 √ λ P m w(λ) m | m⟩, were demonstrably localised around the origin, validating the effectiveness of the real-space localisation procedure. This framework establishes a direct link between the exclusion patterns and the formation of stable composite bosons, offering a novel real-space interpretation of FCI physics.

Real-space localisation of orbitals defines fractional Chern insulator states

A radially ordered set of maximally localised orbitals forms the foundation of this work’s theoretical framework for understanding fractional Chern insulators. Beginning with a topologically flat band, the research team constructed an orthogonal basis by diagonalising the band projection operator, identifying eigenvectors associated with nonzero eigenvalues to span the band.

To establish a real-space hierarchy within this basis, a lattice analogue of the squared position operator was employed, measuring the mean squared distance of each orbital from a chosen origin. Diagonalising this operator automatically sorted the orbitals by distance, yielding the desired radially ordered and maximally localised basis set. This methodology diverges from conventional approaches that prioritise band properties by focusing instead on the real-space arrangement of electrons and their interactions.

Within this framework, the study postulates that fractional Chern insulator states arise from the condensation of composite bosons, which are formed when electrons bind to their energetically excluded surrounding orbitals. For example, in a ν = 1/3 state, occupation of a central orbital necessitates the exclusion of the subsequent two orbitals, effectively attaching them to the central electron and creating a composite boson.

The stability of these composite bosons, and thus the FCI state, is determined by a simple energy criterion. The two-body interaction energy between an electron in the central orbital and a test electron in each surrounding orbital was calculated, with stable composite bosons defined by maximising this interaction energy in the excluded orbitals. This approach offers a novel way to evaluate and compare the energy of different phases, and to predict the emergence of both Abelian and non-Abelian topologically ordered phases. The Haldane model was used to validate this framework, providing direct numerical evidence of composite boson formation and FCI emergence.

Composite bosons unlock a real-space understanding of fractional Chern insulators

Scientists have long sought a unifying framework to understand the exotic behaviour of fractional Chern insulators, materials poised to revolutionise areas like quantum computing and materials science. This work represents a significant step forward by shifting the focus from abstract band topology to a more intuitive, real-space picture of interacting electrons.

For decades, the challenge has been to reconcile the theoretical elegance of fractional quantum Hall physics, traditionally described in continuous systems, with the discrete, lattice-based structures found in real materials. This new approach, framing FCIs in terms of ‘composite bosons’, essentially electrons bundled with their surrounding environment, offers a potential bridge between these worlds.

The power of this framework lies in its simplicity and predictive capability. By identifying stable FCIs as those where electrons maximise interactions with their immediate surroundings, researchers provide a clear criterion for both diagnosing existing materials and designing novel ones. This isn’t merely an academic exercise; it opens avenues for tailoring materials with specific topological properties, crucial for building robust quantum devices less susceptible to environmental noise.

However, the real test will be extending this real-space understanding to more complex, less symmetrical systems. The current validation relies on the relatively simple Haldane model, and applying this criterion to materials with strong disorder or anisotropic interactions will undoubtedly present challenges. Furthermore, while the framework elegantly describes Abelian FCIs, its applicability to the more sought-after non-Abelian phases, those with even greater potential for quantum information processing, remains an open question. Future research will likely focus on refining the composite boson picture to encompass these more intricate states, and on developing computational tools to efficiently screen potential materials using these new criteria.