Continuum Limit of Bipartite Lattices Replicates SSH Model’s Spectral Features and Zak Phase

The behaviour of electrons in certain materials hinges on the precise arrangement of their constituent atoms, and understanding these arrangements is crucial for designing new technologies, particularly in the field of topological matter. Fotios K. Diakonos from the National and Kapodistrian University of Athens, along with P. Schmelcher, now presents a new theoretical model that accurately mimics the complex behaviour of the well-known Su-Schrieffer-Heeger model, a cornerstone for understanding topological insulators. This achievement lies in creating a continuous, non-local model that replicates key features such as the energy spectrum and topological properties of the original, discrete model, while also introducing a way to control the range of interactions between electrons. Importantly, this new approach preserves crucial symmetries absent in previous continuous models and predicts the existence of unique, localized energy states, paving the way for exploring a broader range of complex materials and potentially inspiring novel experimental designs.

SSH Model and Topological Edge States

This research comprehensively examines the Su-Schrieffer-Heeger (SSH) model, its inherent topological properties, and extensions to continuous systems. The study explores the fundamental concepts underpinning topological insulators, materials that conduct electricity on their surfaces but remain insulating within their bulk, emphasizing the importance of topological invariants, mathematical quantities that characterize these phases and protect edge states from disruption. The research investigates extending the discrete SSH model to continuous systems, crucial for accurately representing real materials which are not perfectly structured lattices. The team surveyed a variety of approaches to creating continuous analogues of the SSH model, including using functional integrals, exploring curved geometries to induce long-range interactions, and methods to simplify the system while retaining key topological features. The research also discusses potential experimental realizations of these models using ultracold atoms, water waves, photonic crystals, and mechanical systems. This work contributes to a deeper understanding of topological phases of matter and the conditions under which they can exist in both discrete and continuous systems, guiding the design of new materials with tailored topological properties for applications in quantum computing, spintronics, and energy harvesting.

Continuous Non-Locality Recreates SSH Model Topology

Scientists developed a continuous non-local model that faithfully replicates the topological and spectral features of the Su-Schrieffer-Heeger (SSH) model, a cornerstone for understanding topological properties in solid-state systems. This work addresses the challenge of extending discrete models like SSH to continuous frameworks, enabling exploration of topological phenomena in diverse systems such as photonic crystals and acoustic metamaterials. The team engineered a model incorporating a tunable parameter, which quantifies the degree of non-locality and allows controlled interpolation between non-local and local regimes. Crucially, for a specific value of this parameter, the model establishes exact spectral equivalence to the discrete SSH model, a significant advancement over previous continuous analogues.

This research diverges from earlier approaches based on Schrödinger or Dirac-type Hamiltonians, which typically require external potentials or exhibit discrepancies in the bulk energy spectrum. The team’s model maintains chiral symmetry without needing an external potential and features periodic energy bands, mirroring the discrete SSH model more accurately. Experiments employing finite domains demonstrate the model supports a flat band with zero energy, formed by a countable infinite set of exponentially localized zero-energy edge states of topological origin. This achievement establishes a clear one-to-one relationship between the discrete SSH model and its continuous counterpart, both in spectral and topological properties. The study pioneered a method for constructing non-local, continuous analogues of bipartite and multipartite lattices, extending beyond the SSH model itself.

Tunable Non-Locality Recreates SSH Model Spectrum

Scientists have developed a continuous non-local model that faithfully replicates the topological and spectral features of the well-known Su-Schrieffer-Heeger (SSH) model, a cornerstone in the study of topological materials. This new model shares the SSH model’s bulk energy spectrum, eigenstates, and Zak phase, all hallmarks of its topological character, while introducing a tunable length-scale, which quantifies the degree of non-locality. The team demonstrated that by adjusting this parameter, they could smoothly interpolate between non-local and local regimes, offering unprecedented control over the system’s behavior. Remarkably, the researchers established that for a specific value of this parameter, the model achieves exact spectral equivalence to the discrete SSH model.

Experiments revealed that the continuous model supports a flat band with zero energy, formed by a countable infinite set of exponentially localized zero-energy edge states of topological origin. The data shows that, unlike the discrete SSH lattice which exhibits only two such states, the continuous model accumulates an infinite number of states near zero energy. Numerical results confirm that this flat miniband lies exactly at zero energy, a distinct feature from the discrete SSH lattice where states near the midgap possess energies that are approximately, but not exactly, zero. The team observed that the model possesses an entire flat mini band at zero energy in its finite-size version, with corresponding eigenstates localized at the left and right edges of the system.

Further analysis demonstrated the existence of an infinite, countable set of eigenspinors classified by integer numbers, each component exhibiting an exponential envelope leading to localization at the system edges. Measurements confirm that the localization length is controlled by the parameter, directly linking the spatial scale to the non-locality of the system. The researchers established that the model preserves exact chiral symmetry, unlike other continuous analogues based on Schrödinger-type Hamiltonians, and exhibits periodic energy bands, a feature absent in continuous SSH analogues derived from Dirac-type Hamiltonians. This breakthrough delivers a new platform for exploring topological phenomena and opens avenues for designing materials with tailored properties.

Continuous Model Replicates SSH Topology and Chirality

This research presents a continuous model that accurately replicates the topological properties of the well-known Su-Schrieffer-Heeger (SSH) model, a cornerstone in understanding topological insulators. The team successfully created a model sharing the SSH model’s key characteristics, its energy spectrum, the behaviour of its electronic states, and a quantity called the Zak phase which defines its topological nature, while introducing a parameter that controls the degree of non-locality within the system. Notably, the model achieves complete spectral equivalence to the discrete SSH model for a specific value of this parameter, demonstrating a powerful connection between continuous and discrete topological systems. Distinct from previous continuous analogues, this model maintains chiral symmetry, eliminating the need for external potentials and exhibiting periodic energy bands. Investigations of the model on finite systems reveal a unique flat band of zero-energy states, consisting of an infinite set of exponentially localized edge states with topological protection. This achievement extends beyond simply replicating the SSH model, laying the groundwork for constructing continuous analogues of a broader range of complex lattices.