Non-Hermitian Fermion Models Generated from Unidirectional Hopping and Chiral Symmetry

The behaviour of electrons in unusual materials often defies conventional physics, and researchers are increasingly exploring non-Hermitian systems to explain these phenomena. Edward McCann from Lancaster University, along with colleagues, investigates how introducing directed, or ‘unidirectional’, movement of electrons fundamentally alters their energy levels and creates novel states of matter. The team demonstrates a surprising connection between these non-Hermitian systems and exotic particles called parafermions, revealing that the energy levels within a well-known model, the Su-Schrieffer-Heeger model, precisely match those predicted for free parafermions. This discovery provides a new framework for understanding and potentially harnessing the unique properties of non-Hermitian systems, opening doors to advancements in areas like topological electronics and quantum computation, as the research highlights the emergence of exceptional points and unusual edge states within these materials.

Topological and Non-Hermitian Condensed Matter Systems

This extensive collection of references details research into topological physics, non-Hermitian physics, graphene, carbon nanotubes, and related materials. The work focuses on understanding the fundamental properties of these systems and exploring their potential applications, including topological insulators and semimetals, electron behaviour, and the role of symmetry in protecting unique quantum states. The research also delves into non-Hermitian physics, extending traditional quantum mechanics by considering systems where energy is not conserved, leading to complex energy spectra and unusual behaviours. Graphene and carbon nanotubes receive significant attention, with studies exploring their electronic properties, topological phases, and potential for novel devices.

The bibliography covers a range of condensed matter systems, investigating the properties of flat bands and their connection to topological states. The collection is organized around theoretical foundations, including the Berry phase and topology, and the formalisms used to describe non-Hermitian systems. It also covers specific materials, such as graphene and carbon nanotubes, and explores various condensed matter systems, including the Su-Schrieffer-Heeger model. A significant portion of the work focuses on square-root topological insulators and other topological materials with unique properties, alongside experimental studies focused on material growth and fabrication. This research demonstrates an interdisciplinary approach, combining concepts from condensed matter physics, topology, and materials science. The focus on emerging areas like square-root topological insulators and non-Hermitian physics indicates that this work is at the forefront of current condensed matter physics, balancing fundamental understanding with the development of new materials and technologies.

Creating Non-Hermitian Systems from Hermitian Parents

Researchers have developed a method for exploring non-Hermitian physics by constructing models from existing, well-understood Hermitian systems. This approach allows for a systematic investigation of how deviations from standard quantum mechanics emerge and influence material properties. The core of the method involves introducing unidirectional hopping, a directed movement of electrons, fundamentally altering the energy landscape and creating complex energy bands. The team began with a bipartite lattice and a Hermitian tight-binding model describing electron behaviour. By adding orbitals and implementing unidirectional hopping, they generated a non-Hermitian Hamiltonian, carefully controlling the degree of unidirectionality to observe a continuous transition between bidirectional and complete unidirectional hopping.

The resulting models exhibit complex energy bands, meaning the energy values have both real and imaginary components, a hallmark of non-Hermitian systems. A key innovation lies in constructing these non-Hermitian Hamiltonians as generalized permutation matrices, simplifying analysis and revealing underlying symmetries, particularly a complex generalization of chiral symmetry. The energy bands of the daughter model are directly related to those of the parent model, scaled by complex numbers, providing a clear link between the two. This allows them to predict and control the emergence of unusual phenomena, such as exceptional points, singularities in the energy spectrum. For the Su-Schrieffer-Heeger model, the researchers found a connection to solutions in Baxter’s non-Hermitian clock model, revealing a deep mathematical relationship between seemingly disparate physical systems. They also extended this approach to graphene, demonstrating that unidirectional hopping transforms bilayer graphene into a square root Hamiltonian of monolayer graphene, suggesting potential pathways for manipulating electronic properties.

Non-Hermitian Models from Hermitian Symmetry Breaking

Researchers have developed a method for generating non-Hermitian models, systems that extend traditional quantum mechanics, starting from well-understood Hermitian systems with specific symmetries. This approach allows them to create complex energy landscapes and explore novel quantum phenomena. The core idea involves introducing a degree of “unidirectionality” into the hopping of electrons within the material, effectively controlling how electrons move between atoms, transforming energy levels into complex numbers. The team demonstrated this process using the Su-Schrieffer-Heeger model, finding a surprising connection to solutions found in the Baxter clock model.

Furthermore, they showed that increasing the unidirectionality in the system leads to a predictable evolution of the energy spectrum, transitioning from real energies to complex energies as the system becomes increasingly non-Hermitian, linked to the topology of the original material and the number of orbitals considered. Expanding on this, the researchers applied their method to graphene, discovering that introducing unidirectionality effectively creates a “square root” model of graphene. Near the Dirac point, this process leads to the emergence of exceptional points, locations where standard quantum mechanical rules break down. Electrons traversing a loop around these exceptional points acquire a specific phase shift, known as a Berry phase, equal to π, a value that is distinct and robust, potentially harnessed for novel quantum technologies.

Non-Hermitian Physics From Tight-Binding Models

This work demonstrates a method for generating non-Hermitian models from Hermitian tight-binding models, specifically focusing on systems with multiple orbitals per unit cell and unidirectional hopping. The researchers show that by introducing this unidirectional hopping, the energy bands of the original model evolve into complex forms, dictated by the topology of the parent system and the number of orbitals present. When the initial model is the Su-Schrieffer-Heeger model, the resulting energy levels correspond to those found in solutions of Baxter’s non-Hermitian clock model. The study reveals that these non-Hermitian models exhibit exceptional points, singularities in the energy spectrum, which appear at edge states and solitons in the SSH model and at the Dirac point of graphene-like systems. The researchers also demonstrate that fully unidirectional hopping can effectively create nth root models of the original system, offering a pathway to engineer materials with tailored electronic properties. The models consistently respect time-reversal symmetry, ensuring energy spectra are either real or appear in complex-conjugate pairs, and for an even number of orbitals, also exhibit sub-lattice symmetry.