Topological Band Theory for Non-Hermitian Hamiltonians (1706.07435v2)

Abstract: We develop the topological band theory for systems described by non-Hermitian Hamiltonians, whose energy spectra are generally complex. After generalizing the notion of gapped band structures to the non-Hermitian case, we classify "gapped" bands in one and two dimensions by explicitly finding their topological invariants. We find nontrivial generalizations of the Chern number in two dimensions, and a new classification in one dimension, whose topology is determined by the energy dispersion rather than the energy eigenstates. We then study the bulk-edge correspondence and the topological phase transition in two dimensions. Different from the Hermitian case, the transition generically involves an extended intermediate phase with complex-energy band degeneracies at isolated "exceptional points" in momentum space. We also systematically classify all types of band degeneracies.

• The paper establishes a novel framework for classifying non-Hermitian systems by introducing topological invariants based on energy vorticity.
• The paper demonstrates that 2D non-Hermitian systems retain a Chern number structure while revealing unique edge state behaviors.
• The paper identifies exceptional points as key markers of phase transitions, reinforcing the bulk-edge correspondence in complex spectra.

Topological Band Theory for Non-Hermitian Hamiltonians

This paper investigates the extension of topological band theory to systems governed by non-Hermitian Hamiltonians. Traditional studies in topological phases have largely been confined to Hermitian systems, where the eigenvalues are real. In contrast, non-Hermitian systems exhibit complex spectra, necessitating a novel approach to their topological classification. The authors of this paper systematically develop a framework to understand and characterize the topological aspects of non-Hermitian systems.

Generalization of Topological Concepts

The authors begin by generalizing the notion of band gaps to systems with non-Hermitian Hamiltonians, allowing for energy dispersion in complex spaces. Two-dimensional (2D) systems exhibit nontrivial generalizations of the Chern number, while a unique one-dimensional (1D) classification based on energy dispersion is proposed. Specifically, non-Hermitian systems allow for the definition of a new topological invariant linked to the vorticity of energy eigenvalues, a feature absent in Hermitian cases. This vorticity, quantified by winding numbers, characterizes band degeneracies and captures the phenomena unique to non-Hermitian physics, such as exceptional points, where eigenstates coalesce.

Significant Findings and Implications

The paper identifies several key findings:

• Chern Numbers for Non-Hermitian Bands: In 2D non-Hermitian systems, a uniform Chern number, computable from different Berry curvatures, still dictates the topology, maintaining structural parallels to Hermitian systems while highlighting discrepancies due to non-Hermitian effects.
• Bulk-Edge Correspondence: Despite the differences in underlying mathematics, the bulk-edge correspondence principle, a cornerstone in topological band theory, remains intact. Protected edge states emerge at interfaces between topologically distinct non-Hermitian regimes, with trajectories in complex energy space that link two bulk bands.
• Exceptional Points: The manuscript identifies non-Hermitian exceptional points as robust entities within the band structure. These points serve as transition markers between topological phases, driven by the complex nature of spectral degeneracies.

Theoretical and Practical Implications

From a theoretical perspective, this work opens new vistas in the paper of complex eigenvalue problems, showing that band topology extends beyond the confines of Hermitian boundaries. The implications of their findings impact a range of systems—quantum and classical alike—including photonic structures and optical lattices, often characterized by gain and loss or open boundary conditions. The introduction of unique topological markers, such as exceptional points, suggests rich avenues for technological advancement in non-reciprocal devices and other applications reliant on non-Hermitian dynamics.

Speculation on Future Directions

The implications for future research are manifold. Potential initiatives include delving deeper into the interplay between symmetries and topological features in higher dimensions, expanding the classification schemes tailored to various symmetry classes. Furthermore, as technological advances channel the abstract into the attainable, non-Hermitian systems may be engineered to harness the topological properties elucidated in this work, paving the way for innovations in information processing, sensor technologies, and possibly quantum computing.

In conclusion, this paper contributes significantly to the comprehension of non-Hermitian systems through the extension of topological band theory, elucidating both foundational concepts and new phenomena such as exceptional points. The paper's insights offer a valuable framework for both theorists and experimentalists exploring the burgeoning field of non-Hermitian physics.