New topological invariants in non-Hermitian systems (1902.07972v4)

Abstract: Both theoretical and experimental studies of topological phases in non-Hermitian systems have made a remarkable progress in the last few years of research. In this article, we review the key concepts pertaining to topological phases in non-Hermitian Hamiltonians with relevant examples and realistic model setups. Discussions are devoted to both the adaptations of topological invariants from Hermitian to non-Hermitian systems, as well as origins of new topological invariants in the latter setup. Unique properties such as exceptional points and complex energy landscapes lead to new topological invariants including winding number/vorticity defined solely in the complex energy plane, and half-integer winding/Chern numbers. New forms of Kramers degeneracy appear here rendering distinct topological invariants. Modifications of adiabatic theory, time-evolution operator, biorthogonal bulk-boundary correspondence lead to unique features such as topological displacement of particles, `skin-effect', and edge-selective attenuated and amplified topological polarizations without chiral symmetry. Extension and realization of topological ideas in photonic systems are mentioned. We conclude with discussions on relevant future directions, and highlight potential applications of some of these unique topological features of the non-Hermitian Hamiltonians.

• The paper presents the main contribution by extending topological invariants to non-Hermitian systems through the definition of winding numbers in the complex energy plane and half-integer Chern numbers.
• It analyzes exceptional points and non-diagonalizable Hamiltonians to adapt classical Berry phase and Chern number concepts into complex and, under PT-symmetry, real quantized forms.
• The study underscores practical implications for photonics and quantum optics, where robust edge-state transport and directional amplification open avenues for innovative technologies.

New Topological Invariants in Non-Hermitian Systems

The paper under review presents an in-depth analysis of the emerging field of topological phases in non-Hermitian (NH) systems. The authors, Ananya Ghatak and Tanmoy Das, offer a comprehensive overview of how topological invariants, traditionally studied within the Hermitian framework, can be adapted and extended to NH systems. They explore the unique properties of NH Hamiltonians, such as exceptional points and complex energy landscapes, which give rise to novel topological invariants, including winding numbers defined in the complex energy plane and half-integer Chern numbers.

Overview of NH Systems in Quantum Mechanics

NH Hamiltonians, which often arise in open, dissipative systems like quantum optics, differ fundamentally from Hermitian Hamiltonians, primarily because they yield complex eigenvalues. This complexity allows for phenomena absent in Hermitian systems, such as exceptional points (EPs), where eigenstates coalesce, leading to non-diagonalizable Hamiltonians. Historically, NH systems have been associated with non-conserved probabilities and complex eigenvalues, but recent studies have demonstrated that they can also exhibit real energy spectra and conserved laws under certain conditions, such as PT-symmetry or pseudo-Hermitian operators.

Topological Invariants in NH Systems

The paper identifies ways in which topological invariants known from Hermitian systems, such as the Chern number and winding number, manifest in NH settings. In particular, NH Hamiltonians import complexity into previously straightforward calculations, resulting in new topological phenomena:

• Winding Numbers in Complex Energy Planes: The presence of EPs allows for the definition of winding numbers in the complex energy plane, distinct from those defined by the behavior of eigenstates. This vorticity counts the number of EPs encircled by a path in the energy plane, offering a new invariant distinct from conventional Hermitian systems.
• Complex Berry Phases and Chern Numbers: The adaptation of these invariants in NH systems frequently involves complex-valued Berry connections and curvatures. However, in pseudo-Hermitian or PT-symmetric systems, these quantities can become purely real and quantized, maintaining their role as topological invariants.

Practical and Theoretical Implications

The paper of NH topological phases holds substantial promise for both theoretical exploration and practical application. NH systems, particularly those with PT-symmetry, are pertinent to fields like photonics and quantum optics, where they have already led to the realization of novel states not possible in Hermitian lattices. Photonic systems, for instance, benefit from NH dynamics by enabling robust edge-state transport and topological protection, overcoming traditional limitations associated with gain and loss.

Speculations for Future Developments

Looking forward, the insights provided by NH topological phases may inspire new technologies in communication and computation, particularly in systems that operationalize the unique properties of NH dynamics, such as directional amplification. Furthermore, the realization of these phases in experimental settings beyond photonic lattices, including condensed matter systems, remains a vibrant avenue for exploration, with potential applications in areas like energy-efficient devices and novel quantum technologies.

In conclusion, the paper by Ghatak and Das paves the way for a deeper understanding of topological invariants in NH systems. By extending the concept of topology into the NH domain, the authors demonstrate new avenues for research that challenge existing paradigms in both mathematics and physics, fostering advancements in various technological fields.