Non-Hermitian topological phenomena: A review (2205.10379v1)

Abstract: The past decades have witnessed an explosion of interest in topological materials, and a lot of mathematical concepts have been introduced in condensed matter physics. Among them, the bulk-boundary correspondence is the central topic in topological physics, which has inspired researchers to focus on boundary physics. Recently, the concepts of topological phases have been extended to non-Hermitian Hamiltonians, whose eigenvalues can be complex. Besides the topology, non-Hermiticity can also cause a boundary phenomenon called the non-Hermitian skin effect, which is an extreme sensitivity of the spectrum to the boundary condition. In this article, we review developments in non-Hermitian topological physics by focusing mainly on the boundary problem. As well as the competition between non-Hermitian and topological boundary phenomena, we discuss the topological nature inherent in non-Hermiticity itself.

Non-Hermitian Topological Phenomena: A Technical Overview

The paper of non-Hermitian topological phenomena has rapidly developed, driven by the needs of various physical systems where gain and loss or other non-conservative forces are present. This paper, authored by Nobuyuki Okuma and Masatoshi Sato, offers a comprehensive review of advancements in non-Hermitian topological physics, presenting a critical analysis focused primarily on boundary phenomena, such as the bulk-boundary correspondence and the emergent non-Hermitian skin effect. This review is of particular interest to researchers familiar with conventional topological phases, offering an exploration into how these concepts are altered and extended in non-Hermitian regimes.

Non-Hermitian Topological Phases

Non-Hermitian Hamiltonians introduce complex eigenvalues, broadening the traditional understanding of topology. The paper analyses how typical bulk-boundary correspondence, a cornerstone of Hermitian topological phases, is significantly affected by non-Hermiticity. This correspondence, which correlates bulk topological invariants with boundary states, is shown to be disrupted by phenomena such as the non-Hermitian skin effect, where eigenstates at the boundaries display extreme sensitivity to boundary conditions.

Non-Hermitian Skin Effect

The non-Hermitian skin effect is a peculiar boundary localization of eigenstates, displacing the bulk-boundary correspondence by localizing bulk modes at the boundaries. This effect is exemplified in the analysis of the Hatano-Nelson model, where asymmetric hopping terms cause more than mathematical novelty—they underscore challenges in defining spectra under different boundary conditions. The paper crucially distinguishes conventional skin effects from symmetry-protected phenomena, establishing the relevance of symmetries and extending topological classification to non-Hermitian systems.

Mathematical Framework

Non-Hermitian matrices and their spectral properties form the mathematical foundation for examining these phenomena. The paper cites significant work from spectral theory, contributing to the understanding of non-Hermitian physical models. One of the central insights is recognizing that non-Hermitian systems do not solely rely on the traditional bulk-band theory; instead, generalized concepts such as the generalized Brillouin zone provide new interpretations and predictions.

Topological Implications and Symmetry

The paper dissects the implications of non-Hermiticity for boundary phenomena, arguing that non-Hermitian skin effects display topological characteristics akin to Hermitian topological phases. Symmetry remains crucial; for instance, transpose-type time-reversal symmetry introduces protection against non-Hermitian perturbations, and symmetry-protected skin effects become possible. Such results are vital for understanding the emergence of topological numbers that reflect these non-Hermitian properties.

Computational and Experimental Outlook

While the paper primarily presents theoretical frameworks, it also suggests implications for computational techniques and experimental approaches. Exploring non-Hermitian phases in higher-dimensional systems or in presence of topological defects navigates new territory for quantum materials and photonic systems. Critical computational methods are acknowledged for their ability to simulate systems where analytical solutions are not feasible.

Future Directions in Non-Hermitian Physics

The review proposes speculative directions for future research, indicating how foundational understanding in non-Hermitian topological phenomena could lead to advancements in sensor technology, topological quantum computing, and dynamic systems exhibiting complex behaviors. It is anticipated that continued exploration will further bridge traditional Hermitian physics with its non-Hermitian counterparts, unveiling new phases and technological potentials.

Overall, "Non-Hermitian topological phenomena: A review" provides an invaluable resource for researchers in condensed matter physics and related fields, laying the groundwork for understanding these advanced concepts and motivating further research into the intriguing properties of non-Hermitian systems.