Geometric classification of non-Hermitian topological systems through the singularity ring

This work unveils how geometric features of two-band non-Hermitian Hamiltonians can completely classify the topology of their eigenstates and energy manifolds. Our approach generalizes the Bloch sphere visualization of Hermitian systems to a ``Bloch torus'' picture for non-Hermitian systems, where a singularity ring (SR) captures the degeneracy structure of generic exceptional points. The SR picture affords convenient visualization of various symmetry constraints and reduces their topological characterization to the classification of simple intersection or winding behavior, as detailed by our explicit study of chiral, sublattice, particle-hole and conjugated particle-hole symmetries. In 1D, the winding number about the SR corresponds to the band vorticity measurable through the Berry phase. In 2D, more complicated winding behavior leads to a variety of phases that illustrate the richness of the interplay between SR topology and geometry beyond mere Chern number classification. Through a normalization procedure that puts generic 2-band non-Hermitian Hamiltonians on equal footing, our SR approach also allows for vivid visualization of the non-Hermitian skin effect.Comment: 17 pages, 15 figures, Comments are welcom

Hybrid higher-order skin-topological modes in non-reciprocal systems

Higher-order phases are characterized by corner or hinge modes that arise due to the interesting interplay of localization mechanisms along two or more dimensions. In this work, we introduce and construct a novel class of "hybrid" higher-order skin-topological boundary modes in non-reciprocal systems with two or more open boundaries. Their existence crucially relies on non-reciprocal pumping in addition to topological localization. Unlike usual non-Hermitian "skin" modes, they can exist in lattices with \emph{vanishing} net reciprocity due to the selective nature of non-reciprocal pumping: While the bulk modes remain extended due to the cancellation of non-reciprocity within each unit cell, boundary modes experience a curious \emph{spontaneous breaking} of reciprocity in the presence of topological localization, thereby experiencing the non-Hermitian skin effect. The number of possible hybridization channels increases rapidly with dimensionality, leading to a proliferation of distinct phases. In addition, skin modes or hybrid skin-topological modes can restore unitarity and are hence stable, allowing for experimental observations and manipulations in non-Hermitian photonic and electrical metamaterials.Comment: 13 pages, 12 figure

Dilemmatic Deliberations In Kierkegaard’s Fear and Trembling

My central claim in this paper is that Kierkegaard’s Fear and Trembling is governed by the basic aim to articulate a real dilemma, and to elicit its proper recognition as such. I begin by indicating how Kierkegaard’s works are shaped in general by this aim, and what the aim involves. I then show how the dilemmaticstructure of Fear and Trembling is obscured in a recent dispute between Michelle Kosch and John Lippitt regarding the basic aims and upshot of the book. Finally, I consider two critical questions: Why does Kierkegaard present his dilemmatic reasoning in the form of a “dialectical lyric”? And why does he write a book that aims only to articulate a dilemma, and not also to resolve it