We unveil the geometrical meaning of winding number and utilize it to characterize the topological phases in one-dimensional chiral non-Hermitian systems. While chiral symmetry ensures the winding number of Hermitian systems being integers, it can take half integers for non-Hermitian systems. We give a geometrical interpretation of the half integers by demonstrating that the winding number ν of a non-Hermitian system is equal to half of the summation of two winding numbers ν1 and ν2 associated with two exceptional points respectively. The winding numbers ν1 and ν2 represent the times of real part of the Hamiltonian in momentum space encircling the exceptional points and can only take integers. We further find that the difference of ν1 and ν2 is related to the second winding number or energy vorticity. By applying our scheme to a non-Hermitian Su-Schrieffer-Heeger model and an extended version of it, we show that the topologically different phases can be well characterized by winding numbers. Furthermore, we demonstrate that the existence of left and right zero-mode edge states is closely related to the winding number ν1 and ν2.

We study the extended Su-Schrieffer-Heeger model with both the nearest-neighbor and nextnearest-neighbor hopping strengths being cyclically modulated and find the family of the model system exhibiting topologically nontrivial phases, which can be characterized by a nonzero Chern number defined in a two-dimensional space spanned by the momentum and modulation parameter. It is interesting that the model has a similar phase diagram as the well-known Haldane's model. We propose to use photonic crystal systems as the idea systems to fabricate our model systems and probe their topological properties. Some other models with modulated on-site potentials are also found to exhibit similar phase diagrams and their connection to the Haldane's model is revealed.

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 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 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.

Critical systems represent physical boundaries between different phases of matter and have been intensely studied for their universality and rich physics. Yet, with the rise of non-Hermitian studies, fundamental concepts underpinning critical systems - like band gaps and locality - are increasingly called into question. This work uncovers a new class of criticality where eigenenergies and eigenstates of non-Hermitian lattice systems jump discontinuously across a critical point in the thermodynamic limit, unlike established critical scenarios with spectrum remaining continuous across a transition. Such critical behavior, dubbed the “critical non-Hermitian skin effect”, arises whenever subsystems with dissimilar non-reciprocal accumulations are coupled, however weakly. This indicates, as elaborated with the generalized Brillouin zone approach, that the thermodynamic and zero-coupling limits are not exchangeable, and that even a large system can be qualitatively different from its thermodynamic limit. Examples with anomalous scaling behavior are presented as manifestations of the critical non-Hermitian skin effect in finite-size systems. More spectacularly, topological in-gap modes can even be induced by changing the system size. We provide an explicit proposal for detecting the critical non-Hermitian skin effect in an RLC circuit setup, which also directly carries over to established setups in non-Hermitian optics and mechanics.

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We show how to define a dynamical topological invariant for general one-dimensional topological systems after a quantum quench. Focusing on two-band topological insulators, we demonstrate that the reduced momentum-time manifold can be viewed as a series of submanifold S 2 , and thus we are able to define a dynamical topological invariant on each of the sphere. We also unveil the intrinsic relation between the dynamical topological invariant and the difference of topological invariant of the initial and final static Hamiltonian. By considering some concrete examples, we illustrate the calculation of the dynamical topological invariant and its geometrical meaning explicitly.Introduction.-In the last decade, the study of topological quantum matter is one of the most attractive topic in condensed matter physics [1][2][3][4][5], and our knowledge of topological properties for various quantum systems has been widely expanded. In contrast to equilibrium systems, what we know about the topological quantum matter out of equilibrium is quite rare [6]. The topology properties far from equilibrium have been studied in different ways, such as the dynamics of edge states [7][8][9], dynamical quantum phase transition [10][11][12], Floquet topological states [13][14][15], etc. The rapid development of cold atom experiments provides a powerful tool to study the dynamics far from equilibrium [16][17][18][19], and the evolution of a quantum state can be visualized with the method of Bloch state tomography [20][21][22].

Biologically active gases that occur naturally in the body include nitric oxide (NO), carbon monoxide (CO) and hydrogen sulfide (H(2)S). Each of these molecules is synthesized by enzymes which have been characterized biochemically and pharmacologically, and each acts, via well-established molecular targets, to effect physiological and/or pathophysiological functions within the body. Major biological roles that appear to be common to all three gases include the regulation of vascular homoeostasis and central nervous system function. It is becoming increasingly clear that both the synthesis and the biological activity of each gas are, to some extent, regulated by the presence of the others, and as such it is necessary to consider these molecules not in isolation but acting together to control cell function. Additional, more speculative candidates for gaseous cell signalling molecules include ammonia, acetaldehyde, sulfur dioxide and nitrous oxide. Whether such molecules also play a role in regulating body function remains to be determined.