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Square-root topology is one of the newest additions to the ever expanding field of topological insulators (TIs). It characterizes systems that relate to their parent TI through the squaring of their Hamiltonians. Extensions to 2 -root topology, where is the number of squaring operations involved in retrieving the parent TI, were quick to follow. Here, we go one step further and develop the framework for designing general -root TIs, with any positive integer, using the Su–Schrieffer–Heeger (SSH) model as the parent TI from which the higher-root versions are constructed. The method relies on using loops of unidirectional couplings as building blocks, such that the resulting model is non-Hermitian and embedded with a generalized chiral symmetry. Edge states are observed at the branches of the complex energy spectrum, appearing within what we designate as a ring gap, shown to be irreducible to the usual point or line gaps. We further detail on how such an -root model can be realistically implemented in photonic ring systems. Near perfect unidirectional effective couplings between the main rings can be generated via mediating link rings with modulated gains and losses. These induce high imaginary gauge fields that strongly suppress couplings in one direction, while enhancing them in the other. We use these photonic lattices to validate and benchmark the analytical predictions. Our results introduce a new class of high-root topological models, as well as a route for their experimental realization.

The exceptional points (EPs) of non-Hermitian systems, where n different energy eigenstates merge into an identical one, have many intriguing properties that have no counterparts in Hermitian systems. In particular, the ϵ 1 n dependence of the energy level splitting on a perturbative parameter ϵ near an nth order EP stimulates the idea of metrology with arbitrarily high sensitivity, since the susceptibility dϵ1/n/dϵ diverges at the EP. Here we theoretically study the sensitivity of parameter estimation near the EPs, using the exact formalism of quantum Fisher information (QFI). The QFI formalism allows the highest sensitivity to be determined without specifying a specific measurement approach. We find that the EP bears no dramatic enhancement of the sensitivity. Instead, the coalescence of the eigenstates exactly counteracts the eigenvalue susceptibility divergence and makes the sensitivity a smooth function of the perturbative parameter.

The past few years have witnessed a surge of interest in non-Hermitian Floquet topological matter due to its exotic properties resulting from the interplay between driving fields and non-Hermiticity. The present review sums up our studies on non-Hermitian Floquet topological matter in one and two spatial dimensions. We first give a bird’s-eye view of the literature for clarifying the physical significance of non-Hermitian Floquet systems. We then introduce, in a pedagogical manner, a number of useful tools tailored for the study of non-Hermitian Floquet systems and their topological properties. With the aid of these tools, we present typical examples of non-Hermitian Floquet topological insulators, superconductors, and quasicrystals, with a focus on their topological invariants, bulk-edge correspondences, non-Hermitian skin effects, dynamical properties, and localization transitions. We conclude this review by summarizing our main findings and presenting our vision of future directions.

The study of non-Hermitian degeneracies—called exceptional points (EPs)—has become an exciting frontier at the crossroads of optics, photonics, acoustics, and quantum physics. Here, we introduce the Newton polygon method as a general algebraic framework for characterizing and tuning EPs. Newton polygons, first described by Isaac Newton, are conventionally used in algebraic geometry, with deep roots in various topics in modern mathematics. We propose and illustrate how the Newton polygon method can enable the prediction of higher-order EPs, using a recently experimentally realized optical system. Using the paradigmatic Hatano-Nelson model, we demonstrate how our method can predict the presence of the non-Hermitian skin effect. As further application of our framework, we show the presence of tunable EPs of various orders in PT -symmetric one-dimensional models. We further extend our method to study EPs in higher number of variables and demonstrate that it can reveal rich anisotropic behaviour around such degeneracies. Our work provides an analytic recipe to understand exceptional physics.

We propose a scheme to realize various non-Hermitian topological phases in a topolectrical (TE) circuit network consisting of resistors, inductors, and capacitors. These phases are characterized by topologically protected exceptional points and lines. The positive and negative resistive couplings R g in the circuit provide loss and gain factors which break the Hermiticity of the circuit Laplacian. By controlling R g , the exceptional lines of the circuit can be modulated, e.g. from open curves to closed ellipses in the Brillouin zone. In practice, the topology of the exceptional lines can be detected by the impedance spectra of the circuit. We also considered finite TE systems with open boundary conditions, the admittance spectra of which exhibit highly tunable zero-admittance states demarcated by boundary points (BPs). The phase diagram of the system shows topological phases that are characterized by the number of their BPs. The transition between different phases can be controlled by varying the circuit parameters and tracked via the impedance readout between the terminal nodes. Our TE model offers an accessible and tunable means of realizing different topological phases in a non-Hermitian framework and characterizing them based on their boundary point and exceptional line configurations.

Non-Hermitian systems can exhibit exotic topological and localization properties. Here we elucidate the non-Hermitian effects on disordered topological systems using a nonreciprocal disordered Su-Schrieffer-Heeger model. We show that the non-Hermiticity can enhance the topological phase against disorders by increasing bulk gaps. Moreover, we uncover a topological phase which emerges under both moderate non-Hermiticity and disorders, and is characterized by localized insulating bulk states with a disorder-averaged winding number and zero-energy edge modes. Such topological phases induced by the combination of non-Hermiticity and disorders are dubbed non-Hermitian topological Anderson insulators . We reveal that the system has unique non-monotonous localization behavior and the topological transition is accompanied by an Anderson transition. These properties are general in other non-Hermitian models.

We investigate non-Hermitian elastic lattices characterized by non-local feedback interactions. In one-dimensional lattices, proportional feedback produces non-reciprocity associated with complex dispersion relations characterized by gain and loss in opposite propagation directions. For non-local controls, such non-reciprocity occurs over multiple frequency bands characterized by opposite non-reciprocal behavior. The dispersion topology is investigated with focus on winding numbers and non-Hermitian skin effect, which manifests itself through bulk modes localized at the boundaries of finite lattices. In two-dimensional lattices, non-reciprocity is associated with directional wave amplification. Moreover, the combination of skin effect in two directions produces modes that are localized at the corners of finite two-dimensional lattices. Our results describe fundamental properties of non-Hermitian elastic lattices, and suggest new possibilities for the design of meta materials with novel functionalities related to selective wave filtering, amplification and localization. The considered non-local lattices also provide a platform for the investigation of topological phases of non-Hermitian systems.

We take non-Hermitian Aubry–André–Harper models and quasiperiodic Kitaev chains as examples to demonstrate the equivalence and superposition of real and imaginary quasiperiodic potentials (QPs) on inducing localization of single-particle states. We prove this equivalence by analytically computing Lyapunov exponents (or inverse of localization lengths) for systems with purely real and purely imaginary QPs. Moreover, when superposed and with the same frequency, real and imaginary QPs are coherent on inducing the localization, in a way which is determined by the relative phase between them. The localization induced by a coherent superposition can be simulated by the Hermitian model with an effective strength of QP, implying that models are in the same universality class. When their frequencies are different and relatively incommensurate, they are incoherent and their superposition leads to less correlation effects. Numerical results show that the localization happens earlier and there is an intermediate mixed phase lacking of mobility edge.

Non-Hermitian (NH) physics predicts open quantum system dynamics with unique topological features such as exceptional points and the NH skin effect. We show that this new paradigm of topological systems can serve as probes for bulk Hamiltonian parameters with quantum-enhanced sensitivity reaching Heisenberg scaling. Such enhancement occurs close to a spectral topological phase transition, where the entire spectrum undergoes a delocalization transition. We provide an explanation for this enhanced sensitivity based on the closing of point gap, which is a genuinely NH energy gap with no Hermitian counterpart. This establishes a direct connection between energy-gap closing and quantum enhancement in the NH realm. Our findings are demonstrated through several paradigmatic NH topological models in various dimensions and potential experimental implementations.

In this work, we have studied the non-Hermitian nonlinear Landau–Zener–Stückelberg–Majorana (LZSM) interferometry in a non-Hermitian N -body interacting boson system in which the non-Hermiticity is from the nonreciprocal tunnelings between the bosons. By using the mean-field approximation and projective Hilbert space, the effect of nonreciprocity and nonlinearity on the energy spectrum, the dynamics, and the formation of the interference fringes have been studied. The different symmetries and the impact of the two different types of reciprocity, i.e. the in-phase tunneling and anti-phase tunneling, on the energy spectrum and the phase transition between the Josephson oscillations and the self-trapping have been investigated. For the LZSM interferometry, the strength of the nonreciprocity is found to take an essential role in the population of the projective state and the strengths of the interference patterns in the projective space. While the conditions of destructive and constructive interference under the weak-coupling approximation still only depend on the strength of nonlinearity. Our result provides an application of the nonlinear non-Hermitian LZSM interferometry in studying the parameters of a two-level system which related to the nonlinearity and the non-Hermiticity.