A scheme for realizing nonreciprocal interlayer coupling in bilayer topological systems

Topological insulators possess strong topological protection properties and can manipulate the wave propagation to combat disorder and defects. And now they have grown into a large research field in photonic and phononic crystals. However, the conventional topological band theory is used to describe a closed photonic/phononic crystal that is assumed to be a Hermitian system. In fact, actual physical systems often couple with external environment, and generate non-Hermitian Hamiltonians with complex eigenvalues. Recently, many novel topological properties have been induced by the interaction between non-Hermitian phase and topological phase. A prominent example is non-Hermitian skin effect that all eigenstates are localized to the boundary in open system, which is different from the conventional topological edge state. This unique physical phenomenon has inspired various applications, such as wave funneling, enhanced sensing, and topological lasing. In this work, we describe the non-Hermitian skin effect by using winding number domains. The sign of the winding number domain determines the rotation direction of the loops in the complex frequency plane, whose sign can be controlled by the nonreciprocal coupling direction. In this work, we design different topological skin interfaces between different domains with opposite winding numbers to manipulate the energy focusing on middle or two-end of non-Hermitian one-dimensional acoustic cavity chain. In experiment, we use an electroacoustic coupling method, in which a unidirectional coupler composed of microphones, speakers, phase shifters, and amplifiers is used to introduce positive and negative non-reciprocal couplings between the two acoustic cavities, and study the characteristics of these non-reciprocal couplings. Then, the non-reciprocal coupling cavities are extended into a chain structure, and the magnitudes and signs of the non-reciprocal couplings are flexibly controlled by using phase shifters and amplifiers. Through this method, we successfully construct the interfaces between different winding number domains, achieving a one-dimensional non-Hermitian skin effect at various interfaces. The experimental results indicate that the sound can be focused on the middle interface or two-end interfaces for different nonreciprocal coupling distributions, and the skin interface can also be switched from middle to two-end by exchanging the nonreciprocal coupling direction of the domains. Our research results provide greater flexibility for designing acoustic devices and also a new platform for exploring advanced topological acoustic systems for controlling sound propagation.

Parity-time ($\mathcal{PT}$) symmetry plays an important role both in non-Hermitian and topological systems. In non-Hermitian systems $\mathcal{PT}$ symmetry can lead to an entirely real energy spectrum, while in topological systems $\mathcal{PT}$ symmetry gives rise to stable and protected Dirac points. Here, we study a $\mathcal{PT}$-symmetric system which is both non-Hermitian and topological, namely a $\mathcal{PT}$-symmetric Dirac semimetal with non-Hermitian perturbations in three dimensions. We find that, in general, there are only two types of symmetry allowed non-Hermitian perturbations, namely non-Hermitian kinetic potentials, and non-Hermitian anti-commuting potentials. For both of these non-Hermitian potentials we investigate the band topology for open and periodic boundary conditions, determine the exceptional points, and compute the surface states. We find that with periodic boundary conditions, the non-Hermitian kinetic potential leads to exceptional rings, while the non-Hermitian anti-commuting potential generates exceptional spheres, which separate regions with broken $\mathcal{PT}$ symmetry from regions with unbroken $\mathcal{PT}$ symmetry. With open boundary conditions, the non-Hermitian kinetic potential induces a non-Hermitian skin effect which is localized on both sides of the sample due to symmetry, while the non-Hermitian anticommuting potential leads to Fermi ribbon surface states.

In condensed matter physics, non-Abelian statistics for Majorana zero modes (or Majorana Fermions) is very important, really exotic, and completely robust. The race for searching Majorana zero modes and verifying the corresponding non-Abelian statistics becomes an important frontier in condensed matter physics. In this paper we generalize the Majorana zero modes to non-Hermitian (NH) topological systems that show universal but quite different properties from their Hermitian counterparts. Based on the NH Majorana zero modes, the orthogonal and nonlocal Majorana qubits are well defined. In particular, due to the particle-hole-symmetry breaking, NH Majorana zero modes have anomalous non-Abelian statistics with continuously tunable braiding Berry phase from $\ensuremath{\pi}/8$ to $3\ensuremath{\pi}/8$. This is quite different from the usual non-Abelian statistics with fixed braiding Berry phase $\ensuremath{\pi}/4$. The one-dimensional NH Kitaev model is taken as an example to numerically verify the anomalous non-Abelian statistics for two NH Majorana zero modes. The numerical results are exactly consistent with the theoretical prediction. With the help of braiding these two zero modes, the $\ensuremath{\pi}/8$ gate can be reached and thus universal topological quantum computation becomes possible.

Spin-momentum locking is a direct consequence of bulk topological order and provides a basic concept to control a carrier's spin and charge flow for new exotic phenomena in condensed matter physics. However, up to date the research on spin-momentum locking solely focuses on its in-plane transport properties. Here, we report an emerging out-of-plane radiation feature of spin-momentum locking in a non-Hermitian topological photonic system and demonstrate a high performance topological vortex laser based on it. We find that the gain saturation effect lifts the degeneracy of the paired counterpropagating spin-momentum-locked edge modes enabling lasing from a single topological edge mode. The near-field spin and orbital angular momentum of the topological edge mode lasing has a one-to-one far-field radiation correspondence. The methodology of probing the near-field topology feature by far-field lasing emission can be used to study other exotic phenomena. The device can lead to applications in superresolution imaging, optical tweezers, free-space optical sensing, and communication.

In recent years, exploring new topological quantum model structures and in depth analyzing the novel physical properties in various multimerized topological lattices have become a hot topic in the field of quantum optics. Among the different model structures, the multimerized non-Hermitian lattice controlled by different parameters in the future research of topological quantum materials, we believe, can exhibit more meaningful novel topological properties. As one of the most classic topological models, the one-dimensional Aubry-André-Harper (AAH) model has received more and more attention in the study of multimerized lattices. In this paper, we focus on the novel topological properties of a trimerized non-Hermitian lattice, and extend the trimer model structure from a one-dimensional chain to a quasi-one-dimensional zigzag structure. The results show that firstly, if the nearest-neighbor forward coupling coefficient in the unit cell is not equal to the backward coupling coefficient, the chiral inversion symmetry of the system is destroyed. It can be observed that the bulk states and the edge states in the trimerization model will be localized on the same edge of the lattice, and the skin effect will appear in the system. With the increase of the nearest-neighbor coupling coefficient, the width of the edge state changes in which the lower edge state of the imaginary part of the spectrum is narrowed until it disappears. The degree of degeneracy of the system changes, and the number of edge states is reduced from four to two. Remarkably, the generalized bulk-boundary correspondence is shown in certain non-Hermitian topological systems. Secondly, when the trimerization model considers the influence of the next-nearest-neighbor coupling, the numerical results show that the upper and lower energy gaps in the energy spectrum and the edge states in the energy spectrum are asymmetrical as the next-nearest-neighbor coupling coefficient is modulated in an appropriate range. The upper energy gaps and the edge states are narrowed, and the edge states of the lower energy gaps are widened. At the same time, the novel topology features of the system can also be used to achieve the quantitative control of the energy spectrum edge states, and other interesting directions are worth exploring.

The topological one-way waveguide mode of the acoustic wave has recently been demonstrated in various meta-structure systems. Here, we show that in a topological phononic crystal with a symmetry-broken acoustic unit cell, the topological state possesses not just a “spin-like” pseudospin mode but also real spin angular momentum. By rotating the double-square units, the band of the phononic crystal will become inverse and induce both topological phase transition and spin angular momentum reversal. As such, by putting two topologically different systems together, the spin angular momentum dependent one-way interface modes can be selectively excited by acoustic spin sources, exhibiting robust transport protected by tight spin-momentum locking. The spin angular momentum density distribution in the unit cell and edge states shows that in addition to the pseudospin, there is a strong correlation between the real spin angular momentum and topological properties in this acoustic system, producing the topologically selective acoustic torque. Revealing the real spin angular momentum and associated acoustic spin torque properties of these topological phononic and acoustic systems will give people a more general understanding about symmetric breaking wave systems and help people to explore more potential applications of acoustic spins in various topological systems.

Non-Hermitian physics has greatly enriched our understanding of nonequilibrium phenomena and uncovered novel effects such as the non-Hermitian skin effect (NHSE) that has profoundly revolutionized the field. NHSE has been predicted in systems with nonreciprocal couplings which, however, are challenging to realize in experiments. Without nonreciprocal couplings, the NHSE can also emerge in systems with coexisting gauge fields and loss or gain (e.g., in Floquet non-Hermitian systems). However, such Floquet NHSE remains largely unexplored in experiments. Here, we realize the Floquet NHSEs in periodically modulated optical waveguides integrated on a silicon photonic platform. By engineering the artificial gauge fields induced by the periodical modulation, we observe various Floquet NHSE phases and unveil their rich topological transitions. Remarkably, we discover the transitions between the unipolar NHSE phases and an unconventional bipolar NHSE phase, which is accompanied by the directional reversal of the NHSEs. The underlying physics is revealed by the band winding in complex quasienergy space which undergoes a topology change from isolated loops with the same winding to linked loops with opposite windings. Our work unfolds a new route toward Floquet NHSEs originating from the interplay between gauge fields and dissipation effects, and thus offers fundamentally new ways for steering light and other waves.

The Su–Schrieffer–Heeger (SSH) system is a popular model for exploring topological insulators and topological phases in one dimension. Recent interest in exceptional points has led to re-examination of non-Hermitian generalizations of many physical models, including the SSH model. In such non-Hermitian systems, singular points called exceptional points (EPs) appear that are of interest for applications in super-resolution sensing systems and topological lasers. Here, a non-Hermitian and non-PT-symmetric variation of the SSH model is introduced, in which the hopping amplitudes are nonreciprocal and vary monotonically along the chain. It is found that, while the existence of the EPs is due to the nonreciprocal couplings, the number, position, and order of the EPs can all be altered by the addition of the hopping amplitude gradient, adding a new, to the best of our knowledge, tool for tailoring the spectrum of a non-Hermitian system.

A notable feature of systems with non-Hermitian skin effects is the sensitivity to boundary conditions. In this work, we introduce one type of boundary condition provided by a coupling impurity. We consider a system where a two-level system as an impurity couples to a nonreciprocal Su-Schrieffer-Heeger chain under periodic boundary conditions at two points with asymmetric couplings. We first study the spectrum of the system and find that asymmetric couplings lead to topological phase transitions. Meanwhile, a striking feature is that the coupling impurity can act as an effective boundary, and asymmetric couplings can also induce a flexibly adjusted zero mode. It is localized at one of the two effective boundaries or both of them by tuning coupling strengths. Moreover, we uncover three types of localization behaviors of eigenstates for this non-Hermitian impurity system with on-site disorder. These results corroborate the potential for control of a class of non-Hermitian systems with coupling impurities.

We study non-Bloch bulk-boundary correspondence in a non-Hermitian Su-Schieffer-Heeger model in a domain-wall configuration where the left and right bulks have different parameters. Focusing on the case where chiral symmetry is still conserved, we show that non-Hermitian skin effects of bulk states persist in the system, while the definition of the non-Bloch winding number of either bulk depends on parameters on both sides of the boundary. Under these redefined non-Bloch topological invariants, we confirm non-Bloch bulk-boundary correspondence under the domain-wall configuration, which exemplifies the impact of boundary conditions in non-Hermitian topological systems.

The topological photonics plays an important role in the fields of fundamental physics and photonic devices. The traditional method of designing topological system is based on the momentum space, which is not a direct and convenient way to grasp the topological properties, especially for the perturbative structures or coupled systems. Here, we propose an interdisciplinary approach to study the topological systems in real space through combining the information entropy and topological photonics. As a proof of concept, the Kagome model has been analyzed with information entropy. We reveal that the bandgap closing does not correspond to the topological edge state disappearing. This method can be used to identify the topological phase conveniently and directly, even the systems with perturbations or couplings. As a promotional validation, Su–Schrieffer–Heeger model and the valley-Hall photonic crystal have also been studied based on the information entropy method. This work provides a method to study topological photonic phase based on information theory, and brings inspiration to analyze the physical properties by taking advantage of interdisciplinarity.Graphical

1D topological systems for next-generation electronics

As first demonstrated by the characterization of the quantum Hall effect by the Chern number, topology provides a guiding principle to realize the robust properties of condensed-matter systems immune to the existence of disorder. The bulk–boundary correspondence guarantees the emergence of gapless boundary modes in a topological system whose bulk exhibits non-zero topological invariants. Although some recent studies have suggested a possible extension of the notion of topology to nonlinear systems, the nonlinear counterpart of a topological invariant has not yet been understood. Here we propose a nonlinear extension of the Chern number based on the nonlinear eigenvalue problems in two-dimensional systems and show the existence of bulk–boundary correspondence beyond the weakly nonlinear regime. Specifically, we find nonlinearity-induced topological phase transitions, in which the existence of topological edge modes depends on the amplitude of oscillatory modes. We propose and analyse a minimal model of a nonlinear Chern insulator whose exact bulk solutions are analytically obtained. The model exhibits the amplitude dependence of the nonlinear Chern number, for which we confirm the nonlinear extension of the bulk–boundary correspondence. Thus, our result reveals the existence of genuinely nonlinear topological phases that are adiabatically disconnected from the linear regime.

By designing tailor-made resonance modes with structured atoms, metamaterials allow us to obtain constitutive parameters outside their limited range from natural materials. Nonetheless, tuning the constitutive parameters depends on our ability to modify the physical structure or external circuits attached to the metamaterials, posing a fundamental challenge to the range of tunability in many real-time applications. Here, we propose the concept of virtualized metamaterials on their signal response function to escape the boundary inherent in the physical structure of metamaterials. By replacing the resonating physical structure with a designer mathematical convolution kernel with a fast digital signal processing circuit, we demonstrate a decoupled control of the effective bulk modulus and mass density of acoustic metamaterials on-demand through a software-defined frequency dispersion. Providing freely software-reconfigurable amplitude, center frequency, bandwidth of frequency dispersion, our approach adds an additional dimension to constructing non-reciprocal, non-Hermitian, and topological systems with time-varying capability as potential applications.

Topological states of matter in non-Hermitian systems have attracted a lot of attention due to their intriguing dynamical and transport properties. In this study, we propose a periodically driven non-Hermitian lattice model in one-dimension, which features rich Floquet topological phases. The topological phase diagram of the model is derived analytically. Each of its non-Hermitian Floquet topological phases is characterized by a pair of integer winding numbers, counting the number of real 0- and \pi-quasienergy edge states at the boundaries of the lattice. Non-Hermiticity induced Floquet topological phases with unlimited winding numbers are found, which allow arbitrarily many real 0- and \pi-quasienergy edge states to appear in the complex quasienergy bulk gaps in a well-controlled manner. We further suggest to probe the topological winding numbers of the system by dynamically imaging the stroboscopic spin textures of its bulk states.