#### Special topics

SPECIAL TOPIC — Non-Hermitian physics

 Select Geometry of time-dependent $\mathcal{PT}$-symmetric quantum mechanics Da-Jian Zhang(张大剑), Qing-hai Wang(王清海), and Jiangbin Gong(龚江滨) Chin. Phys. B, 2021, 30 (10): 100307.   DOI: 10.1088/1674-1056/ac0ba8 Abstract （334）   HTML （7）    PDF （528KB）（183）       A new type of quantum theory known as time-dependent $\mathcal{PT}$-symmetric quantum mechanics has received much attention recently. It has a conceptually intriguing feature of equipping the Hilbert space of a $\mathcal{PT}$-symmetric system with a time-varying inner product. In this work, we explore the geometry of time-dependent $\mathcal{PT}$-symmetric quantum mechanics. We find that a geometric phase can emerge naturally from the cyclic evolution of a $\mathcal{PT}$-symmetric system, and further formulate a series of related differential-geometry concepts, including connection, curvature, parallel transport, metric tensor, and quantum geometric tensor. These findings constitute a useful, perhaps indispensible, tool to investigate geometric properties of $\mathcal{PT}$-symmetric systems with time-varying system's parameters. To exemplify the application of our findings, we show that the unconventional geometric phase [Phys. Rev. Lett. 91 187902 (2003)], which is the sum of a geometric phase and a dynamical phase proportional to the geometric phase, can be expressed as a single geometric phase unveiled in this work.
 Select Non-Hermitian quasicrystal in dimerized lattices Longwen Zhou(周龙文) and Wenqian Han(韩雯岍) Chin. Phys. B, 2021, 30 (10): 100308.   DOI: 10.1088/1674-1056/ac1efc Abstract （365）   HTML （12）    PDF （3117KB）（193）       Non-Hermitian quasicrystals possess $\mathcal{PT}$ and metal-insulator transitions induced by gain and loss or nonreciprocal effects. In this work, we uncover the nature of localization transitions in a generalized Aubry-André-Harper model with dimerized hopping amplitudes and complex onsite potential. By investigating the spectrum, adjacent gap ratios and inverse participation ratios, we find an extended phase, a localized phase and a mobility edge phase, which are originated from the interplay between hopping dimerizations and non-Hermitian onsite potential. The lower and upper bounds of the mobility edge are further characterized by a pair of topological winding numbers, which undergo quantized jumps at the boundaries between different phases. Our discoveries thus unveil the richness of topological and transport phenomena in dimerized non-Hermitian quasicrystals.
 Select Observation of the exceptional point in superconducting qubit with dissipation controlled by parametric modulation Zhan Wang(王战), Zhongcheng Xiang(相忠诚), Tong Liu(刘桐), Xiaohui Song(宋小会), Pengtao Song(宋鹏涛), Xueyi Guo(郭学仪), Luhong Su(苏鹭红), He Zhang(张贺), Yanjing Du(杜燕京), and Dongning Zheng(郑东宁) Chin. Phys. B, 2021, 30 (10): 100309.   DOI: 10.1088/1674-1056/ac0520 Abstract （253）   HTML （1）    PDF （2957KB）（139）       Open physical systems described by the non-Hermitian Hamiltonian with parity-time-reversal (PT) symmetry show peculiar phenomena, such as the presence of an exceptional point (EP) at which the PT symmetry is broken and two resonant modes of the Hamiltonian become degenerate. Near the EP, the system could be more sensitive to external perturbations and this may lead to enhanced sensing. In this paper, we present experimental results on the observation of PT symmetry broken transition and the EP using a tunable superconducting qubit. The quantum system of investigation is formed by the two levels of the qubit and the energy loss of the system to the environment is controlled by a method of parametric modulation of the qubit frequency. This method is simple with no requirements for additional elements or qubit device modifications. We believe it can be easily implemented on multi-qubit devices that would be suitable for further exploration of non-Hermitian physics in more complex and diverse systems.
 Select Non-Hermitian Weyl semimetals: Non-Hermitian skin effect and non-Bloch bulk-boundary correspondence Xiaosen Yang(杨孝森), Yang Cao(曹阳), and Yunjia Zhai(翟云佳) Chin. Phys. B, 2022, 31 (1): 010308.   DOI: 10.1088/1674-1056/ac3738 Abstract （180）   HTML （5）    PDF （990KB）（106）       We investigate novel features of three-dimensional non-Hermitian Weyl semimetals, paying special attention to the unconventional bulk-boundary correspondence. We use the non-Bloch Chern numbers as the tool to obtain the topological phase diagram, which is also confirmed by the energy spectra from our numerical results. It is shown that, in sharp contrast to Hermitian systems, the conventional (Bloch) bulk-boundary correspondence breaks down in non-Hermitian topological semimetals, which is caused by the non-Hermitian skin effect. We establish the non-Bloch bulk-boundary correspondence for non-Hermitian Weyl semimetals: the topological edge modes are determined by the non-Bloch Chern number of the bulk bands. Moreover, these topological edge modes can manifest as the unidirectional edge motion, and their signatures are consistent with the non-Bloch bulk-boundary correspondence. Our work establishes the non-Bloch bulk-boundary correspondence for non-Hermitian topological semimetals.
 Select Two-body exceptional points in open dissipative systems Peize Ding(丁霈泽) and Wei Yi(易为) Chin. Phys. B, 2022, 31 (1): 010309.   DOI: 10.1088/1674-1056/ac3396 Abstract （245）   HTML （2）    PDF （994KB）（159）       We study two-body non-Hermitian physics in the context of an open dissipative system depicted by the Lindblad master equation. Adopting a minimal lattice model of a handful of interacting fermions with single-particle dissipation, we show that the non-Hermitian effective Hamiltonian of the master equation gives rise to two-body scattering states with state- and interaction-dependent parity-time transition. The resulting two-body exceptional points can be extracted from the trace-preserving density-matrix dynamics of the same dissipative system with three atoms. Our results not only demonstrate the interplay of parity-time symmetry and interaction on the exact few-body level, but also serve as a minimal illustration on how key features of non-Hermitian few-body physics can be probed in an open dissipative many-body system.
 Select Topological properties of non-Hermitian Creutz ladders Hui-Qiang Liang(梁辉强) and Linhu Li(李林虎) Chin. Phys. B, 2022, 31 (1): 010310.   DOI: 10.1088/1674-1056/ac3991 Abstract （239）   HTML （4）    PDF （7101KB）（156）       We study topological properties of the one-dimensional Creutz ladder model with different non-Hermitian asymmetric hoppings and on-site imaginary potentials, and obtain phase diagrams regarding the presence and absence of an energy gap and in-gap edge modes. The non-Hermitian skin effect (NHSE), which is known to break the bulk-boundary correspondence (BBC), emerges in the system only when the non-Hermiticity induces certain unbalanced non-reciprocity along the ladder. The topological properties of the model are found to be more sophisticated than that of its Hermitian counterpart, whether with or without the NHSE. In one scenario without the NHSE, the topological winding is found to exist in a two-dimensional plane embedded in a four-dimensional space of the complex Hamiltonian vector. The NHSE itself also possesses some unusual behaviors in this system, including a high spectral winding without the presence of long-range hoppings, and a competition between two types of the NHSE, with the same and opposite inverse localization lengths for the two bands, respectively. Furthermore, it is found that the NHSE in this model does not always break the conventional BBC, which is also associated with whether the band gap closes at exceptional points under the periodic boundary condition.
 Select Disorder in parity-time symmetric quantum walks Peng Xue(薛鹏) Chin. Phys. B, 2022, 31 (1): 010311.   DOI: 10.1088/1674-1056/ac3741 Abstract （189）   HTML （3）    PDF （3149KB）（122）       We experimentally investigate the impact of static disorder and dynamic disorder on the non-unitary dynamics of parity-time (PT)-symmetric quantum walks. Via temporally alternating photon losses in an interferometric network, we realize the passive PT-symmetric quantum dynamics for single photons. Controllable coin operations allow us to simulate different environmental influences, which result in three different behaviors of quantum walkers: a standard ballistic spread, a diffusive behavior, and a localization, respectively, in a PT-symmetric quantum walk architecture.
 Select Topology of a parity-time symmetric non-Hermitian rhombic lattice Shumai Zhang(张舒迈), Liang Jin(金亮), and Zhi Song(宋智) Chin. Phys. B, 2022, 31 (1): 010312.   DOI: 10.1088/1674-1056/ac364a Abstract （230）   HTML （4）    PDF （888KB）（151）       We investigate the topological properties of a trimerized parity-time ($\mathcal{PT}$) symmetric non-Hermitian rhombic lattice. Although the system is $\mathcal{PT}$-symmetric, the topology is not inherited from the Hermitian lattice; in contrast, the topology can be altered by the non-Hermiticity and depends on the couplings between the sublattices. The bulk-boundary correspondence is valid and the Bloch bulk captures the band topology. Topological edge states present in the two band gaps and are predicted from the global Zak phase obtained through the Wilson loop approach. In addition, the anomalous edge states compactly localize within two diamond plaquettes at the boundaries when all bands are flat at the exceptional point of the lattice. Our findings reveal the topological properties of the $\mathcal{PT}$-symmetric non-Hermitian rhombic lattice and shed light on the investigation of multi-band non-Hermitian topological phases.
 Select Exact solutions of non-Hermitian chains with asymmetric long-range hopping under specific boundary conditions Cui-Xian Guo(郭翠仙) and Shu Chen(陈澍) Chin. Phys. B, 2022, 31 (1): 010313.   DOI: 10.1088/1674-1056/ac3228 Abstract （210）   HTML （3）    PDF （679KB）（150）       We study the one-dimensional general non-Hermitian models with asymmetric long-range hopping and explore how to analytically solve the systems under some specific boundary conditions. Although the introduction of long-range hopping terms prevents us from finding analytical solutions for arbitrary boundary parameters, we identify the existence of exact solutions when the boundary parameters fulfill some constraint relations, which give the specific boundary conditions. Our analytical results show that the wave functions take simple forms and are independent of hopping range, while the eigenvalue spectra display rich model-dependent structures. Particularly, we find the existence of a special point coined as pseudo-periodic boundary condition, for which the eigenvalues are the same as those of the periodical system when the hopping parameters fulfill certain conditions, whereas the eigenstates display the non-Hermitian skin effect.
 Select Anti-$\mathcal{PT}$-symmetric Kerr gyroscope Huilai Zhang(张会来), Meiyu Peng(彭美瑜), Xun-Wei Xu(徐勋卫), and Hui Jing(景辉) Chin. Phys. B, 2022, 31 (1): 014215.   DOI: 10.1088/1674-1056/ac3988 Abstract （282）   HTML （4）    PDF （1441KB）（141）       Non-Hermitian systems can exhibit unconventional spectral singularities called exceptional points (EPs). Various EP sensors have been fabricated in recent years, showing strong spectral responses to external signals. Here we propose how to achieve a nonlinear anti-parity-time ($\mathcal{APT}$) gyroscope by spinning an optical resonator. We show that, in the absence of any nonlinearity, the sensitivity or optical mode splitting of the linear device can be magnified up to 3 orders compared to that of the conventional device without EPs. Remarkably, the $\mathcal{APT}$ symmetry can be broken when including the Kerr nonlinearity of the materials and, as a result, the detection threshold can be significantly lowered, i.e., much weaker rotations which are well beyond the ability of a linear gyroscope can now be detected with the nonlinear device. Our work shows the powerful ability of $\mathcal{APT}$ gyroscopes in practice to achieve ultrasensitive rotation measurement.
 Select Majorana zero modes, unconventional real-complex transition, and mobility edges in a one-dimensional non-Hermitian quasi-periodic lattice Shujie Cheng(成书杰) and Xianlong Gao(高先龙) Chin. Phys. B, 2022, 31 (1): 017401.   DOI: 10.1088/1674-1056/ac3222 Abstract （212）   HTML （3）    PDF （2739KB）（87）       A one-dimensional non-Hermitian quasiperiodic p-wave superconductor without $\mathcal{PT}$-symmetry is studied. By analyzing the spectrum, we discovered that there still exists real-complex energy transition even if the inexistence of $\mathcal{PT}$-symmetry breaking. By the inverse participation ratio, we constructed such a correspondence that pure real energies correspond to the extended states and complex energies correspond to the localized states, and this correspondence is precise and effective to detect the mobility edges. After investigating the topological properties, we arrived at a fact that the Majorana zero modes in this system are immune to the non-Hermiticity.
 Select Efficient and stable wireless power transfer based on the non-Hermitian physics Chao Zeng(曾超), Zhiwei Guo(郭志伟), Kejia Zhu(祝可嘉), Caifu Fan(范才富), Guo Li(李果), Jun Jiang(江俊), Yunhui Li(李云辉), Haitao Jiang(江海涛), Yaping Yang(羊亚平), Yong Sun(孙勇), and Hong Chen(陈鸿) Chin. Phys. B, 2022, 31 (1): 010307.   DOI: 10.1088/1674-1056/ac3815 Abstract （312）   HTML （6）    PDF （912KB）（154）       As one of the most attractive non-radiative power transfer mechanisms without cables, efficient magnetic resonance wireless power transfer (WPT) in the near field has been extensively developed in recent years, and promoted a variety of practical applications, such as mobile phones, medical implant devices and electric vehicles. However, the physical mechanism behind some key limitations of the resonance WPT, such as frequency splitting and size-dependent efficiency, is not very clear under the widely used circuit model. Here, we review the recently developed efficient and stable resonance WPT based on non-Hermitian physics, which starts from a completely different avenue (utilizing loss and gain) to introduce novel functionalities to the resonance WPT. From the perspective of non-Hermitian photonics, the coherent and incoherent effects compete and coexist in the WPT system, and the weak stable of energy transfer mainly comes from the broken phase associated with the phase transition of parity-time symmetry. Based on this basic physical framework, some optimization schemes are proposed, including using nonlinear effect, using bound states in the continuum, or resorting to the system with high-order parity-time symmetry. Moreover, the combination of non-Hermitian physics and topological photonics in multi-coil system also provides a versatile platform for long-range robust WPT with topological protection. Therefore, the non-Hermitian physics can not only exactly predict the main results of current WPT systems, but also provide new ways to solve the difficulties of previous designs.
 Select Filling up complex spectral regions through non-Hermitian disordered chains Hui Jiang and Ching Hua Lee Chin. Phys. B, 2022, 31 (5): 050307.   DOI: 10.1088/1674-1056/ac4a73 Abstract （200）   HTML （11）    PDF （9183KB）（174）       Eigenspectra that fill regions in the complex plane have been intriguing to many, inspiring research from random matrix theory to esoteric semi-infinite bounded non-Hermitian lattices. In this work, we propose a simple and robust ansatz for constructing models whose eigenspectra fill up generic prescribed regions. Our approach utilizes specially designed non-Hermitian random couplings that allow the co-existence of eigenstates with a continuum of localization lengths, mathematically emulating the effects of semi-infinite boundaries. While some of these couplings are necessarily long-ranged, they are still far more local than what is possible with known random matrix ensembles. Our ansatz can be feasibly implemented in physical platforms such as classical and quantum circuits, and harbors very high tolerance to imperfections due to its stochastic nature.
 Select Real non-Hermitian energy spectra without any symmetry Boxue Zhang(张博学), Qingya Li(李青铔), Xiao Zhang(张笑), and Ching Hua Lee(李庆华) Chin. Phys. B, 2022, 31 (7): 070308.   DOI: 10.1088/1674-1056/ac67c6 Abstract （273）   HTML （4）    PDF （1506KB）（97）       Non-Hermitian models with real eigenenergies are highly desirable for their stability. Yet, most of the currently known ones are constrained by symmetries such as PT-symmetry, which is incompatible with realizing some of the most exotic non-Hermitian phenomena. In this work, we investigate how the non-Hermitian skin effect provides an alternative route towards enforcing real spectra and system stability. We showcase, for different classes of energy dispersions, various ansatz models that possess large parameter space regions with real spectra, despite not having any obvious symmetry. These minimal local models can be quickly implemented in non-reciprocal experimental setups such as electrical circuits with operational amplifiers.
 Select Quantum simulation of τ-anti-pseudo-Hermitian two-level systems Chao Zheng(郑超) Chin. Phys. B, 2022, 31 (10): 100301.   DOI: 10.1088/1674-1056/ac8738 Abstract （155）   HTML （4）    PDF （3498KB）（51）       Different from the Hermitian case, non-Hermitian (NH) systems have novel properties and strongly relate to open and dissipative quantum systems. In this work, we investigate how to simulate τ-anti-pseudo-Hermitian systems in a Hermitian quantum device using linear combinations of unitaries and duality quantum algorithm. Specifying the τ to time-reversal (T) and parity-time-reversal (PT) operators, we construct the two NH two-level systems, design quantum circuits including three qubits, and decide the quantum gates explicitly in detail. We also calculate the success probabilities of the simulation. Experimental implementation can be expected in small quantum simulator.