Omar Mustafa, S. Habib Mazharimousavi
A transformation of the form x to iy; x,y in R, or an equivalent similarity transformation with a metric operator $η$ are shown to transform non-Hermitian PT-symmetric Hamiltonians into Hermitian partner Hamiltonians in Hilbert space. Isospectrality and mass signature are discussed.
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A complexified von Roos Hamiltonian is considered and a Hermitian first-order intertwining differential operator is used to obtain the related position dependent mass $η$-weak-pseudo-Hermitian Hamiltonians. Using a Liouvillean-type change of variables, the $η$-weak-pseudo-Hermitian von Roos Hamiltonians H(x) are mapped into the traditional Schrodinger Hamiltonian form H(q), where exact isospectral correspondence between H(x) and H(q) is obtained. Under a user-friendly position dependent mass settings, it is observed that for each exactly-solvable $η$-weak-pseudo-Hermitian reference-Hamiltonian…
The generalized Swanson Hamiltonian $H_{GS} = w (\tilde{a}\tilde{a}^†+ 1/2) + α\tilde{a}^2 + β\tilde{a}^{†^2}$ with $\tilde{a} = A(x)d/dx + B(x)$, can be transformed into an equivalent Hermitian Hamiltonian with the help of a similarity transformation. It is shown that the equivalent Hermitian Hamiltonian can be further transformed into the harmonic oscillator Hamiltonian so long as $[\tilde{a},\tilde{a}^†]=$ constant. However, the main objective of this paper is to show that though the commutator of $\tilde{a}$ and $\tilde{a}^†$ is constant, the generalized Swanson Hamiltonian is not necessar…
Breakdown of Non-Bloch Bulk-Boundary Correspondence and Emergent Topology in Floquet Non-Hermitian Systems
研究揭示非厄密Floquet系统中体-边界对应关系的破坏及拓扑保护边态的脆弱性，提出基于热力学极限的稳定零模奇异态与拓扑保护边态的对应关系。
We systematically study the topology of the exceptional point (EP) in the finite non-Hermitian system. Based on the concrete form of the Berry connection, we demonstrate that the exceptional line (EL), at which the eigenstates coalesce, can act as a vortex filament. The direction of the EL can be identified by the corresponding Berry curvature. In this context, such a correspondence makes the topology of the EL clear at a glance. As an example, we apply this finding to the non-Hermitian Rice-Mele (RM) model, the non-Hermiticity of which arises from the staggered on-site complex potential. The …
We in this paper demonstrate that the $PT$-symmetric non-Hermitian Hamiltonian for a periodically driven system can be generated from a kernel Hamiltonian by a generalized gauge transformation. The kernel Hamiltonian is Hermitian and static, while the time-dependent transformation operator has to be $PT$ symmetric and non-unitary in general. Biorthogonal sets of eigenstates appear necessarily as a consequence of non-Hermitian Hamiltonian. We obtain analytically the wave functions and associated non-adiabatic Berry phase $γ_{n}$ for the $n$th eigenstate. The classical version of the non-Hermiti…
The classical and quantum mechanical correspondence for constant mass settings is used, along with some point canonical transformation, to find the position-dependent mass (PDM) classical and quantum Hamiltonians. The comparison between the resulting quantum PDM-Hamiltonian and the von Roos PDM-Hamiltonian implied that the ordering ambiguity parameters of von Roos are strictly determined. Eliminating, in effect, the ordering ambiguity associated with the von Roos PDM-Hamiltonian. This, consequently, played a vital role in the construction and identification of the PDM-momentum operator. The sa…
Quantum-classical correspondence of non-Hermitian spin-orbit coupled bosonic junction
本文研究非厄米自旋-轨道耦合玻色结的经典-量子对应，推导出非厄米离散非线性薛定谔方程，发现半整数SOC强度下无损阱中必然发生原子自俘获，并揭示SOC促进自旋分量同步振荡的对称性保护机制。
Unified Bulk-Entanglement Correspondence in Non-Hermitian Systems
该研究通过建立非布洛赫极化与纠缠极化之间的对应关系，揭示了非厄米特系统中体-边界对应的新机制，解决了传统拓扑不变量的局限性。
Symmetry-Engineered Multiple Bulk-Boundary Correspondences and Anomalous Modes in a Non-Hermitian Creutz Ladder
研究非厄密Creutz阶梯中多重体边界对应关系及异常模，揭示对称性调控下的拓扑相变机制。
Recently, the study of non-Hermitian physics has attracted considerable attention. The modified bulk-boundary correspondence has been proposed to understand topological edge states in non-Hermitian static systems. Here we report a new experimental observation of edge states in non-Hermitian periodically driven systems. Some unconventional edge states are found not to be satisfied with the bulk-boundary correspondence when the system belongs to the broken parity-time (PT) symmetric phase. The experiments are performed in our constructed non-Hermitian light quantum walk platform with left and ri…
The bulk-boundary correspondence (BBC), i.e. the direct relation between bulk topological invariants defined for infinite periodic systems and the occurrence of protected zero-energy surface states in finite samples, is a ubiquitous and widely observed phenomenon in topological matter. In non-Hermitian generalizations of topological systems, however, this fundamental correspondence has recently been found to be qualitatively altered, largely owing to the sensitivity of non-Hermitian eigenspectra to changing the boundary conditions. In this work, we report on two contributions towards comprehen…
Generalized bulk-boundary correspondence in periodically driven non-Hermitian systems
本文教学性地综述了周期驱动非厄米系统，重点探讨非厄米趋肤效应与拓扑的相互作用，建立广义体-边界对应，并涵盖高维新现象及实验进展。
Asymmetric coupling amplitudes effectively create an imaginary gauge field, which induces a non-Hermitian Aharonov-Bohm (AB) effect. Nonzero imaginary magnetic flux invalidates the bulk-boundary correspondence and leads to a topological phase transition. However, the way of non-Hermiticity appearance may alter the system topology. By alternatively introducing the non-Hermiticity under symmetry to prevent nonzero imaginary magnetic flux, the bulk-boundary correspondence recovers and every bulk state becomes extended; the bulk topology of Bloch Hamiltonian predicts the (non)existence of edge sta…
Quantum mechanics of inverted potential well -- Hermitian Hamiltonian with imaginary eigenvalues, quantum-classical correspondence
本文利用虚频率升降玻色子算符的代数方法求解倒置势阱的量子化问题，构造具有虚本征值的厄米哈密顿量的对偶本征态与相干态，并证明相干态中海森堡方程的概率平均与经典运动方程精确一致，从而揭示虚本征值系统中的量子-经典对应。
We study a non-Hermitian Rice-Mele model without breaking time-reversal symmetry, with the non-Hermiticity arising from imbalanced hopping rates. The Berry connection, Berry curvature and Chern number are introduced in the context of biorthonormal inner product. It is shown that for a bulk system, although the Berry connection can be complex numbers, the Chern number is still quantized, as topological invariant. For an opened chain system, the mid-gap edge modes are obtained exactly, obeying the bulk-edge correspondence. Furthermore, we also introduce a local current in the context of biorthon…
Non-Hermitian systems exhibit striking exceptions from the paradigmatic bulk-boundary correspondence, including the failure of bulk Bloch band invariants in predicting boundary states and the (dis)appearance of boundary states at parameter values far from those corresponding to gap closings in periodic systems without boundaries. Here, we provide a comprehensive framework to unravel this disparity based on the notion of biorthogonal quantum mechanics: While the properties of the left and right eigenstates corresponding to boundary modes are individually decoupled from the bulk physics in non-H…
Topological invariants of complex energy plane in non-Hermitian systems
本文提出非厄米系统中复能量平面赝边界态与拓扑不变量的广义局域-全局对应关系，并在非厄米陈绝缘体模型中发现了超出陈数和涡度预测的新拓扑不变量，丰富了相图。
The conventional bulk-boundary correspondence breaks down in non-Hermitian systems. In this paper, we reestablish the bulk-boundary correspondence in one-dimensional non-Hermitian systems by applying the scattering theory, which is a systematical way in various symmetry classes. Based on the scattering theory, it is discovered that the topological invariant can be obtained by solving a generalized eigenproblem without calculating the generalized Brillouin zone. As a direct consequence, we unveil a new type of topological phase transition without typical bulk enengy gap closing and an unstable …
In Hermitian systems, according to the bulk-edge correspondence interfacing two topological optical media with different bulk topological numbers implies the existence of edge states, which can trap light at the interface. However, such a general scenario can be violated when dealing with non-Hermitian systems. Here we show that interfacing two semi-infinite Hatano-Nelson chains with different bulk topological numbers can result in the existence of infinitely many edge (interface) states, however light waves cannot be rather generally trapped at the interface.
The generalized Brillouin zone (GBZ), which is the core concept of the non-Bloch band theory to rebuild the bulk boundary correspondence in the non-Hermitian topology, appears as a closed loop generally. In this work, we find that even if the GBZ itself collapses into a point, the recovery of the open boundary energy spectrum by the continuum bands remains unchanged. Contrastively, if the bizarreness of the GBZ occurs, the winding number will become illness. Namely, we find that the bulk boundary correspondence can still be established whereas the GBZ has singularities from the perspective of …
Genuinely non-Hermitian topological phases can be realized in open systems with sufficiently strong gain and loss; in such phases, the Hamiltonian cannot be deformed into a gapped Hermitian Hamiltonian without energy bands touching each other. Comparing Green functions for periodic and open boundary conditions we find that, in general, there is no correspondence between topological invariants computed for periodic boundary conditions, and boundary eigenstates observed for open boundary conditions. Instead, we find that the non-Hermitian winding number in one dimension signals a topological pha…
Non-Hermitian second-order topological insulator with point gap
研究揭示非厄米特系统中二阶拓扑绝缘体的零模式奇异态与能量谱顶点态的拓扑保护关系，建立 bulk-boundary 对应关系。
Bulk-boundary correspondence, a central principle in topological matter relating bulk topological invariants to edge states, breaks down in a generic class of non-Hermitian systems that have so far eluded experimental effort. Here we theoretically predict and experimentally observe non-Hermitian bulk-boundary correspondence, a fundamental generalization of the conventional bulk-boundary correspondence, in discrete-time non-unitary quantum-walk dynamics of single photons. We experimentally demonstrate photon localizations near boundaries even in the absence of topological edge states, thus conf…
Topological edge modes are excitations that are localized at the materials' edges and yet are characterized by a topological invariant defined in the bulk. Such bulk-edge correspondence has enabled the creation of robust electronic, electromagnetic and mechanical transport properties across a wide range of systems, from cold atoms to metamaterials, active matter and geophysical flows. Recently, the advent of non-Hermitian topological systems---wherein energy is not conserved---has sparked considerable theoretical advances. In particular, novel topological phases that can only exist in non-Herm…
Bulk boundary correspondence is crucial to topological insulator as it associates the boundary states (with zero energy, chiral or helical) to topological numbers defined in bulk. The application of this correspondence needs a prerequisite condition which is usually not mentioned explicitly: the boundaries themselves cannot alter the bulk states, so as to the topological numbers defined on them. In non-hermitian models with fractional winding number, we prove that such precondition fails and the bulk boundary correspondence is cut out. We show that, as eliminating the hopping between the bound…
The hallmark of symmetry-protected topological (SPT) phases is the existence of anomalous boundary states, which can only be realized with the corresponding bulk system. In this work, we show that for every Hermitian anomalous boundary mode of the ten Altland-Zirnbauer classes, a non-Hermitian counterpart can be constructed, whose long time dynamics provides a realization of the anomalous boundary state. We prove that the non-Hermitian counterpart is characterized by a point-gap topological invariant, and furthermore, that the invariant exactly matches that of the corresponding Hermitian anoma…
In this study, we introduce a two dimensional complex harmonic oscillator potential with space and time reflection symmetries. The corresponding time independent Schrödinger equation yields real eigenvalues with complex eigenfunctions. We also construct the coherent state of the system by using a superposition of 12 eigenfunctions. Using the complex correspondence principle for the probability density we investigate the possible modifications in the probability densities due to the non-Hermitian aspect of the Hamiltonian.
Non-Hermitian Hamiltonians, which describe a wide range of dissipative systems, and higher-order topological phases, which exhibit novel boundary states on corners and hinges, comprise two areas of intense current research. Here we investigate systems where these frontiers merge and formulate a generalized biorthogonal bulk-boundary correspondence, which dictates the appearance of boundary modes at parameter values that are, in general, radically different from those that mark phase transitions in periodic systems. By analyzing the interplay between corner/hinge, edge/surface and bulk degrees …
Topological characterization of non-Hermitian band structures demands more than a straightforward generalization of the Hermitian cases. Even for one-dimensional tight-binding models with nonreciprocal hopping, the appearance of point gaps and the skin effect leads to the breakdown of the usual bulk-boundary correspondence. Luckily, the correspondence can be resurrected by introducing a winding number for the generalized Brillouin zone for systems with even number of bands and chiral symmetry. Here, we analyze the topological phases of a nonreciprocal hopping model on the stub lattice, where o…
We explore the Jacobi Last Multiplier as a means for deriving the Lagrangian of a fourth-order differential equation. In particular we consider the classical problem of the Pais-Uhlenbeck oscillator and write down the accompanying Hamiltonian. We then compare such an expression with an alternative derivation of the Hamiltonian that makes use of the Ostrogradski's method and show that a mapping from the one to the other is achievable by variable transformations. Assuming canonical quantization procedure to be valid we go for the operator version of the Hamiltonian that represents a pair of unco…
We investigate the many-particle and mean-field correspondence for a non-Hermitian N-particle Bose-Hubbard dimer where a complex onsite energy describes an effective decay from one of the modes. Recently a generalized mean-field approximation for this non-Hermitian many-particle system yielding an alternative complex nonlinear Schrödinger equation was introduced. Here we give details of this mean-field approximation and show that the resulting dynamics can be expressed in a generalized canonical form that includes a metric gradient flow. The interplay of nonlinearity and non-Hermiticity introd…
Point-gap topology in amorphous non-Hermitian quantum systems
针对非厄米非晶系统中谱不稳定导致难以识别拓扑边缘态的问题，建立稳定零模奇异态与隙间态的对应关系，基于哈密顿量奇异值分解构建实空间拓扑不变量，为点隙拓扑探索提供通用策略并重新定义非厄米皮效应。
The existence of a hermitian time operator is proposed in the framework of non-relativistic quantum mechanics.The Heisenberg equation of motion is shown to yield constant rate of flow of time.It is shown to yield results consistent with classical expectations,when Bohr's correspondence principle is invoked.Further,it is shown that a massless free particle with non-zero momentum experiences timelessness in its own frame of reference whereas a negatively massive particle perceives forwardness of time.Moreover,application to quantum harmonic oscillators shows instantaneous transition of electrons…
Observation of non-Hermitian bulk-boundary correspondence in non-chiral non-unitary quantum dynamics of single photons
研究手性对称性破缺下非厄米系统体边对应的重构问题，通过单光子离散时间非手性非幺正量子行走实验，构建畴壁构型观察到拓扑边缘模，确立非厄米体边对应的一般原理。
Non-Hermiticity enriches the 10-fold Altland-Zirnbauer symmetry class into the 38-fold symmetry class, where critical behavior of the Anderson transitions (ATs) has been extensively studied recently. Here, we propose a correspondence of the universality classes of the ATs between Hermitian and non-Hermitian systems. We illustrate that the critical exponents of the length scale in non-Hermitian systems coincide with the critical exponents in the corresponding Hermitian systems with additional chiral symmetry. A remarkable consequence of the correspondence is superuniversality, i.e., the ATs in …
Topological phases and the associated multiple edge states are studied for parity and time-reversal $(\mathcal{PT})$ symmetric non-Hermitian open quantum systems by constructing a non-unitary three-step quantum walk retaining $\mathcal{PT}$ symmetry in one dimension. We show that the non-unitary quantum walk has large topological numbers of the $\mathbb{Z}$ topological phase and numerically confirm that multiple edge states appear as expected from the bulk-edge correspondence. Therefore, the bulk-edge correspondence is valid in this case. Moreover, we study the stability of the multiple edge s…
We identify and investigate two classes of non-Hermitian systems, i.e., one resulting from Lorentz-symmetry violation (LSV) and the other from a complex mass (CM) with Lorentz invariance, from the perspective of quantum field theory. The mechanisms to break, and approaches to restore, the bulk-boundary correspondence in these two types of non-Hermitian systems are clarified. The non-Hermitian system with LSV shows a non-Hermitian skin effect, and its topological phase can be characterized by mapping it to the Hermitian system via a non-compact $U(1)$ gauge transformation. In contrast, there ex…
Emergent non-Hermitian conservation laws at exceptional points
本文提出并验证了非厄米系统在例外点处会涌现非厄米守恒定律，通过非厄米海森堡链构建通用理论，揭示其物理起源为辅助厄米系统的涌现对称性，并在量子电路模拟中证实该现象可被观测。
Topological phases of Hermitian systems are known to exhibit intriguing properties such as the presence of robust boundary states and the famed bulk-boundary correspondence. These features can change drastically for their non-Hermitian generalizations, as exemplified by a general breakdown of bulk-boundary correspondence and a localization of all states at the boundary, termed the non-Hermitian skin effect. In this article, we present a completely analytical unifying framework for studying these systems using generalized transfer matrices -- a real-space approach suitable for systems with peri…
The breakdown of the conventional bulk-boundary correspondence due to non-Hermitian skin effect leads to the non-Bloch bulk-boundary correspondence in the generalized Brillouin zone. Inspired by the case of the equivalence between the non-reciprocal hopping and imaginary gauge field, we propose a method to construct the topological equivalent models of the non-Hermitian dimerized lattices with the similarity transformations. The idea of the constructions is from that the imaginary magnetic flux vanishes under the open boundary condition and the period boundary spectra can be well approximated …
Symmetric non-Hermitian skin effect with emergent nonlocal correspondence
本文预言并理论证明了在局域粒子-空穴对称性下，非厄米趋肤效应会对称地分布于不同边界，并通过四重希尔伯特空间映射建立非局域对应理论，提出光学拉曼晶格模型以供冷原子实验观测。
Non-Hermitian topology in driven-dissipative systems: correspondence with quantum correlations and a resource for entanglement
本文研究驱动-耗散系统中的非厄米拓扑，发现其与量子关联对应，并为纠缠提供资源，推导解析表达式，揭示关联指数增长及拓扑相变特征。
Bulk-boundary correspondence in topological two-dimensional non-Hermitian systems: Toeplitz operators and singular values
该研究通过Toeplitz算子和奇异值建立了二维非厄米特系统中体-边对应关系，揭示了奇异值在拓扑保护中的关键作用，并展示了拓扑保护的边缘态和角落态。
In this work, a classical-quantum correspondence for two-level pseudo-Hermitian systems is proposed and analyzed. We show that the presence of a complex external field can be described by a pseudo-Hermitian Hamiltonian if there is a suitable canonical transformation that links it to a real field. We construct a covariant quantization scheme which maps canonically related pseudoclassical theories to unitarily equivalent quantum realizations, such that there is a unique metric-inducing isometry between the distinct Hilbert spaces. In this setting, the pseudo-Hermiticity condition for the operato…
Many-body Non-Hermitian Skin Effect At Finite Temperatures
本研究揭示有限温度热力学极限下多体非厄米趋肤效应与线性电势效应的对应关系，阐明其与单体效应的差异，及在费米、玻色系统中引发的独特相变与对称性破缺特性。
Quantum-Classical Correspondence of Non-Hermitian Symmetry Breaking
本研究通过复路径积分和广义Gutzwiller迹公式，建立了非厄米系统中能级实复性与半经典轨道对称性之间的普适量子-经典对应，统一解释了谱跃迁和PT对称破缺，并揭示奇异点为量子现象。
Bulk-Edge Correspondence for Point-Gap Topological Phases in Junction Systems
本文将非布洛赫能带理论扩展至结系统，建立了非厄米点隙拓扑相的体-边对应关系，并揭示了界面附近本征态的“非厄米邻近效应”及其独特的局域化长度性质。
Entanglement Spectrum Dynamics as a Probe for Non-Hermitian Bulk-Boundary Correspondence in Systems with Periodic Boundaries
本文通过周期性开放量子费米子系统的纠缠谱淬火动力学，证明即使在非厄米/刘维尔趋肤效应存在时，由开放边界条件定义的拓扑相变也能在周期性边界系统的动力学中被观测到。
Correspondence between non-Hermitian topology and directional amplification in the presence of disorder
本文解析证明驱动-耗散腔阵列中非厄米拓扑与定向放大的对应关系在无序下仍成立，并给出散射矩阵元概率界，揭示非互易增益特征的稳健性。
Observation of dynamical degeneracy splitting for the non-Hermitian skin effect
该研究在二维声学晶体中实验观测非厄米趋肤效应与等频线动力学简并分裂，并通过单频激发测量证实二者对应，为表征非厄米趋肤效应提供新途径。
For decades, the topological phenomena in quantum systems have always been catching our attention. Recently, there are many interests on the systems where topologically protected edge states exist, even in the presence of non-Hermiticity. Motivated by these researches, the topological properties of a non-Hermitian dice model are studied in two non-Hermitian cases, viz. in the imbalanced and the balanced dissipations. Our results suggest that the topological phases are protected by the real gaps and the bulk-edge correspondence readily seen in the real edge-state spectra. Besides, we show that …
Restoration of the non-Hermitian bulk-boundary correspondence via topological amplification
本文针对具有复能带且无对称性的非厄米哈密顿量，通过驱动-耗散腔阵列与奇异值分解恢复了体-边界对应关系，指出非平庸拓扑体现为随系统大小指数增长的方向性放大，解决了非厄米拓扑相理论的突出问题。
Non-Hermitian Aharonov-Bohm Cage in Bosonic Bogoliubov-de Gennes Systems
针对非厄米AB笼简并类型分类方法不明确的问题，在玻色子BdG系统中构建含多类简并平带的非厄米AB笼，借助转移矩阵阐明局域机制，推导极小多项式建立三者对应关系，提出高简并平带实现方案。
Non-Hermitian second-order topological superconductors
基于非布洛赫能带理论，研究了二维非厄米二阶拓扑超导体中非厄米性与拓扑的相互作用，发现非厄米性驱动体态和边缘模式向相反角落聚集并打破马约拉纳角模式简并，揭示了$Z_2$趋肤效应等普适特征。
Bulk-boundary correspondence in point-gap topological phases
本文建立了非厄米系统点隙拓扑中的体-边对应关系，揭示了开放边界与周期边界条件下体点隙拓扑的差异，并基于实空间拓扑不变量和K-理论给出完整分类，证明非平凡开放边界拓扑导致稳健且奇异的表面态。
Unifying framework for non-Hermitian and Hermitian topology in driven-dissipative systems
本文将单复能带非厄米拓扑与定向放大的对应关系框架扩展至多能带和高维系统，引入广义奇异谱定义相关能带及拓扑不变量，证明体-边对应并通过模型验证，为相关实验应用奠定基础。
Symmetry classification correspondence between quadratic Lindbladians and their steady states
针对Lindbladian与稳态对称性分类分离的问题，本文建立二者联系，将守恒粒子的二次Lindbladian分为27个非厄米类，其中5类与稳态厄米类一一对应，数值模拟验证了长时间收敛性。
Recently, the search for topological states of matter has turned to non-Hermitian systems, which exhibit a rich variety of unique properties without Hermitian counterparts. Lattices modeled through non-Hermitian Hamiltonians appear in the context of photonic systems, where one needs to account for gain and loss, circuits of resonators, and also when modeling the lifetime due to interactions in condensed matter systems. Here we provide a brief overview of this rapidly growing subject, the search for topological states and a bulk-boundary correspondence in non-Hermitian systems.
A non-Hermitian extension of a Chern insulator and its bulk-boundary correspondence are investigated. It is shown that in addition to the robust chiral edge states that reflect the nontrivial topology of the bulk (nonzero Chern number), anomalous helical edge states localized only at one edge can appear, which are unique to the non-Hermitian Chern insulator.
We propose a non-Hermitian generalization of the correspondence between the spectral flow and the topological charges of band crossing points (Berry-Chern monopoles). A class of non-Hermitian Hamiltonians that display a complex-valued spectral flow is built by deforming an Hermitian model while preserving its analytical index. We relate those spectral flows to a generalized Chern number that we show to be equal to that of the Hermitian case, provided a line gap exists. We demonstrate the homotopic invariance of both the non-Hermitian Chern number and the spectral flow index, making explicit th…
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