A. Ramirez, B. Mielnik
We present a brief review of physical problems leading to indefinite Hilbert spaces and non-hermitian Hamiltonians. With the exception of pseudo-Riemannian manifolds in GR, the problem of a consistent physical interpretation of these structures still waits to be faced. In print in Rev. Mex. Fis.
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In arXiv:0709.0483 Gunther and Samsonov outline a ``generalization'' of quantum mechanics that involves simultaneous consideration of Hermitian and non-Hermitian operators and promises to be ``capable to produce effects beyond those of standard Hermitian quantum mechanics.'' We give a simple physical interpretation of Hermiticity and discuss in detail the shortcomings of the above-mentioned composite quantum theory. In particular, we show that the corresponding ``generalization of measurement theory'' suffers from a dynamical inconsistency and that it is by no means adequate to replace the sta…
We explore an alternative way of finding the link between a PT non-Hermitian Hamiltonian and a Hermitian one. Based on the analysis of the scattering problem for an imaginary potential and its time reversal process, it is shown that any real-energy eigenstate of a PT tight-binding lattice with on-site imaginary potentials shares the same wave function with a resonant transmission state of the corresponding Hermitian lattice embedded in a chain. It indicates that the PT eigenstate of a PT non-Hermitian Hamiltonian has connection to the resonance transmission state of the extended Hermitian Hami…
We explore a way of finding the link between a non-Hermitian Hamiltonian and a Hermitian one. Based on the analysis of Bethe Ansatz solutions for a class of non-Hermitian Hamiltonians and the scattering problems for the corresponding Hermitian Hamiltonians. It is shown that a scattering state of an arbitrary Hermitian lattice embedded in a chain as the scattering center shares the same wave function with the corresponding non-Hermitian tight binding lattice, which consists of the Hermitian lattice with two additional on-site complex potentials, no matter the non-Hermitian is broken PT symmetry…
The non-Hermitian formalism is used at present in many papers for the description of open quantum systems. A special language developed in this field of physics which makes it difficult for many physicists to follow and to understand the corresponding papers. We show that the characteristic features of the non-Hermitian formalism are nothing but nonlinearities that may appear in the equations when the Hamiltonian is non-Hermitian. They are related directly to singular points (called mostly exceptional points, EPs). At low level density, they may cause counterintuitive physical results which al…
Thermodynamics of quasianti-Hermitian quaternionic systems with constant number of particles in equilibrium is studied. A toy model is introduced and the physically relevant quantities are derived. The energy fluctuation which shows that for large N the relative r.m.s fluctuation in the values of E is quite negligible is derived. The negative temperature for such systems is also studied. Finally a physical example is discussed and physical explanations in the context of quantum physics are given.
We present an evaluation of some recent attempts at understanding the role of pseudo-Hermitian and PT-symmetric Hamiltonians in modeling unitary quantum systems and elaborate on a particular physical phenomenon whose discovery originated in the study of complex scattering potentials.
In the context of traditional quantum-control considerations it is conjectured that one of the promising new strategies of the constructive model building could be sought in a non-stationary upgrade of the formalism of PT-symmetric quantum mechanics. This upgrade is briefly reviewed and characterized by the simultaneous participation of a triplet of Hilbert spaces in the representation of a single quantum system.
For a non-Hermitian Hamiltonian H possessing a real spectrum, we introduce a canonical orthonormal basis in which a previously introduced unitary mapping of H to a Hermitian Hamiltonian h takes a simple form. We use this basis to construct the observables O of the quantum mechanics based on H. In particular, we introduce pseudo-Hermitian position and momentum operators and a pseudo-Hermitian quantization scheme that relates the latter to the ordinary classical position and momentum observables. These allow us to address the problem of determining the conserved probability density and the under…
The metric associated with a quasi-Hermitian Hamiltonian and its physical implications are scrutinised. Consequences of the non-uniqueness such as the question of the probability interpretation and the possible and forbidden choices of additional observables are investigated and exemplified by specific illustrative examples. In particular it is argued that the improper identification of observables lies at the origin of the claimed violation of the brachistchrone transition time between orthogonal states. The need for further physical input to remove ambiguities is pointed out.
Transfer learning from Hermitian to non-Hermitian quantum many-body physics
本文提出利用仅基于厄米关联函数训练的机器学习方法，无需额外训练即可识别非厄米相互作用模型的相边界，确立了迁移学习作为从厄米物理到非厄米现象学习的通用策略。
Physically, one tends to think of non-Hermitian systems in terms of gain and loss: the decay or amplification of a mode is given by the imaginary part of its energy. Here, we introduce an alternative avenue to the realm of non-Hermitian physics, which involves neither gain nor loss. Instead, complex eigenvalues emerge from the amplitudes and phase-differences of waves backscattered from the boundary of insulators. We show that for any strong topological insulator in a Wigner-Dyson class, the reflected waves are characterized by a reflection matrix exhibiting the non-Hermitian skin effect. This…
In contrast to classical systems, actual implementation of non-Hermitian Hamiltonian dynamics for quantum systems is a challenge because the processes of energy gain and dissipation are based on the underlying Hermitian system-environment dynamics that is trace preserving. Recently, a scheme for engineering non-Hermitian Hamiltonians as a result of repetitive measurements on an anicillary qubit has been proposed. The induced conditional dynamics of the main system is described by the effective non-Hermitian Hamiltonian arisng from the procedure. In this paper we demonstrate the effectiveness o…
We explain why the main conclusion of Bender et al, hep-th/0511229 [J. Phys. A 39 (2006) 1657] regarding the practical superiority of the non-Hermitian description of PT-symmetric quantum systems over their Hermitian description is not valid. Recalling the essential role played by the Hermitian description in the characterization and interpretation of the physical observables, we maintain that as far as the physical aspects of the theory are concerned the Hermitian description is not only unavoidable but also indispensable.
Observables are believed that they must be Hermitian in quantum theory. Based on the obviously physical fact that only eigenstates of observable and its corresponding probabilities, i.e., spectrum distribution of observable are actually observed, we argue that observables need not necessarily to be Hermitian. More generally, observables should be reformulated as normal operators including Hermitian operators as a subclass. This reformulation is consistent with the quantum theory currently used and does not change any physical results. The Clauser-Horne-Shimony-Holt (CHSH) inequality is taken a…
Non-Hermitian physics in the dissipative many-body system of Rydberg atoms
本文综述了Rydberg原子系统中非厄米特物理的研究进展，探讨了非厄米特哈密顿量的构造、耗散机制及其对能谱和对称性破缺的影响。
We review the methodology to theoretically treat parity-time- ($\mathcal{PT}$-) symmetric, non-Hermitian quantum many-body systems... (For the full abstract see paper)
Physics and computation: An insight from non-Hermitian quantum computing
本文提出并研究了非厄米特量子计算机模型，展示其能解决NP及P^#P类问题，其计算能力源于所需指数级物理资源。
A longstanding tool to characterize the evolution of open Markovian quantum systems is the GKSL (Gorini-Kossakowski-Sudarshan-Lindblad) master equation. However, in some cases, open quantum systems can be effectively described with non-Hermitian Hamiltonians, which have attracted great interest in the last twenty years due to a number of unconventional properties, such as the appearance of exceptional points. Here, we present a short review of these two different approaches aiming in particular to highlight their relation and illustrate different ways of connecting non-Hermitian Hamiltonian to…
We show that the physical principle "the adjoint associates to each state a `test' for that state" fully characterises the Hermitian adjoint for pure quantum theory, therefore providing the adjoint with operational meaning beyond its standard mathematical definition. Also, we show that for general process theories, which all admit a diagrammatic representation, this physical principle induces a reflection operation.
Emergent non-Hermitian physics in generalized Lotka-Volterra model
本文研究广义Lotka-Volterra捕食者-猎物模型中的非厄米物理，发现混沌与局域动力学由临界点分隔，其源于有效哈密顿量的周期性与非厄米性，临界点行为对应非厄米物理中的奇异点。
Local and Global Master Equations through the Lens of Non-Hermitian Physics
研究非厄米特哈密顿量与开放量子系统非平衡朗之万动力学的关系，探讨量子跃迁和例外点在非平衡开放量子系统中的作用，并提出实验可实现的架构。
The formalism for non-Hermitian quantum systems sometimes blurs the underlying physics. We present a systematic study of the vielbein-like formalism which transforms the Hilbert space bundles of non-Hermitian systems into the conventional ones, rendering the induced Hamiltonian to be Hermitian. In other words, any non-Hermitian Hamiltonian can be "transformed" into a Hermitian one without altering the physics. Thus we show how to find a reference frame (corresponding to Einstein's quantum elevator) in which a non-Hermitian system, equipped with a non-trivial Hilbert space metric, reduces to a …
Topological phases are enriched in non-equilibrium open systems effectively described by non-Hermitian Hamiltonians. While several properties unique to non-Hermitian topological systems were uncovered, the fundamental role of symmetry in non-Hermitian physics has yet to be fully understood, and it has remained unclear how symmetry protects non-Hermitian topological phases. Here we show that two fundamental anti-unitary symmetries, time-reversal and particle-hole symmetries, are topologically equivalent in the complex energy plane and hence unified in non-Hermitian physics. A striking consequen…
We discuss non-Hermitian field theories where the spectrum of the Hamiltonian involves only real energies. We make three observations. (i) The theories obtained from supersymmetric theories by nonanticommutative deformations belong in many cases to this class. (ii) When the deformation parameter is small, the deformed theory enjoys the same supersymmetry algebra as the undeformed one. Half of the supersymmetries are manifest and the existence of another half can be deduced from the structure of the spectrum. (iii) Generically, the conventionally defined S--matrix is not unitary for such theori…
While classic quantum chaos originated from the idea to set into context nonlinear physics and Hermitian quantum mechanics, non-Hermitian models have enhanced the field in recent years. At the same time, low-dimensional effective matrix models have proven to be a powerful tool in accessing the physical properties of a system in a semiquantitative manner. Here, we focus on two realizations of non-Hermitian physics in mesoscopic systems. First, we consider spiral optical microcavities in which the asymmetric scattering between whispering gallery modes induces the non-Hermitian behaviour. Second,…
Physical constraints on effective non-Hermitian systems
本文研究了非厄米特系统的物理约束，指出将反厄米特部分直接引入马斯巴拉格林函数的方法与标准框架不兼容，并通过伪厄米特量子力学描述了这些系统，同时分析了零温度分布函数和非厄米特系统的电磁响应。
In this paper, based on a one-dimensional non-Hermitian spin model with $\mathcal{RT}$-invariant term, we study the non-Hermitian physics for the two (nearly) degenerate ground states. By using the high-order perturbation method, an effective pseudo-spin model is obtained to describe non-Hermitian physics for the two (nearly) degenerate ground states, which are precisely consistent with the numerical calculations. We found that there may exist effective (anti) $\mathcal{PT}$ symmetry for the effective pseudo-spin model of the two (nearly) degenerate ground states. In particular, there exists s…
In recent years, non-Hermitian quantum physics has gained a lot in popularity in the quantum optics and condensed matter communities in order to model quantum systems with varying symmetries. In this paper, we identify a non-standard inner product that implies bosonic commutator relations for local electric and magnetic field observables and leads to a natural local biorthogonal description of the quantised electromagnetic field. When comparing this description with an alternative local Hermitian description, in which the states of local photonic particles - i.e. of so-called bosons localised …
Non-Hermitian-Hamiltonian-induced unitarity and optional physical inner products in Hilbert space
本研究在准厄米量子力学框架下，提出通过弱化希尔伯特空间几何各向同性来扩大幺正演化参数域的方法，并利用简化度量子集与两个精确可解模型验证了该构想。
The surface Hamiltonian for a spin zero particle that is pinned to a surface by letting the thickness of a layer surrounding the surface go to zero -- assuming a strong normal force -- is constructed. The new approach we follow to achieve this is to start with an expression for the 3D momentum operators whose components along the surface and the normal to the surface are separately Hermitian. The normal part of the kinetic energy operator is a Hermitian operator in this case. When this operator is dropped and the thickness of the layer is set to zero, one automatically gets the Hermitian surfa…
In this letter, we investigate the effects of non-Hermitian driving on quantum coherence in a bipartite system. The results that the dynamical localization destroyed by the Hermitian interaction revives are an evidence of the restoration of quantum coherence by non-Hermitian driving. Besides, the entanglement between the two subsystems also decays with the boosting of non-hermitian driving strength, which provides another evidence that non-Hermitian driving will protect quantum coherence. The physics behind this phenomenon is the domination of the quasieigenstate with maximum imaginary value o…
Definition of Current Density in Non-Hermitian Quantum Systems
本文定义了非厄米特量子系统中电流密度的新概念，以确保连续性方程成立并具有物理意义。
Light propagation in distributed feedback optical structures with gain/loss regions is shown to provide an accessible laboratory tool to visualize in optics the spectral properties of the one-dimensional Dirac equation with non-Hermitian interactions. Spectral singularities and PT symmetry breaking of the Dirac Hamiltonian are shown to correspond to simple observable physical quantities and related to well-known physical phenomena like resonance narrowing and laser oscillation.
Advances in non-Hermitian dynamics of quadratic bosonic systems
研究非厄米特性在二次玻色子系统中的动态演化及拓扑现象，探索其在信号放大和量子效应中的应用。
A review is given on the foundations and applications of non-Hermitian classical and quantum physics. First, key theorems and central concepts in non-Hermitian linear algebra, including Jordan normal form, biorthogonality, exceptional points, pseudo-Hermiticity and parity-time symmetry, are delineated in a pedagogical and mathematically coherent manner. Building on these, we provide an overview of how diverse classical systems, ranging from photonics, mechanics, electrical circuits, acoustics to active matter, can be used to simulate non-Hermitian wave physics. In particular, we discuss rich a…
Experimental investigation of direct non-Hermitian measurement and uncertainty relation towards high-dimensional quantum domain
本文提出非厄米投影协议并推导赝厄米可观测量不确定性关系，在量子光学线路中实现单光子qutrit上的3维非厄米量子测量，实验验证了不同PH度规下的不确定性关系，为直接获取非厄米量子投影统计提供了态无关方法。
We extend the definition of generalized parity $P$, charge-conjugation $C$ and time-reversal $T$ operators to nondiagonalizable pseudo-Hermitian Hamiltonians, and we use these generalized operators to describe the full set of symmetries of a pseudo-Hermitian Hamiltonian according to a fourfold classification. In particular we show that $TP$ and $CTP$ are the generators of the antiunitary symmetries; moreover, a necessary and sufficient condition is provided for a pseudo-Hermitian Hamiltonian $H$ to admit a $P$-reflecting symmetry which generates the $P$-pseudounitary and the $P$-pseudoantiunit…
Locality and Exceptional Points in Pseudo-Hermitian Physics
本论文研究拟厄米与赝厄米物理，证明局部可观测量及非定域关联边界受限，并分析例外点、非厄米缺陷晶格、PT 相变及互缠算子构造。
Experimental Measurement of Non-Hermitian Left Eigenvectors
本研究提出基于非厄米格林函数的方法，从实验测量的稳态响应中直接提取非厄米左本征矢（LEVs）和右本征矢（REVs），并通过声学实验验证其有效性，揭示了LEVs在非厄米现象中被低估的可观测效应。
Physical relevance of time-independent scattering calculations in non-Hermitian systems: The role of time-growing bound states
本文研究了非厄米系统中不含时散射方法的适用性，指出随时间增长的束缚态会指数放大并主导动力学，导致常规透反射系数失效，强调分析 $S$ 矩阵极点是验证散射计算物理相关性的必要步骤。
Non-Hermitian Chiral Skin Effect
本文提出非厄米手性趋肤效应（NHCSE），在周期边界下使手性模式局域于拓扑缺陷，并以此阐明Chern绝缘体中混合趋肤-拓扑效应的机制。
Why finite-size non-Hermitian eigenvalues can miss asymptotic eigenstates in topological physics
非厄米特系统中本征值谱无法检测某些本征态，Hatano-Nelson模型揭示了体-边对应失败的根源。
The dynamics of open quantum systems is determined by avoided and true crossings of eigenvalue trajectories of a non-Hermitian Hamiltonian. The phases of the eigenfunctions are not rigid so that environmentally induced spectroscopic redistribution processes may take place and a dynamical phase transition may occur. Due to the formal equivalence between the quantum mechanical Schrödinger equation and the optical wave equation in PT symmetric lattices, the dynamics of the system is determined also in this case by avoided and true crossings of eigenvalue trajectories of the non-Hermitian Hamilton…
We relate topological properties of non-Hermitian systems and observables of quantum open systems by using the Keldysh path-integral method. We express Keldysh Green's functions in terms of effective non-Hermitian Hamiltonians that contain all the relevant topological information. We arrive at a frequency dependent topological index that is linked to the response of the system to perturbations at a given frequency. We show how to detect a transition between different topological phases by measuring the response to local perturbations. Our formalism is exemplified in a 1D Hatano-Nelson model, h…
We investigate a non-Hermitian Aharonov-Bohm (AB) ring system with a quantum dot (QD) embedded in each of its two arms. The energy levels of the QDs are complex in order to take into account the physical gain or loss of the ring system during its interacting processes with the environment. When there is magnetic flux threading through the ring, by allocating the flux phase factor into the tunneling amplitudes between the QDs and the leads in different ways, the Hamiltonian of the system can be written into different formalisms. We calculate the transmission through the ring by using these diff…
Real critical exponents from the $\varepsilon$-expansion in an interacting $U(1)$ model with non-Hermitian $Z_4$ anisotropy
本文研究了一个具有非厄米特$Z_4$各向异性的$U(1)$不变拉格朗日量的临界行为，发现无论是未破坏还是破坏$\mathcal{PT}$对称性区域，均得到实临界指数，揭示了非厄米特系统中$U(1)$对称性和厄米特性作为涌现特征的重要性。
Fractional Quantum Zeno Effect Emerging from Non-Hermitian Physics
本文预言了非厄米量子系统中的分数阶量子芝诺效应及其诱导的光子反聚束，通过耗散能带边缘发散的态密度实现分数代数标度的自发辐射抑制，并阐明非厄米性在产生无经典对应的亚泊松光子统计中的关键作用。
A non-commuting measurement transfers, via the apparatus, information encoded in a system's state to the external "observer". Classical measurements determine properties of physical objects. In the quantum realm, the very same notion restricts the recording process to orthogonal states as only those are distinguishable by measurements. Therefore, even a possibility to describe physical reality by means of non-hermitian operators should \emph{volens nolens} be excluded as their eigenstates are not orthogonal. Here, we show that non-hermitian operators with real spectrum can be treated within th…
Non-Hermitian Hamiltonians are relevant to describe the features of a broad class of physical phenomena, ranging from photonics and atomic and molecular systems to nuclear physics and mesoscopic electronic systems. An important question relies on the understanding of the influence of curved background on the static and dynamical properties of non-Hermitian systems. In this work, we study the interplay of geometry and non-Hermitian dynamics by unveiling the existence of curvature-dependent non-Hermitian phase transitions. We investigate a prototypical model of Dirac fermions on a sphere with an…
Non-stationary non-Hermitian "wrong-sign'' quantum oscillators and their meaningful physical interpretation
本文研究了非稳态非厄密itian量子振荡子，探讨了其物理意义及生成元的复数谱特性。
A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools and elaborate on a number of rel…
Quantum phase transitions also occur in non-Hermitian systems. In this work we show that density functional theory, for the first time, uncovers universal behaviors for phase transitions in non-Hermitian many-body systems. To be specific, we first prove that the non-degenerate steady state of a non-Hermitian quantum many-body system is a universal function of the first derivative of the steady state energy with respect to the control parameter. This finding has far-reaching consequences for non-Hermitian systems: (i) It bridges the nonanalytic behavior in physical observable and nonanalytic be…
The past decades have witnessed an explosion of interest in topological materials, and a lot of mathematical concepts have been introduced in condensed matter physics. Among them, the bulk-boundary correspondence is the central topic in topological physics, which has inspired researchers to focus on boundary physics. Recently, the concepts of topological phases have been extended to non-Hermitian Hamiltonians, whose eigenvalues can be complex. Besides the topology, non-Hermiticity can also cause a boundary phenomenon called the non-Hermitian skin effect, which is an extreme sensitivity of the …
We develop a complete theory of symmetry and topology in non-Hermitian physics. We demonstrate that non-Hermiticity ramifies the celebrated Altland-Zirnbauer symmetry classification for insulators and superconductors. In particular, charge conjugation is defined in terms of transposition rather than complex conjugation due to the lack of Hermiticity, and hence chiral symmetry becomes distinct from sublattice symmetry. It is also shown that non-Hermiticity enables a Hermitian-conjugate counterpart of the Altland-Zirnbauer symmetry. Taking into account sublattice symmetry or pseudo-Hermiticity a…
Kinked linear response from non-Hermitian cold-atom pumping
该研究揭示非厄米泵浦在超冷原子波包轨迹中引发扭结线性响应，源于非局域性导致的能带奇点，并提出二维光晶格实验方案，无需精细调节即可观测，拓展了非厄米动力学研究。
Perspective: Interactions and Nonlinearity in Non-Hermitian Physics
本文探讨非厄运物理从单粒子线性模型到多体相互作用系统的演变，分析非厄运动力学的物理起源，讨论相互作用诱导的拓扑相、非线性 regime 中的集体现象及测量诱导纠缠相变。
For a specific exactly solvable 2 by 2 matrix model with a PT-symmetric Hamiltonian possessing a real spectrum, we construct all the eligible physical metrics and show that none of them admits a factorization CP in terms of an involutive charge operator C. Alternative ways of restricting the physical metric to a unique form are briefly discussed.
We consider a generalization of the non-Hermitian ${\mathcal PT}$ symmetric Jaynes-Cummings {Hamiltonian, recently introduced for studying optical phenomena with time-dependent physical parameters, that includes environment-induced decay}. In particular, we investigate the interaction of a two-level fermionic system (such as a two-level atom) with a single bosonic field mode in a cavity. The states of {the} two-level system are allowed to decay because of the interaction with the environment, {and this is included phenomenologically in our non-Hermitian Hamiltonian by introducing complex energ…
A pseudo-Hermitian coupled-channel square-well model is proposed, solved and discussed. The domain of parameters is determined where all the bound-state energies (twice degenerate with respect to the second observable which we call "spin") remain real. The standard probabilistic interpretation of the wave functions is achieved by the transition from the original elementary non-physical indefinite pseudo-metric $θ$ to another, more involved but correct positive-definite physical metric $Θ$. In our model the latter step remains comparatively easily feasible. Another fortunate circumstance emerge…
Symmetry plays fundamental role in physics and the nature of symmetry changes in non-Hermitian physics. Here the symmetry-protected scattering in non-Hermitian linear systems is investigated by employing the discrete symmetries that classify the random matrices. The even-parity symmetries impose strict constraints on the scattering coefficients: the time-reversal (C and K) symmetries protect the symmetric transmission or reflection; the pseudo-Hermiticity (Q symmetry) or the inversion (P) symmetry protects the symmetric transmission and reflection. For the inversion-combined time-reversal symm…
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