Eigenstates exhibit localization at an open edge in a non-Hermitian lattice due to non-Hermitian skin effect. We here explore another interesting feature of non-Hermitian skin effect and predict quasi-stationary solutions, which are approximately time-independent. We show that the transition from such states to eigenstates is dramatically non-perturbative. We discuss that mathematically extending the boundary of a long non-Hermitian lattice to infinity can lead to nontrivial solution. We consider a non-Hermitian variant of the Su-Schrieffer-Heeger (SSH) model and predict non-topological but robust quasi-stationary zero energy modes.
A reformulation of quantum mechanics called ``hybrid interaction picture'' (HIP) is proposed. In contrast to its current non-Hermitian-interaction-picture (NIP) predecessor the new formalism combines a simplification of the NIP Hilbert-space metric with an innovative preconditioning of the observables of interest. Via an illustrative example it is shown that the idea might open new ways towards optimal model-building strategies in quantum theory of unitary systems.
The interaction picture in a non-Hermitian realization is discussed in detail and considered for its practical use in many-body quantum physics. The resulting non-Hermitian interaction-picture (NHIP) description of dynamics, in which both the wave functions and operators belonging to physical observables cease to remain constant in time, is a non-Hermitian generalization of the traditional Dirac picture of standard quantum mechanics, which itself is widely used in quantum field theory calculations. Particular attention is paid here to the variational (or, better, bivariational) and dynamical (…
A harmonic oscillator Hamiltonian augmented by a non-Hermitian \pt-symmetric part and its su(1,1) generalizations, for which a family of positive-definite metric operators was recently constructed, are re-examined in a supersymmetric context. Quasi-Hermitian supersymmetric extensions of such Hamiltonians are proposed by enlarging su(1,1) to a ${\rm su}(1,1/1) \sim {\rm osp}(2/2, \R)$ superalgebra. This allows the construction of new non-Hermitian Hamiltonians related by similarity to Hermitian ones. Some examples of them are reviewed.
The phenomenon of quantum phase transition is considered in the special case in which the evolution laws remain unitary and in which the bound-state energies remain observable. The conventional Hermiticity of observables is lost at the interface, replaced by the so called quasi-Hermiticity. Several features of the passage of the system through the interface are discussed and illustrated by elementary illustrative ${\cal PT}-$symmetric examples.
We construct a previously unknown $E_2$-quasi-exactly solvable non-Hermitian model whose eigenfunctions involve weakly orthogonal polynomials obeying three-term recurrence relations that factorize beyond the quantization level. The model becomes Hermitian when one of its two parameters is fixed to a specific value. We analyze the double scaling limit of this model leading to the complex Mathieu equation. The norms, Stieltjes measures and moment functionals are evaluated for some concrete values of one of the two parameters.
We investigate quasi-hermitian quantum mechanics in phase space using standard deformation quantization methods: Groenewold star products and Wigner transforms. We focus on imaginary Liouville theory as a representative example where exact results are easily obtained. We emphasize spatially periodic solutions, compute various distribution functions and phase-space metrics, and explore the relationships between them.
Using the method of point canonical transformation, we derive some exactly solvable rationally extended quantum Hamiltonians which are non-Hermitian in nature and whose bound state wave functions are associated with Laguerre- or Jacobi-type $X_1$ exceptional orthogonal polynomials. These Hamiltonians are shown, with the help of imaginary shift of co-ordinate: $ e^{-αp} x e^{αp} = x+ i α$, to be both quasi and pseudo-Hermitian. It turns out that the corresponding energy spectra is entirely real.
We give a simple proof of the fact that every diagonalizable operator that has a real spectrum is quasi-Hermitian and show how the metric operators associated with a quasi-Hermitian Hamiltonian are related to the symmetry generators of an equivalent Hermitian Hamiltonian.
In this talk I present a simple and unified approach to both exact and quasi-exact solvabilities of the one-dimensional Schrödinger equation. It is based on the prepotential together with Bethe ansatz equations. This approach gives the potential as well as the eigenfunctions and eigenvalues simultaneously. In this approach the system is completely defined by the choice of the change of variables, and the so-called zero-th order prepotential. We illustrate the approach by several examples of Hermitian and non-Hermitian Hamiltonians with real energies. The method can be easily extended to the co…
We construct six multi-parameter families of Hermitian quasi-exactly solvable matrix Schroedinger operators in one variable. The method for finding these operators relies heavily upon a special representation of the Lie algebra o(2,2) whose representation space contains an invariant finite-dimensional subspace. Besides that we give several examples of quasi-exactly solvable matrix models that have square-integrable eigenfunctions. These examples are in direct analogy with the quasi-exactly solvable scalar Schroedinger operators obtained by Turbiner and Ushveridze.
The discovery of topological phases in non-Hermitian open classical and quantum systems challenges our current understanding of topological order. Non-Hermitian systems exhibit unique features with no counterparts in topological Hermitian models, such as failure of the conventional bulk-boundary correspondence and non-Hermitian skin effect. Advances in the understanding of the topological properties of non-Hermitian lattices with translational invariance have been reported in several recent studies, however little is known about non-Hermitian quasicrystals. Here we disclose topological phases …
We show how generic non-Hermitian tight-binding lattice models can be realized in an unconditional, quantum-mechanically consistent manner by constructing an appropriate open quantum system. We focus on the quantum steady states of such models for both fermionic and bosonic systems. Surprisingly, key features and spatial structures in the steady state cannot be simply understood from the non-Hermitian Hamiltonian alone. Using the 1D Hatano-Nelson model as a paradigmatic example, we show that the steady state has a marked sensitivity to boundary conditions. This dependence however is qualitativ…
We propose the notion of $E_{2}$-quasi-exact solvability and apply this idea to find explicit solutions to the eigenvalue problem for a non-Hermitian Hamiltonian system depending on two parameters. The model considered reduces to the complex Mathieu Hamiltonian in a double scaling limit, which enables us to compute the exceptional points in the energy spectrum of the latter as a limiting process of the zeros for some algebraic equations. The coefficient functions in the quasi-exact eigenfunctions are univariate polynomials in the energy obeying a three-term recurrence relation. The latter prop…
An elucidation of the current state of art in quasi-Hermitian quantum theory (QHQT) as inspired by the recent paper by Alase et al (J. Phys. A: Math. Theor. 55 (2022) 244003, paper [1]) is offered. We point out that the author's main discovery (viz., that the QHQT does not extend the standard quantum theory) is not new. In a related comment on the author's method of proof performed in ``the framework of general probabilistic theories'' (GPT) we add that also in this context a few other, mathematically consistent GPT-like theories are already available in the literature (pars pro toto we mentio…
We show that a non-Hermitian operator with a tridiagonal matrix representation in a finite-dimensional vector space is similar to an Hermitian operator. The required condition is sufficient and simple examples show that it is not necessary. We derive quite general features of the eigenvalues and eigenvectors for a somewhat particular case.
The rationale for introducing non-hermitian Hamiltonians and other observables is reviewed and open issues identified. We present a new approach based on Moyal products to compute the metric for quasi-hermitian systems. This approach is not only an efficient method of computation, but also suggests a new perspective on quasi-hermitian quantum mechanics which invites further exploration. In particular, we present some first results which link the Berry connection and curvature to non-perturbative properties and the metric.
Many features of Bloch oscillations in one-dimensional quantum lattices with a static force can be described by quasiclassical considerations for example by means of the acceleration theorem, at least for Hermitian systems. Here the quasiclassical approach is extended to non-Hermitian lattices, which are of increasing interest. The analysis is based on a generalised non-Hermitian phase space dynamics developed recently. Applications to a single-band tight-binding system demonstrate that many features of the quantum dynamics can be understood from this classical description qualitatively and ev…
We associate the stationary harmonic oscillator with time-dependent systems exhibiting non-Hermiticity by means of point transformations. The new systems are exactly solvable, with all-real spectrum, and transit to the Hermitian configuration for the appropriate values of the involved parameters. We provide a concrete generalization of the Swanson oscillator that includes the Caldirola-Kanai model as a particular case. Explicit solutions are given in both, the classical and quantum pictures.
We give an explicit characterization of the most general quasi-Hermitian operator H, the associated metric operators η_+, and η_+-pseudo-Hermitian operators acting in two-dimensional complex Euclidean space C^2. These operators represent the physical observables of a model whose Hamiltonian and Hilbert space are respectively H and C^2 endowed with the inner product defined by η_+. Our calculations allows for a direct demonstration of the fact that the choice of an irreducible family of observables fixes the metric operator up to a multiplicative factor.
It has been argued that it is incompatible to maintain unitary time-evolution for time-dependent non-Hermitian Hamiltonians when the metric operator is explicitly time-dependent. We demonstrate here that the time-dependent Dyson equation and the time-dependent quasi-Hermiticity relation can be solved consistently in such a scenario for a time-dependent Dyson map and time-dependent metric operator, respectively. These solutions are obtained at the cost of rendering the non-Hermitian Hamiltonian to be a non-observable operator as it ceases to be quasi-Hermitian when the metric becomes time-depen…
Time-periodic driving fields could endow a system with peculiar topological and transport features. In this work, we find dynamically controlled localization transitions and mobility edges in non-Hermitian quasicrystals via shaking the lattice periodically. The driving force dresses the hopping amplitudes between lattice sites, yielding alternate transitions between localized, mobility edge and extended non-Hermitian quasicrystalline phases. We apply our Floquet engineering approach to five representative models of non-Hermitian quasicrystals, obtain the conditions of photon-assisted localizat…
We show that the consequences of an introduction of a manifest time-dependence in a pseudo-Hermitian Hamiltonian H=H(t) are by far less drastic than suggested by A. Mostafazadeh in Phys. Lett. B 650 (2007) 208 (arXiv:0706.1872v2 [quant-ph]). In particular, the unitarity of the evolution does not necessitate the time-independence of the metric $η_+=η_+(t)$.
Non-Hermitian effects could create rich dynamical and topological phase structures. In this work, we show that the collaboration between lattice dimerization and non-Hermiticity could generally bring about mobility edges and multiple localization transitions in one-dimensional quasicrystals. Non-Hermitian extensions of the Aubry-André-Harper (AAH) model with staggered onsite potential and dimerized hopping amplitudes are introduced to demonstrate our results. Reentrant localization transitions due to the interplay between quasiperiodic gain/loss and lattice dimerization are found. Quantized wi…
Non-Hermiticity significantly enriches the properties of topological models, leading to exotic features such as the non-Hermitian skin effects and non-Bloch bulk-boundary correspondence that have no counterparts in Hermitian settings. Its impact is particularly illustrating in non-Hermitian quasicrystals where the interplay between non-Hermiticity and quasiperiodicity results in the concurrence of the delocalization-localization transition, the parity-time (PT)-symmetry breaking, and the onset of the non-Hermitian skin effects. Here we experimentally simulate non-Hermitian quasicrystals using …
As a hallmark of pure quantum effect, quantum entanglement has provided unconventional routes to condensed matter systems. Here, from the perspective of quantum entanglement, we disclose exotic quantum physics in non-Hermitian quasicrystals. We present a class of experimentally realizable models for non-Hermitian quasicrystal chains, in which asymmetric hopping and complex potential coexist. We diagnose global phase diagram by means of entanglement from both real-space and momentum-space partition. By measuring entanglement entropy, we numerically determine the metal-insulator transition point…
Topological metal-insulator transitions in one-dimensional non-Hermitian quasicrystals: beyond PT-symmetry
College of Physics and Optoelectronic Engineering, Ocean University of China（中国海洋大学物理与光电工程学院）；Qingdao Key Laboratory of Advanced Optoelectronics（青岛先进光电子重点实验室）；Engineering Research Center of Advanced Marine Physical Instruments and Equipment of MOE（教育部海洋高端装备工程研究中心）；Department of Physics, National University of Singapore（新加坡国立大学物理系）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；MajuLab, CNRS-UCA-SU-NUS-NTU International Joint Research Unit（马久实验室，CNRS-南法大学-苏黎世联邦理工学院-新加坡国立大学-南洋理工大学国际联合研究单位）；Center for Phononics and Thermal Energy Science, China-EU Joint Lab on Nanophononics, Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, School of Physics Science and Engineering, Tongji University（同济大学物理科学与技术学院，声子学与热能科学中心，中欧纳米声子学联合实验室，上海市特殊人工微结构材料与技术重点实验室）
The Moyal product is used to cast the equation for the metric of a non-hermitian Hamiltonian in the form of a differential equation. For Hamiltonians of the form $p^2+V(ix)$ with $V$ polynomial this is an exact equation. Solving this equation in perturbation theory recovers known results. Explicit criteria for the hermiticity and positive definiteness of the metric are formulated on the functional level.
An open U_q(sl_2)-invariant spin chain of spin S and length N with inhomogeneous coupling is investigated as an example of a non-Hermitian (quasi-Hermitian) model. For several particular cases of such a chain, the ranges of the deformation parameter gamma are determined for which the spectrum of the model is real. For a certain range of gamma, a universal metric operator is constructed and thus the quasi-Hermiticity of the model is established. The constructed metric operator is non-dynamical, its structure is determined only by the symmetry of the model. The results apply, in particular, to a…
We study a general class of PT-symmetric tridiagonal Hamiltonians with purely imaginary interaction terms in the quasi-hermitian representation of quantum mechanics. Our general Hamiltonian encompasses many previously studied lattice models as special cases. We provide numerical results regarding domains of observability and exceptional points, and discuss the possibility of explicit construction of general metric operators (which in turn determine all the physical Hilbert spaces). The condition of computational simplicity for the metrics motivates the introduction of certain one-parametric sp…
Non-Hermitian quasicrystals possess 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-Andre-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 ed…
The cooperation between time-periodic driving fields and non-Hermitian effects could endow systems with distinctive spectral and transport properties. In this work, we uncover an intriguing class of non-Hermitian Floquet matter in one-dimensional quasicrystals, which is characterized by the emergence of multiple driving-induced ${\cal PT}$-symmetry breaking/restoration, mobility edges, and reentrant localization transitions. These findings are demonstrated by investigating the spectra, level statistics, inverse participation ratios and wavepacket dynamics of a periodically quenched nonreciproc…
A non-hermitian deformation of the one-dimensional transverse Ising model is shown to have the property of quasi-hermiticity. The transverse Ising chain is obtained from the starting non-hermitian Hamiltonian through a similarity transformation. Consequently, both the models have identical eigen-spectra, although the eigen-functions are different. The metric in the Hilbert space, which makes the non-hermitian model unitary and ensures the completeness of states, has been constructed explicitly. Although the longitudinal correlation functions are identical for both the non-hermitian and the her…
Starting on the basis of the non-commutative q-differential calculus, we introduce a generalized q-deformed Schrödinger equation. It can be viewed as the quantum stochastic counterpart of a generalized classical kinetic equation, which reproduces at the equilibrium the well-known q-deformed exponential stationary distribution. In this framework, q-deformed adjoint of an operator and q-hermitian operator properties occur in a natural way in order to satisfy the basic quantum mechanics assumptions.
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 π-quasienergy edge states at the boundaries of the lattice. Non-Hermiticity induced Floquet topolog…
In 1956 Dyson analyzed the low-energy excitations of a ferromagnet using a Hamiltonian that was non-Hermitian with respect to the standard inner product. This allowed for a facile rendering of these excitations (known as spin waves) as weakly interacting bosonic quasi-particles. More than 50 years later, we have the full denouement of non-Hermitian quantum mechanics formalism at our disposal when considering Dyson's work, both technically and contextually. Here we recast Dyson's work on ferromagnets explicitly in terms of two inner products, with respect to which the Hamiltonian is always self…
It is well known that an (in general, non-commutative) set of non-Hermitian operators $Λ_j$ with real eigenvalues need not necessarily represent observables. We describe a specific class of quantum models in which these operators plus the underlying physical Hilbert-space metric $Θ$ are all represented in terms of an auxiliary operator $(N+1)-$plet $Z_k$, $k=0,1,\ldots,N$. Our formalism degenerates to the ${\cal PT}-$symmetric quantum mechanics at $N=2$, with metric $Θ=Z_2Z_1$, parity ${\cal P}=Z_2$, charge ${\cal C}=Z_1$ and Hamiltonian $H=Z_0$.
Anomalous Localization and Mobility Edges in Non-Hermitian Quasicrystals with Disordered Imaginary Gauge Fields
Sanli Honors College, School of Physics and Electronics Engineering, Shanxi University（山西大学物理与电子工程学院三立荣誉学院）；Institute of Theoretical Physics and State Key Laboratory of Quantum Optics Technologies and Devices, Shanxi University（山西大学理论物理研究所量子光学技术与器件国家重点实验室）；State Key Laboratory of Quantum Optics Technologies and Devices, Institute of Laser Spectroscopy, Shanxi University（山西大学激光光谱研究所量子光学技术与器件国家重点实验室）
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 investigate the topology and localization of one-dimensional Hermitian and non-Hermitian Su-Schrieffer-Heeger chains with quasiperiodic hopping modulations. In the Hermitian case, phase diagrams are obtained by numerically and analytically calculating various topological and localization characters. We show the presence of topological extended, intermediate, and localized phases due to the coexistence of independent topological and localization phase transitions driven by the quasiperiodic disorder. Unlike the gapless and localized TAI phase in one-dimensional random disordered systems, we …
