We propose that the real spectrum and the orthogonality of the states for several known complex potentials of both types, PT-symmetric and non-PT-symmetric can be understood in terms of currently proposed $η$-pseudo-Hermiticity (Mostafazadeh, quant-ph/0107001) of a Hamiltonian, provided the Hermitian linear automorphism, $η$, is introduced as $e^{-θp}$ which affects an imaginary shift of the co-ordinate : $e^{-θp} x e^{θp} = x+iθ$.
Exact solvability of some non-Hermitian $η$-weak-pseudo-Hermitian Hamiltonians is explored as a byproduct of $η$-weak-pseudo-Hermiticity generators. A class of V_{eff}(x)=V(x)+iW(x) potentials is considered, where the imaginary part W(x) is used as an $η$-weak-pseudo-Hermiticity generator to obtain exactly solvable $η$-weak-pseudo-Hermitian Hamiltonian models.
For a given pseudo-Hermitian Hamiltonian of the standard form: H=p^2/2m+v(x), we reduce the problem of finding the most general (pseudo-)metric operator ηsatisfying H^\dagger=ηH η^{-1} to the solution of a differential equation. If the configuration space is the real line, this is a Klein-Gordon equation with a nonconstant mass term. We obtain a general series solution of this equation that involves a pair of arbitrary functions. These characterize the arbitrariness in the choice of η. We apply our general results to calculate ηfor the PT-symmetric square well, an imaginary scattering potentia…
Imaginary eigenvalues of Hermitian Hamiltonian with an inverted potential well and transition to the real spectrum at exceptional point by a non-Hermitian interaction
When a non-hermitian hamiltonian has a certain symmetry, such as the PT pseudo-hermiticity, it is still possible that the hamiltonian has a real spectrum. In this note, by adding an imaginary potential proportional to ip_1p_2 to the hamiltonian of an anisotropic planar oscillator, we construct a PT pseudo-hermitian hamiltonian and obtain its real spectrum. In addition, we find that the reality of the energy spectrum remains when the model is extended to the canonical noncommutative space.
Time continues to be an intriguing physical property in the modern era. On the one hand, we have the Classical and Relativistic notion of time, where space and time have the same hierarchy, which is essential in describing events in spacetime. On the other hand, in Quantum Mechanics, time appears as a classical parameter, meaning that it does not have an uncertainty relation with its canonical conjugate. In this work, we use a recent proposed spacetime-symmetric formalism~\href{https://doi.org/10.1103/PhysRevA.95.032133}{[Phys.~Rev.~A {\bf 95}, 032133 (2017)]} that tries to solve the unbalance…
A new version of the change of the "phase" (i.e., of the set of observable characteristics) of a quantum system is proposed. In a general scenario the evolution is assumed generated, before the phase transition, by some standard Hermitian Hamiltonian $H^{(before)}$, and, after the phase transition, by one of the recently very popular non-standard, non-Hermitian (but hiddenly Hermitian, i.e., still unitarity-guaranteeing) Hamiltonians $H^{(after)}$. For consistency, a smoothness of matching between the two operators as well as between the related physical Hilbert spaces must be guaranteed. The …
A conventional quantum phase transition (QPT) can be accessed by varying a real parameter at absolute zero temperature. Motivated by the discovery of the pseudo-Hermiticity of non-Hermitian systems, we explore the QPT in non-Hermitian PT-symmetric Ising model, which is driven by a staggered complex transverse field. Exact solution shows that the Laplacian of the groundstate energy density, with respect to real and imaginary components of the transverse field, diverges on the boundary in the complex plane. The phase diagram indicate that the imaginary transverse field has the effect of shrinkin…
We deform the real potential of Poeschl and Teller by a shift of its coordinate in imaginary direction. We show that the new model remains exactly solvable. Its bound states are constructed in closed form. Wave functions are complex and proportional to Jacobi polynomials. Some of them diverge in the Hermitian limit. In contrast, all their energies prove real and shift-independent. In this sense the lost Hermiticity of our family of Hamiltonians seems replaced by their accidental PT symmetry.
The energy bands of non-Hermitian systems exhibit nontrivial topological features that arise from the complex nature of the energy spectrum. Under periodic boundary conditions (PBC), the energy spectrum describes rather generally closed loops in complex plane, characterized by integer nonzero winding numbers. Such nontrivial winding provides the topological signature of the non-Hermitian skin effect (NHSE), i.e. the macroscopic condensation of bulk states at the lattice edges under open boundary conditions (OBC). In spite of the great relevance of band winding in the non-Hermitian topological …
Bridging Frustration and Non-Hermiticity via COMPASS: An Adaptive Biorthogonal Neural Quantum State Framework
Max Planck Institute for the Science of Light（马克斯·普朗克光科学研究所）；Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡大学）；University of Waterloo（滑铁卢大学）；Vector Institute（向量研究所）
We study the spectrum in such a PT-symmetric square well of a diameter L where the "strength of the non-Hermiticity" is controlled by the two parameters, viz., by an imaginary coupling ig and by the distance d of its onset from the origin. We solve this problem and confirm that the spectrum is discrete and real in a non-empty interval of g. Surprisingly, a specific distinction between the bound states is found in their asymptotic stability/instability with respect to an unlimited growth of g. In our model, all of the low-lying levels remain asymptotically unstable at the small d and finite L w…
We study how to understand the complex coordinates involved in the non-Hermitian but PT-symmetric systems. We explore a PT-symmetric oscillator model to show that the entire information on the complex position is attainable. Its real part is from the observation while its imaginary part is from the non-Hermiticity parameter. We also propose a new complex extension of P-transformation and T-transformation (the `parity' and `time reflection' respectively). Particularly, the P-transformation realizes the left-right reflection in the complex plane.
The energy spectrum of bosonic excitations from a condensate is given by the spectrum of a non-Hermitian Hamiltonian constructed from a bosonic Bogoliubov-de Gennes (BdG) Hamiltonian in general even though the system is essentially Hermitian. In other words, two types of non-Hermiticity can coexist: one from the bosonic BdG nature and the other from the open quantum nature. In this paper, we propose boundary-dependent dynamical instability. We first define the bosonic dissipative BdG Hamiltonian in terms of Green's function in Nambu space and discuss the correct particle-hole symmetry of the c…
We explore the Hamiltonian operator H=-d^2/dx^2 + z δ(x) where x is real, δ(x) is the Dirac delta function, and z is an arbitrary complex coupling constant. For a purely imaginary z, H has a (real) spectral singularity at E=-z^2/4. For \Re(z)<0, H has an eigenvalue at E=-z^2/4. For the case that \Re(z)>0, H has a real, positive, continuous spectrum that is free from spectral singularities. For this latter case, we construct an associated biorthonormal system and use it to perform a perturbative calculation of a positive-definite inner product that renders H self-adjoint. This allows us to addr…
Topological stability of the edge states is investigated for non-Hermitian systems. We examine two classes of non-Hermitian Hamiltonians supporting real bulk eigenenergies in weak non-Hermiticity: SU(1,1) and SO(3,2) Hamiltonians. As an SU(1,1) Hamiltonian, the tight-binding model on the honeycomb lattice with imaginary on-site potentials is examined. Edge states with ReE=0 and their topological stability are discussed by the winding number and the index theorem, based on the pseudo-anti-Hermiticity of the system. As a higher symmetric generalization of SU(1,1) Hamiltonians, we also consider S…
We extend the application of the techniques developed within the framework of the pseudo-Hermitian quantum mechanics to study a unitary quantum system described by an imaginary PT-symmetric potential v(x) having a continuous real spectrum. For this potential that has recently been used, in the context of optical potentials, for modelling the propagation of electromagnetic waves travelling in a wave guide half and half filed with gain and absorbing media, we give a perturbative construction of the physical Hilbert space, observables, localized states, and the equivalent Hermitian Hamiltonian. I…
We consider the interaction between the Hermitian world, represented by a real delta-function potential $-αδ(x)$, and the non-Hermitian world, represented by a PT-symmetric pair of delta functions with imaginary coefficients $iβ(δ(x-L)-δ(x+L))$. In the context of standard quantum mechanics, the effect of the introduction of the imaginary delta functions on the bound-state energy of the real delta function and its associated wave-function is small for L large. However, scattering from the combined potentials does not conserve probability as conventionally defined. Both these problems can be stu…
A possible method to investigate non-Hermitian Hamiltonians is suggested through finding a Hermitian operator $η_+$ and defining the annihilation and creation operators to be $η_+$-pseudo-Hermitian adjoint to each other. The operator $η_+$ represents the $η_+$-pseudo-Hermiticity of Hamiltonians. As an example, a non-Hermitian and non-$PT$-symmetric Hamiltonian with imaginary linear coordinate and linear momentum terms is constructed and analyzed in detail. The operator $η_+$ is found, based on which, a real spectrum and a positive-definite inner product, together with the probability explanati…
A non-Hermitian PT-symmetric version of the kicked top is introduced to study the interplay of quantum chaos with balanced loss and gain. The classical dynamics arising from the quantum dynamics of the angular momentum expectation values are derived. The presence of PT-symmetry can lead to stable mixed regular chaotic behaviour without sinks or sources for subcritical values of the gain-loss parameter. This is an example of what is known in classical dynamical systems as reversible dynamical systems. For large values of the kicking strength a strange attractor is observed that also persists if…
