non-Hermitian PT Research Articles

Schwinger-Dyson equations and mass generation for an axion theory with a [formula omitted] symmetric Yukawa fermion interaction

A nonperturbative Schwinger-Dyson analysis of mass generation is presented for a non-Hermitian PT -symmetric field theory in four dimensions of an axion coupled to a Dirac fermion. The model is motivated by phenomenological considerations. The axion has a quartic self-coupling λ and a Yukawa coupling g to the fermion. The Schwinger-Dyson equations are derived for the model with generic couplings. In the non-Hermitian case there is an additional nonperturbative contribution to the scalar mass. In a simplified rainbow analysis the solutions for the SD equations, are given for different regimes of the couplings g and λ.

Gain-Loss-Induced Hybrid Skin-Topological Effect.

Non-Hermitian topological effects are of crucial importance both in fundamental physics and applications. Here we discover the gain-loss-induced hybrid second-order skin-topological effect and the PT phase transition in skin-topological modes. By studying a non-Hermitian Haldane model, we find that the topological edge modes are localized on a special type of corner, while the bulk modes remain extended. Such an effect originates from the interplay between gain, loss, and the chiral edge currents induced by the nonlocal flux, which can be characterized by considering the properties of the edge sites as a one-dimensional chain. We establish a relation between the skin-topological effect and the PT symmetries belonging to different edges. Moreover, we discover the PT phase transition with the emergence of exceptional points between pairs of skin-topological modes. Our results pave the way for the investigation of non-Hermitian topological physics and PT phase transition in higher-dimensional systems.

Temporal correlation beyond quantum bounds in non-Hermitian PT- symmetric dynamics of a two level system

We study the dynamics of a non-Hermitian PT- symmetric Hamiltonian of a two level system (TLS) which has real eigenvalues. Within the framework of Hermitian quantum mechanics, it is known that maximal violation of Leggett–Garg inequality (LGI) is bounded by K 3 = 3/2 (Luder’s bound). We show that this absolute bound can be evaded when dynamics is governed by non-Hermitian PT- symmetric Hamiltonians with real eigenvalues. Moreover, the extent of violation can be optimized to asymptotically approach the algebraic maximum of , which is otherwise observed for Hermitian Hamiltonian with infinite dimensional Hilbert space. The extreme violation of LGI is shown to be directly related to the two basic ingredients: (i) the Bloch equation for the TLS having a non-linear terms which allow for accelerated dynamics of states on the Bloch sphere exceeding all known quantum speed limits of state evolution; and (ii) quantum trajectory of states lie on a great circle (geodesic path) on the Bloch sphere at all times. We demonstrate that such extreme temporal correlation of TLS can be simulated in realistic system by embedding the TLS into a higher dimensional Hilbert space such that the composite system obeys unitary dynamics. Specifically we show that a four dimensional embedding of non-Hermitian PT- symmetric TLS is enough to host K 3 → 3 limit. We also discuss the effect of random noise on our results. Finally we conclude with a comparative study of our results with existing experimental realization of embedding.

Non-Hermitian quantum impurity systems in and out of equilibrium: Noninteracting case

We provide systematic analysis on a non-Hermitian PT -symmetric quantum impurity system both in and out of equilibrium, based on exact computations. In order to understand the interplay between non-Hermiticity and Kondo physics, we focus on a prototypical noninteracting impurity system, the resonant level model, with complex coupling constants. Explicitly constructing biorthogonal basis, we study its thermodynamic properties as well as the Loschmidt echo starting from the initially disconnected two free fermion chains. Remarkably, we observe the universal crossover physics in the Loschmidt echo, both in the PT broken and unbroken regimes. We also find that the ground state quantities we compute in the PT broken regime can be obtained by analytic continuation. It turns out that Kondo screening ceases to exist in the PT broken regime, which was also previously predicted in the non-hermitian Kondo model. All the analytical results are corroborated against biorthogonal free fermion numerics.

Parity-Time Symmetry in Non-Hermitian Complex Optical Media.

The exploration of quantum-inspired symmetries in optical and photonic systems has witnessed immense research interest both fundamentally and technologically in a wide range of subject areas in physics and engineering. One of the principal emerging fields in this context is non-Hermitian physics based on parity-time symmetry, originally proposed in the studies pertaining to quantum mechanics and quantum field theory and recently ramified into a diverse set of areas, particularly in optics and photonics. The intriguing physical effects enabled by non-Hermitian physics and PT symmetry have enhanced significant application prospects and engineering of novel materials. In addition, there has been increasing research interest in many emerging directions beyond optics and photonics. Here, the state-of-the art developments in the field of complex non-Hermitian physics based on PT symmetry in various physical settings are brought together, and key concepts, a background, and a detailed perspective on new emerging directions are described. It can be anticipated that this trendy field of interest will be indispensable in providing new perspectives in maneuvering the flow of light in the diverse physical platforms in optics, photonics, condensed matter, optoelectronics, and beyond, and will offer distinctive application prospects in novel functional materials.

PT symmetry protected non-Hermitian topological systems

We study PT symmetry protected topological phase in non-Hermitian 1D and 2D systems. We show that topological phase exist in non-Hermitian PT symmetric systems for both gapped and gapless systems and discuss appearance of exceptional points. We apply our formalism to a complex extension of the SSH model, topological semimetals and nodal superconductors.

On the quantum mechanics of bicomplex Hamiltonian system

We investigate the Schrödinger equation in the framework of bicomplex numbers, which are pairs of complex numbers making up a commutative ring with zero-divisors. We propose an analytical method to solve bicomplex-version of Schrödinger equation corresponding to the systems of Hamiltonians of both hermitian (self-adjoint) and non-hermitian PT symmetric type. In our approach we extend the existing mathematical formulation of quantum system searching for the exact or quasi-exact solution for ground state energy eigenvalues and associated wave functions acting in bicomplex Hilbert space. The model concerning hermitian Hamiltonians is then applied to the problems of two bicomplex valued polynomial oscillators one involving x2 and another of isotonic type. The ground states and associated energy values for both the oscillators are found to be hyperbolic in nature. The model in connection to the unbroken PT symmetric Hamiltonians is then applied to illustrate the problems of complex and bicomplex valued shifted oscillators.

Non-Hermitian neutrino oscillations in matter with PT symmetric Hamiltonians

We introduce and develop a novel approach to extend the ordinary two-flavor neutrino oscillation formalism in matter using a non-Hermitian PT symmetric effective Hamiltonian. The condition of PT symmetry is weaker and less mathematical than that of hermicity, but more physical, and such an extension of the formalism can give rise to sub-leading effects in neutrino flavor transitions similar to the effects by so-called non-standard neutrino interactions. We derive the necessary conditions for the spectrum of the effective Hamiltonian to be real as well as the mappings between the fundamental and effective parameters. We find that the real spectrum of the effective Hamiltonian will depend on all new fundamental parameters introduced in the non-Hermitian PT symmetric extension of the usual neutrino oscillation formalism and that either i) the spectrum is exact and the effective leptonic mixing must always be maximal or ii) the spectrum is approximate and all new fundamental parameters must be small.

Finite-temperature quantum criticality in a complex-parameter plane

A conventional quantum phase transition (QPT) occurs not only at zero temperature, but also exhibits finite-temperature quantum criticality. Motivated by the discovery of the pseudo-Hermiticity of non-Hermitian systems, we explore the finite-temperature quantum criticality in a non-Hermitian PT -symmetric Ising model. We present the complete set of exact eigenstates of the non-Hermitian Hamiltonian, based on which the mixed-state fidelity in the context of biorthogonal bases is calculated. Analytical and numerical results show that the fidelity approach to finite-temperature QPT can be extended to the non-Hermitian Ising model. This paves the way for experimental detection of quantum criticality in a complex-parameter plane.

Non-Hermitian PT Symmetric Hamiltonian with Position-Dependent Masses: Associated Schrödinger Equation and Finite-Norm Solutions

A one-dimensional non-Hermitian PT symmetric Hamiltonian, characterized by position-dependent masses, defines a Schrödinger equation in terms of a field Ψ(x, t). Based on an exact classical field theory, the necessity of an extra field Φ(x, t) (which satisfies a conjugate equation and in general different is from Ψ∗(x, t)) is shown. Simple applications are investigated by solving analytically both equations and it is shown that the effective masses proposed lead to a probability density characterized by a finite norm, typical of the physical situation that occurs with the concentration of electrons in some semiconductor heterojunctions. An extension to a three-dimensional space is also presented.

Wavepacket Dynamics in a Non-Hermitian PT Symmetric Tight-Binding Chain

We examine the time development of a wavepacket in a Non-Hermitian PT symmetric tight-binding chain based on the Bethe ansatz solutions. The complete orthogonal sets in both unbroken and broken PT symmetry regions are established with respect to an inner product in terms of CPT conjugation. The dynamical development of a stationary wavepacket is exhibited, showing the impacts from additional imaginary potentials within unbroken and broken symmetric regimes. Our result indicates that there are distinct dynamic characteristics for both two regions. The unbroken symmetric imaginary end potentials can be regarded as a special kind of boundary condition in the sense of Hermitian quantum mechanics.

The asymptotic iteration method applied to certain quasinormal modes and non Hermitian systems

Abstract We study the Schrödinger equation with potentials admitting quasinormal modes using the asymptotic iteration method (AIM). We also study non-Hermitian PT symmetric potentials using AIM. The spectra, in all cases, are found to be in excellent agreement with exact results.

Non-Hermitian Quantum Systems and Time-Optimal Quantum Evolution

Recently, Bender et al. have considered the quantum brachistochrone problem for the non-Hermitian PT -symmetric quantum system and have shown that the optimal time evolution required to transform a given initial state | ii into a specific final state | fi can be made arbitrarily small. Additionally, it has been shown that finding the shortest possible time requires only the solution of the two-dimensional problem for the quantum system governed by the effective Hamiltonian acting in the subspace spanned by | ii and | fi. In this paper, we study a similar problem for the generic non-Hermitian Hamiltonian, focusing our attention on the geometric aspects of the problem.

Nonunique [formula omitted] operator in [formula omitted] quantum mechanics

The three simultaneous algebraic equations, C 2 = 1 , [ C , PT ] = 0 , [ C , H ] = 0 , which determine the C operator for a non-Hermitian PT -symmetric Hamiltonian H, are shown to have a nonunique solution. Specifically, the C operator for the Hamiltonian H = 1 2 p 2 + 1 2 μ 2 q 2 + i ϵ q 3 is determined perturbatively to first order in ϵ and it is demonstrated that the C operator contains an infinite number of arbitrary parameters. For each different C operator, the corresponding equivalent isospectral Dirac Hermitian Hamiltonian h is calculated.

Eigenvalue Problems of non-Hermitian Systems via Improved Asymptotic Iteration Method

We simply use the relation between the asymptotic iteration method and the Nikiforov–Uvarov method for the analytical solution of the second order linear ordinary differential equations. We apply this relation to study the Schrödinger equation with potentials admitting quasinormal modes. Non-Hermitian PT symmetric potentials have also been studied. Energy eigenvalues in all the cases by the relation are found to be consistent with exact results.

Optical lattices with PT symmetry are not transparent

In diffraction of a plane wave by a non-Hermitian PT symmetric optical lattice, the sum of the Bragg beam intensities need not be conserved, even though the gain and loss are equally distributed: the evolution is not unitary. Instead, different sums are conserved, in which the intensities are weighted with real numbers (positive or negative); several such sum rules are derived. Two-beam diffraction from a refractive index of the form constant −a cos x + ib sin x is studied in detail; the sum rule depends on the balance between the (real) Hermitian parameter a and the (real) anti-Hermitian parameter b.

Faster than Hermitian Time Evolution

For any pair of quantum states, an initial state |Ii and a final quantum state |Fi, in a Hilbert space, there are many Hamiltonians H under which |Ii evolves into |Fi. Let us impose the constraint that the difference between the largest and smallest eigenvalues of H, Emax and Emin, is held fixed. We can then determine the Hamiltonian H that satisfies this constraint and achieves the transformation from the initial state to the final state in the least possible time . For Hermitian Hamiltonians, has a nonzero lower bound. However, among non-Hermitian PT -symmetric Hamiltonians satisfying the same energy constraint, can be made arbitrarily small without violating the time-energy uncertainty principle. The minimum value of can be made arbitrarily small because for PT -symmetric Hamiltonians the path from the vector |Ii to the vector |Fi, as measured using the Hilbert-space metric appropriate for this theory, can be made arbitrarily short. The mechanism described here is similar to that in general relativity in which the distance between two space-time points can be made small if they are connected by a wormhole. This result may have applications in quantum computing.

P T symmetric models with nonlinear pseudosupersymmetry

By applying the higher order Darboux algorithm to an exactly solvable non-Hermitian PT symmetric potential, we obtain a hierarchy of new exactly solvable non-Hermitian PT symmetric potentials with real spectra. It is shown that the symmetry underlying the potentials so generated and the original one is nonlinear pseudosupersymmetry. We also show that this formalism can be used to generate a larger class of new solvable potentials when applied to non-Hermitian systems.

Semiclassical calculation of the [formula omitted] operator in [formula omitted]-symmetric quantum mechanics

To determine the Hilbert space and inner product for a quantum theory defined by a non-Hermitian PT -symmetric Hamiltonian H, it is necessary to construct a new time-independent observable operator called C. It has recently been shown that for the cubic PT -symmetric Hamiltonian H= p 2+ x 2+ iϵx 3 one can obtain C as a perturbation expansion in powers of ϵ. This Letter considers the more difficult case of noncubic Hamiltonians of the form H= p 2+ x 2( ix) δ ( δ⩾0). For these Hamiltonians it is shown how to calculate C by using nonperturbative semiclassical methods.

On an exactly solvable BN type Calogero model with non-Hermitian PT invariant interaction

An exactly solvable many-particle quantum system is proposed by adding some non-Hermitian but PT invariant interactions to the B N Calogero model. We have shown that such extended B N Calogero model leads to a real spectrum which obey generalised exclusion statistics. It is also found that the corresponding exchange statistics parameter exhibit ‘reflection symmetry’ provided the strength of a PT invariant interaction exceeds a critical value.