Non-Hermitian delocalization and eigenfunctions

Abstract

Recent literature on delocalization in non-Hermitian systems has stressed criteria based on sensitivity of eigenvalues to boundary conditions and the existence of a non-zero current. We emphasize here that delocalization also shows up clearly in eigenfunctions, provided one studies the product of left-and right-eigenfunctions, as required on physical grounds, and not simply the squared modulii of the eigenfunctions themselves. We also discuss the right-and left-eigenfunctions of the ground state in the delocalized regime and suggest that the behavior of these functions, when considered separately, may be viewed as "intermediate" between localized and delocalized.

Bloch oscillations in non-Hermitian lattices with trajectories in the complex plane

“…The model (34) was originally introduced by Hatano and Nelson in a pioneering work to study the motion of magnetic flux lines in disordered type-II superconductors [45], showing that an 'imaginary' gauge field in a disordered one-dimensional lattice can induce a delocalization transition, i.e. it can prevent Anderson localization [45][46][47]. A possible implementation in an optical setting of the Hatano-Nelson model (34), based on light transport in coupled resonator optical waveguides, has been recently suggested in Ref.…”

“…However, in strongly non-Hermitian potentials the band structure can become imaginary, and in this case it is not clear whether and how the semiclassical picture of BOs can be extended to account for a complex energy lattice band. Several examples of tight-binding lattice models that show a non-vanishing imaginary part of the band dispersion curve have been discussed in many works, including the Hatano-Nelson model describing the hopping motion of a quantum particle in a linear tight-binding lattice with an imaginary vector potential [45][46][47][48], the nonHermitian extension of the Su-Schrieffer-Heeger tightbinding model [49][50][51], PT -symmetric binary superlattices [36,52], and the PT -symmetric Aubry-Andre model [53]. Lattice bands with a non vanishing imaginary part could be realized in synthetic temporal optical crystals, waveguide lattices, coupled-resonator optical waveguides, microwave resonator chains, etc.…”

Spectral Theory of Pseudo-Ergodic Operators

“…Recent papers have obtained some striking results concerning the spectral properties of the non-self-adjoint (nsa) Anderson model, which models the growth of bacteria in an inhomogeneous environment, [10,11,12,13,5]. To be more precise the authors have determined the asymptotic limit of the spectrum of a nsa random finite periodic chain almost surely as the length of the chain increases to infinity.…”

P T symmetric models with nonlinear pseudosupersymmetry

“…If all the intermediate potentials [11] and possible applications [12]. It is well known by now that a non Hermitian PT symmetric Hamiltonian admits real eigenvalues if the eigenfunctions, too, respect the PT invariance (the so-called unbroken PT symmetry), whereas the eigenvalues occur as complex conjugate pairs if PT symmetry is spontaneously broken (in this case the eigenfunctions are no longer PT invariant).…”