Exponentially FragilePTSymmetry in Lattices with Localized Eigenmodes

Abstract

We study the effect of localized modes in lattices of size N with parity-time (PT) symmetry. Such modes are arranged in pairs of quasidegenerate levels with splitting delta approximately exp(-N/xi) where xi is their localization length. The level "evolution" with respect to the PT breaking parameter gamma shows a cascade of bifurcations during which a pair of real levels becomes complex. The spontaneous PT symmetry breaking occurs at gammaPT approximately min{delta}, thus resulting in an exponentially narrow exact PT phase. As N/xi decreases, it becomes more robust with gammaPT approximately 1/N2 and the distribution P(gammaPT) changes from log-normal to semi-Gaussian. Our theory can be tested in the frame of optical lattices.

Invisibility in non-Hermitian tight-binding lattices

“…Non-Hermitian lattices with complex site energies can be mimicked by considering arrays of evanescently-coupled waveguides in which light propagation in each waveguide is either absorbed or amplified by some loss or gain mechanism (see, for instance, [19,25]), where the z-invariant gain or loss coefficients in the various waveguides determine the imaginary parts of the site energies V n . Such nonHermitian lattices have been intensively investigated in the past few years, especially in connection with PTsymmetric quantum mechanics [17][18][19]25]. However, the non-Hermitian lattices that realize invisibility, discussed in Secs.…”

“…E = µ 1 belongs to the point spectrum of H 2 and its eigenfunction is given by Eq. (17). It should be noted that the synthesis of the partner Hamiltonian H 2 , with spectrum σ 2 = σ 1 ∪ {µ 1 }, is not unique because of some freedom left in the choice of φ (1) n satisfying Eq.…”

“…It is the aim of this work to show that fully invisibility of localized defects can be realized in non-Hermitian tight-binding lattices, which are synthesized by iterated application of the intertwining operator technique (Darboux transformation) to a defect-free tight-binding Hermitian lattice. The study of non-Hermitian tightbinding lattices has received in recent years a great attention (see, e.g., [14][15][16][17][18][19] and references therein); such previous studies have been mainly focused to lattices possessing parity-time (PT ) symmetry and were framed in the context of non-Hermitian quantum mechanics [17][18][19][20], however the possibility to realize invisibility in a non-Hermitian lattice was not investigated in such previous works [21]. It should be noted that the class of non-Hermtian lattices synthesized in the present work by application of the Darboux transformation and showing the property of invisibility are not PT -symmetric.…”

PT symmetric Aubry–Andre model

“…The critical point decreases with increasing N and approaches zero when N is large. The PT symmetric phase is said to be fragile since γ P T is zero as N →∞ [13].…”

“…The critical number of non-Hermitian degree is shown to be different for planar and circular array configurations [11] and it can be increased if impurities and tunneling energy are made position-dependent in an extended lattice [12]. However, γ P T decreases with increasing the lattice sites [13][14][15][16], hence the PT symmetric phase is fragile. An important consequence of PT symmetric optical systems is the power oscillations.…”