Abstract
We study, analytically and numerically, a simple -symmetric tight-binding ring with an onsite energy a at the gain and loss sites. We show that if a ≠ 0, the system generically exhibits an unbroken -symmetric phase. We study the nature of the spectrum in terms of the singularities in the complex parameter space as well as the behavior of the eigenstates at large values of the gain and loss strength. We find that in addition to the usual exceptional points (EPs), there are ‘diabolical points’, and inverse EPs at which complex eigenvalues reconvert into real eigenvalues. We also study the transport through the system. We calculate the total flux from the source to the drain, and how it splits along the branches of the ring. We find that while usually the density flows from the source to the drain, for certain eigenstates a stationary ‘backflow’ of density from the drain to the source along one of the branches can occur. We also identify two types of singular eigenstates, i.e. states that do not depend on the strength of the gain and loss, and classify them in terms of their transport properties.
Highlights
PT -symmetry is a well established field of Quantum Physics, in which many interesting phenomena arise, see [1,2,3,4] and references therein
We find that in addition to the usual exceptional points, there are “diabolical points”, and inverse exceptional points at which complex eigenvalues reconvert into real eigenvalues
We find that in addition to exceptional points (EPs), there are singularities that resemble “diabolical points” [16, 17] and, under certain circumstances, we find inverse exceptional points, at which complex eigenvalues reconvert into real eigenvalues
Summary
PT -symmetry is a well established field of Quantum Physics, in which many interesting phenomena arise, see [1,2,3,4] and references therein. While many properties are known for a one dimensional homogeneous PT -symmetric tightbinding chain with open boundary conditions, see for instance [8,9,10,11], the case with periodic boundary conditions has been less explored [12,13,14,15] One reason for this is that, for constant values of the tunneling couplings and onsite energies, the spectrum typically becomes complex for any constant strength of the gain and loss different from zero [11], which motivated the study of inhomogeneous rings [13]. A different type of singular eigenstates, which we call accidental singular states, only appear for certain configurations and specific values of the onsite energy These states do couple with the gain and loss, and as they have efficient transport, we classify these states as “transparent” (or conducting). We show how some eigenstates vanish on one branch of the ring as the strength of the input and output increases, attaining a “directional character”
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