We study the phase-diagram of a parity and time-reversal (PT ) symmetric tight-binding chain with N sites and hopping energy J, in the presence of two impurities with imaginary potentials ±iγ located at arbitrary (P-symmetric) positions (m,m = N + 1 − m) on the chain where m ≤ N/2.We find that except in the two special cases where impurities are either the farthest or the closest, the PT -symmetric region -defined as the region in which all energy eigenvalues are real -is algebraically fragile. We analytically and numerically obtain the critical impurity potential γ P T and show that γ P T ∝ 1/N → 0 as N → ∞ except in the two special cases. When the PT symmetry is spontaneously broken, we find that the maximum number of complex eigenvalues is given by 2m. When the two impurities are the closest, we show that the critical impurity strength γ P T in the limit N → ∞ approaches J (J/2) provided that N is even (odd). For an even N the PT symmetry is maximally broken whereas for an odd N , it is sequentially broken. Our results show that the phase-diagram of a PT -symmetric tight-binding chain is extremely rich and that, in the continuum limit, this model may give rise to new PT -symmetric Hamiltonians. * yojoglek@iupui.edu

Abstract. Over the last two decades, advances in fabrication have led to significant progress in creating patterned heterostructures that support either carriers, such as electrons or holes, with specific band structure or electromagnetic waves with a given mode structure and dispersion. In this article, we review the properties of light in coupled optical waveguides that support specific energy spectra, with or without the effects of disorder, that are well-described by a Hermitian tight-binding model. We show that with a judicious choice of the initial wave packet, this system displays the characteristics of a quantum particle, including transverse photonic transport and localization, and that of a classical particle. We extend the analysis to non-Hermitian, parity and time-reversal (PT ) symmetric Hamiltonians which physically represent waveguide arrays with spatially separated, balanced absorption or amplification. We show that coupled waveguides are an ideal candidate to simulate PT -symmetric Hamiltonians and the transition from a purely real energy spectrum to a spectrum with complex conjugate eigenvalues that occurs in them.

We investigate the robustness of the parity-and time-reversal (PT ) symmetric phase in an N -site lattice with a position-dependent, parity-symmetric hopping function and a pair of imaginary, PT -symmetric impurities. We find that the "fragile" PT -symmetric phase in these lattices is stronger than its counterpart in a lattice with constant hopping. With an open system in mind, we explore the degrees of broken PT symmetry and their signatures in single-particle wave-packet evolution. We predict that, when the PT -symmetric impurities are closest to each other, the time evolution of a wave packet in an even-N lattice is remarkably different from that in an odd-N lattice. Our results suggest that PT symmetry breaking in such lattices is accompanied by rich, hitherto unanticipated, phenomena.

In systems with "balanced loss and gain", the PT -symmetry is broken by increasing the nonhermiticity or the loss-gain strength. We show that finite lattices with oscillatory, PT -symmetric potentials exhibit a new class of PT -symmetry breaking and restoration. We obtain the PT phase diagram as a function of potential periodicity, which also controls the location complex eigenvalues in the lattice spectrum. We show that the sum of PT -potentials with nearby periodicities leads to PT -symmetry restoration, where the system goes from a PT -broken state to a PT -symmetric state as the average loss-gain strength is increased. We discuss the implications of this novel transition for the propagation of a light in an array of coupled waveguides.

We investigate the properties of an N -site tight-binding lattice with periodic boundary conditions (PBCs) in the presence of a pair of gain and loss impurities ±iγ and two tunneling amplitudes, t 0 and t b , that are constant along the two paths that connect them. We show that the parity and time-reversal (PT )-symmetric phase of the lattice with PBCs are robust, insensitive to the distance between the impurities, and that the critical impurity strength for PT -symmetry breaking is given by γ PT = |t 0 − t b |. We study the time-evolution of a typical wave packet, initially localized on a single site, across the PT -symmetric phase boundary. We find that it acquires nonzero chirality for γ = 0, and the associated momentum reaches a universal maximum value at the threshold, γ = γ PT , irrespective of the initial location of the wave packet and the lattice parameters. Our results imply that PT -symmetry breaking on a lattice with PBCs has consequences that have no counterpart in open chains.