Degrees and signatures of brokenPTsymmetry in nonuniform lattices

Abstract

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.

Origin of maximal symmetry breaking in evenPT-symmetric lattices

“…Since this result is true for all eigenfunctions, it follows that the PT -symmetry breaks maximally and the critical impurity strength is solely determined by the hopping amplitude between the two impurities. This robust result also explains the fragile nature of PT -symmetric phase in lattices with hopping function t α (k) for α < 0 [20]: in this case, the lattice bandwidth ∆ α ∼ N −|α|/2 whereas the hopping amplitude between the two nearest-neighbor impurities scales as t b ∼ N −|α| . Therefore the critical impurity strength γ c /∆ α ∼ N −|α|/2 → 0 as N → ∞.…”

“…In an exceptional contrast, when m = N/2 -nearest neighbor impurities on an even lattice -all eigenvalues simultaneously become complex at the onset of PT -symmetry breaking. This maximal symmetry breaking is accompanied by unique signatures in the time-evolution of a wavepacket [20]. …”

“…The breaking of PT -symmetry, along with the attendant non-reciprocal behavior, was recently observed in two coupled optical waveguides [11,12] and has ignited further interest in PT -symmetric lattice models. These evanescently coupled waveguides provide an excellent realization [13] of an ideal, one-dimensional lattice with tunable hopping [14], disorder [15], and non-Hermitian, on-site, impurity potentials [16,17].Recently nonuniform lattices with site-dependent hop-α/2 and a pair of imaginary impurities ±iγ at positions (m,m) have been extensively explored [17][18][19][20], wherem = N + 1 − m and N 1 is the number of lattice sites. The PT -symmetric phase in such a lattice is robust when α ≥ 0, the loss and gain impurities ±iγ are closest to each other, and γ ≤ γ c where the critical impurity strength is proportional to the bandwidth of the clean lattice, γ c ∝ 4t 0 (N/2) α .…”

“…α/2 and a pair of imaginary impurities ±iγ at positions (m,m) have been extensively explored [17][18][19][20], wherem = N + 1 − m and N 1 is the number of lattice sites. The PT -symmetric phase in such a lattice is robust when α ≥ 0, the loss and gain impurities ±iγ are closest to each other, and γ ≤ γ c where the critical impurity strength is proportional to the bandwidth of the clean lattice, γ c ∝ 4t 0 (N/2) α .…”

PT symmetric Aubry–Andre model

“…It implies the eigenfunctions of the Hamiltonian are no longer simultaneous eigenfunction of PT operator and consequently the energy spectrum becomes either partially or completely complex. 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.…”