-symmetry breaking and maximal chirality in a nonuniform symmetric ring

We study the properties of an N-site tight-binding ring with parity and time-reversal () symmetric, Hermitian, site-dependent tunneling and a pair of non-Hermitian, -symmetric, loss and gain impurities ±iγ. The properties of such lattices with open boundary conditions have been intensely explored over the past 2 years. We numerically investigate the -symmetric phase in a ring with a position-dependent tunneling function tα(k) = [k(N − k)]α/2 that, in an open lattice, leads to a strengthened -symmetric phase, and study the evolution of the -symmetric phase from the open chain to a ring. We show that, generally, periodic boundary conditions weaken the -symmetric phase, although for experimentally relevant lattice sizes N ∼ 50, it remains easily accessible. We show that the chirality, quantified by the (magnitude of the) average transverse momentum of a wave packet, shows a maximum at the -symmetric threshold. Our results show that although the wavepacket intensity increases monotonically across the -breaking threshold, the average momentum decays monotonically on both sides of the threshold. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.