We consider non-Hermitian dynamics of a quantum particle hopping on a
one-dimensional tight-binding lattice made of N sites with asymmetric hopping
rates induced by a time-periodic oscillating imaginary gauge field. A deeply
different behavior is found depending on the lattice topology. While in a
linear chain (open boundary conditions) an oscillating field can lead to a
complex quasi energy spectrum via a multiple parametric resonance, in a ring
topology (Born-von Karman periodic boundary conditions) an entirely real quasi
energy spectrum can be found and the dynamics is pseudo-Hermitian. In the large
N limit, parametric instability and pseudo-Hermitian dynamics in the two
different lattice topologies are physically explained on the basis of a simple
picture of wave packet propagation.Comment: 10 pages, 6 figure
