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We study the properties of a parity- and time-reversal- (PT) symmetrictight-binding chain of size N with position-dependent hopping amplitude. Incontrast to the fragile PT-symmetric phase of a chain with constant hopping andimaginary impurity potentials, we show that, under very general conditions, ourmodel is {\it always} in the PT-symmetric phase. We numerically obtain theenergy spectrum and the density of states of such a chain, and show that theyare widely tunable. By studying the size-dependence of inverse participationratios, we show that although the chain is not translationally invariant, mostof its eigenstates are extended. Our results indicate that tight-binding modelswith non-Hermitian PT-symmetric hopping have a robust PT-symmetric phase andrich dynamics which may be explored in coupled waveguides.

The robust $\mP\mT$-symmetric chain and properties of its Hermitian counterpart