Robust and fragile PT-symmetric phases in a tight-binding chain

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 +/- i gamma located at arbitrary (P-symmetric) positions (m,(m) over bar = 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 is algebraically fragile. We analytically and numerically obtain the critical impurity potential gamma(PT) and show that gamma(PT) alpha 1/N -> 0 as N -> infinity 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 gamma(PT) in the limit N -> infinity 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 hitherto unexplored PT-symmetric Hamiltonians.