Raising the $\mathcal{PT}$ transition threshold by strong coupling to neutral chains

The $\mathcal{PT}$ symmetry breaking threshold in discrete realizations of systems with balanced gain and loss is determined by the effective coupling between the gain and loss sites. In one dimensional chains, this threshold is maximum when the two sites are closest to each other or the farthest. We investigate the fate of this threshold in the presence of parallel, strongly coupled, Hermitian (neutral) chains, and find that it is increased by a factor proportional to the number of neutral chains. We present numerical results and analytical arguments for this enhancement. We then consider the effects of adding neutral sites to $\mathcal{PT}$ symmetric dimer and trimer configurations and show that the threshold is more than doubled, or tripled by their presence. Our results provide a surprising way to engineer the $\mathcal{PT}$ threshold in experimentally accessible samples.