Origin of maximal symmetry breaking in evenPT-symmetric lattices

Abstract

By investigating a parity- and time-reversal- ($\mathcal{P}\mathcal{T}$-) symmetric, $N$-site lattice with impurities $\ifmmode\pm\else\textpm\fi{}i\ensuremath{\gamma}$ and hopping amplitudes ${t}_{0}\phantom{\rule{0.28em}{0ex}}({t}_{b})$ for regions outside (between) the impurity locations, we probe the interimpurity-distance dependence of the critical impurity strength and the origin of maximal $\mathcal{P}\mathcal{T}$-symmetry breaking that occurs when the impurities are nearest neighbors. Through a simple and exact derivation, we prove that the critical impurity strength is equal to the hopping amplitude between the impurities, ${\ensuremath{\gamma}}_{c}={t}_{b}$, and the simultaneous emergence of $N$ complex eigenvalues is a robust feature of any $\mathcal{P}\mathcal{T}$-symmetric hopping profile. Our results show that the threshold strength ${\ensuremath{\gamma}}_{c}$ can be widely tuned by a small change in the global profile of the lattice and thus have experimental implications.