Real spectra, Anderson localization, and topological phases in one-dimensional quasireciprocal systems

We introduce the one-dimensional quasireciprocal lattices where the forward hopping amplitudes between nearest neighboring sites $\{ t+t_{jR} \}$ are chosen to be a random permutation of the backward hopping $\{ t+t_{jL} \}$ or vice versa. The values of $\{ t_{jL} \}$ (or $\{t_{jR} \}$) can be periodic, quasiperiodic, or randomly distributed. We show that the Hamiltonian matrices are pseudo-Hermitian and the energy spectra are real as long as $\{ t_{jL} \}$ (or $\{t_{jR} \}$) are smaller than the threshold value. While the non-Hermitian skin effect is always absent in the eigenstates due to the global cancellation of local nonreciprocity, the competition between the nonreciprocity and the accompanying disorders in hopping amplitudes gives rise to energy-dependent localization transitions. Moreover, in the quasireciprocal Su-Schrieffer-Heeger models with staggered hopping $t_{jL}$ (or $t_{jR}$), topologically nontrivial phases are found in the real-spectra regimes characterized by nonzero winding numbers. Finally, we propose an experimental scheme to realize the quasireciprocal models in electrical circuits. Our findings shed new light on the subtle interplay among nonreciprocity, disorder, and topology.