Non-Hermitian Hamiltonian approach to quantum transport in disordered networks with sinks: validity and effectiveness

TL;DR

This paper assesses the validity of using non-Hermitian Hamiltonians to model quantum transport in disordered networks with sinks, showing it is effective when the energy range of the system is within the lead's energy band.

Contribution

The study derives conditions under which the non-Hermitian approach accurately describes open quantum networks with disorder, validating it against full Hermitian models.

Findings

Non-Hermitian models are valid when the system's eigenvalue energy range is within the lead's energy band.

Results on disorder and opening interplay are consistent between non-Hermitian and full Hermitian models.

The approach extends to generic networks with sinks when coupling energy dependence is smooth.

Abstract

We investigate the validity of the non-Hermitian Hamiltonian approach in describing quantum transport in disordered tight-binding networks connected to external environments, acting as sinks. Usually, non-Hermitian terms are added, on a phenomenological basis, to such networks to summarize the effects of the coupling to the sinks. Here we consider a paradigmatic model of open quantum network for which we derive a non-Hermitian effective model, discussing its limit of validity by a comparison with the analysis of the full Hermitian model. Specifically, we consider a ring of sites connected to a central one-dimensional lead. The lead acts as a sink which absorbs the excitation initially present in the ring. The coupling strength to the lead controls the opening of the ring system. This model has been widely discussed in literature in the context of light-harvesting systems. We analyze the effectiveness of the non-Hermitian description both in absence and in presence of static disorder on the ring. In both cases, the non-Hermitian model is valid when the energy range determined by the eigenvalues of the ring Hamiltonian is smaller than the energy band in the lead. Under such condition, we show that results about the interplay of opening and disorder, previously obtained within the non-Hermitian Hamiltonian approach, remain valid when the full Hermitian model in presence of disorder is considered. The results of our analysis can be extended to generic networks with sinks, leading to the conclusion that the non-Hermitian approach is valid when the energy dependence of the coupling to the external environments is sufficiently smooth in the energy range spanned by the eigenstates of the network.