Unified theory of resonances and bound states in the continuum in Hermitian tight-binding models

TL;DR

This paper develops a unified theoretical framework for understanding resonances and bound states in the continuum within Hermitian tight-binding models, providing exact conditions for perfect transmission, BICs, and their transformations.

Contribution

It introduces a novel approach connecting resonances and BICs through non-Hermitian Hamiltonians and offers design principles for quantum devices with controllable BICs.

Findings

Derived exact energies for perfect transmittance and BICs.

Established a formal link between scattering states and BICs as a phase transition.

Presented design rules for structures with tailored resonance properties.

Abstract

We study transport properties of an arbitrary two terminal Hermitian system within a tight-binding approximation and derive the expression for the transparency in the form, which enables one to determine exact energies of perfect (unity) transmittance, zero transmittance (Fano resonance) and bound state in the continuum (BIC). These energies correspond to the real roots of two energy-dependent functions that are obtained from two non-Hermitian Hamiltonians: the Feshbach's effective Hamiltonian and the auxiliary Hamiltonian, which can be easily deduced from the effective one. BICs and scattering states are deeply connected to each other. We show that transformation of a scattering state into a BIC can be formally described as a "phase transition" with divergent generalized response function. Design rules for quantum conductors and waveguides are presented, which determine structures exhibiting coalescence of both resonances and antiresonances resulting in the formation of almost rectangular transparency and reflection windows. The results can find applications in construction of molecular conductors, broad band filters, quantum heat engines and waveguides with controllable BIC formation.