Abstract
We propose a robust and efficient algorithm for computing bound states of infinite tight-binding systems that are made up of a finite scattering region connected to semi-infinite leads. Our method uses wave matching in close analogy to the approaches used to obtain propagating states and scattering matrices. We show that our algorithm is robust in presence of slowly decaying bound states where a diagonalization of a finite system would fail. It also allows to calculate the bound states that can be present in the middle of a continuous spectrum. We apply our technique to quantum billiards and the following topological materials: Majorana states in 1D superconducting nanowires, edge states in the 2D quantum spin Hall phase, and Fermi arcs in 3D Weyl semimetals.
Highlights
If the state decays fast enough in the leads and a sufficiently large portion of them has been included, this results in a precise determination of the bound states
We focus on the formulation Eq (8) and do not make use of the alternative self-energy formulation
Since perfect convergence is achieved after only a few iterations, this algorithm is capable of obtaining the bound states for a cost comparable to that of calculating the propagating modes
Summary
Simulating quantum devices that are connected to infinite leads is a commonly occurring problem in quantum nanoelectronics. If the state decays fast enough in the leads and a sufficiently large portion of them has been included, this results in a precise determination of the bound states This approach is not always satisfactory due to its significant computational overhead when the decay length of the bound state diverges. (the ability of our algorithm to isolate bound states from the continuum is one of its strengths.) Sections 5.1, 5.2, and 5.3 continue with further applications: the calculation of edge states for three different kinds of topological phases These are, respectively, a 1D Majorana bound state in a superconducting wire, a 2D quantum spin Hall phase within the Bernevig-Hughes-Zhang (BHZ) model, and Fermi arcs in a 3D Weyl semimetal
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