Boundary Green's function approach for spinful single-channel and multichannel Majorana nanowires

Abstract

The boundary Green's function (bGF) approach has been established as a powerful theoretical technique for computing the transport properties of tunnel-coupled hybrid nanowire devices. Such nanowires may exhibit topologically nontrivial superconducting phases with Majorana bound states at their boundaries. We introduce a general method for computing the bGF of spinful multi-channel lattice models for such Majorana nanowires, where the bGF is expressed in terms of the roots of a secular polynomial evaluated in complex momentum space. In many cases, those roots, and thus the bGF, can be accurately described by simple analytical expressions, while otherwise our approach allows for the numerically efficient evaluation of bGFs. We show that from the behavior of the roots, many physical quantities of key interest can be inferred, e.g., the value of bulk topological invariants, the energy dependence of the local density of states, or the spatial decay of subgap excitations. We apply the method to single-and two-channel nanowires of symmetry class D or DIII. In addition, we study the spectral properties of multi-terminal Josephson junctions made out of such Majorana nanowires.

Weyl point immersed in a continuous spectrum: An example from superconducting nanostructures

“…The derivation of expressions for the currents in superconducting and normal leads can be accomplished by standard methods of superconducting nonequilibrium Keldysh Green's functions [42,43,[47][48][49][50]. However, for the sake of comprehensibility we give here an explicit derivation from scratch.…”

2D topological matter from a boundary Green's functions perspective: Faddeev-LeVerrier algorithm implementation

“…Here we extend and extrapolate the bGF approach developed in Refs. [12,19,20] from 1D nearest-neighbour (nn) Hamiltonians to generalized d-dimensional systems with an arbitrary number of degrees of freedom and neighbours. This method performs the Fourier transform (FT) into real space needed to compute the bGF (see Fig.…”

“…1) by the analytic continuation of the momenta into the complex plane followed by residue integration. This approach exhibits better convergence performance compared to recursive approaches for which precision is linked to the number of iterations [20]. However, previous implementations of the method required analytical expressions for the key building blocks of the formalism such as the characteristic polynomial.…”

“…In the present work we complement the method of Refs. [12,19,20] with a straightforward computational approach using the Faddeev-LeVerrier algorithm (FLA) that sorts out diverse disadvantages of the semianalytical calculations. The FLA requires a low computational cost to construct not only the characteristic polynomial but also the adjugate matrix in the same process and for the same price which are, as mentioned before, the main building blocks to obtain the bGF using the residue integration method.…”

“…Furthermore, the semi-analytical approach used in Refs. [12,19,20] suffer from rigidity in the definition of the GF as any new terms that might be inserted in the Hamiltonian impose a redefinition and consequently, all the analytical coefficients of the characteristic polynomial have to be re-obtained from scratch by time consuming symbolic algorithms. In contrast, the only analytical entry for the FLA is the polynomial decomposition of the Hamiltonian in the analytic continuation variable of the momentum perpendicular to the boundary z = e ik ⊥ L ⊥ .…”