One-dimensional spin–orbit coupled Dirac system with extended s-wave superconductivity: Majorana modes and Josephson effects

Abstract

Motivated by the spin–momentum locking of electrons at the boundaries of certain topological insulators, we study a one-dimensional system of spin–orbit coupled massless Dirac electrons with s-wave superconducting pairing. As a result of the spin–orbit coupling, our model has only two kinds of linearly dispersing modes, and we take these to be right-moving spin-up and left-moving spin-down. Both lattice and continuum models are studied. In the lattice model, we find that a single Majorana zero energy mode appears at each end of a finite system provided that the s-wave pairing has an extended form, with the nearest-neighbor pairing being larger than the on-site pairing. We confirm this both numerically and analytically by calculating the winding number. We find that the continuum model also has zero energy end modes. Next we study a lattice version of a model with both Schrödinger and Dirac-like terms and find that the model hosts a topological transition between topologically trivial and non-trivial phases depending on the relative strength of the Schrödinger and Dirac terms. We then study a continuum system consisting of two s-wave superconductors with different phases of the pairing, with a δ-function potential barrier lying at the junction of the two superconductors. Remarkably, we find that the system has a single Andreev bound state (ABS) which is localized at the junction. When the pairing phase difference crosses a multiple of 2π, an ABS touches the top of the superconducting gap and disappears, and a different state appears from the bottom of the gap. We also study the AC Josephson effect in such a junction with a voltage bias that has both a constant V 0 and a term which oscillates with a frequency ω. We find that, in contrast to standard Josephson junctions, Shapiro plateaus appear when the Josephson frequency ω J = 2eV 0/ℏ is a rational fraction of ω. We discuss experiments which can realize such junctions.

Majorana zero mode-soliton duality and in-gap and BIC bound states in modified Toda model coupled to fermion

“…These states are well known to lead to interesting phenomena in condensed matter and we will search for them as fermion bound states. Recently, the model (3.49) has been considered as the continuum limit of a one-dimensional system of spin-orbit coupled Dirac Hamiltonian and a s-wave superconducting pairing, such that the s-wave pairing is represented by −iM e iβΦ [41]. The bosonization techniques have been applied to related fermionic systems with finite length which exhibit Majorana zero modes [42].…”

“…In fact, the case with β = 0 describes a free massive Majorana system [50,51]. Recently, the model (6.1)-(6.2) has been considered as the simplest continuum model of a one-dimensional superconducting fermionic symmetry-protected topological (SPT) phase in condensed matter such that the mean-field superconducting (SC) pairing potential is represented by the field Φ [42,41].…”

Majorana zero mode-soliton duality and in-gap and BIC bound states in modified Toda model coupled to fermion

“…The Hermitian Toda model coupled to matter has been found in a variety of models, such as the continuum limit of s−wave superconductor [22], effective theory describing Majorana bound states [23], a model for high T c superconductivity [24], low-energy effective Lagrangian in QCD 2 [25], etc; so, our study not only becomes of academic interest, but physically motivated. As pointed out above, pseudo-Hermiticity is a constraint unique to non-Hermitian systems and may provide novel topological features.…”

Field Theoretical Study of Disorder in Non-Hermitian Topological Models

“…This type of Hamiltonian represents a more realistic realization of a 1D superconductor than the paradigmatic pwave Kitaev chain, as it takes into account the effect of the electron spin. The Hermitian model exhibits nontrivial topological phases, manifested through the presence of Majorana modes at the boundaries of the open system [23]. Here, we include a non-Hermitian term to extend the topological phase diagram and study its purely non-Hermitian part.…”