Abstract We introduce a new type of one-dimensional Kitaev chain with staggered p-wave superconducting pairing. We find three physical regimes in this model by tuning the p-wave pairing and the chemical potential of the system. In the topologically nontrivial phase, there are two Majorana zero modes localized at the opposite ends of the lattice, which are characterized and protected by nonzero topological invariants. More interestingly, we also find a regime where the system can hold four unprotected nonzero-energy edge modes in the trivial phase, which is analogous to a weak topological phase. The third regime is also trivial but holds no edge modes. The emergence of zero- and nonzero-energy edge modes in the system are analyzed by transforming the lattice model into a ladder consisting of Majorana fermions, where the competition between the intra- and inter-leg couplings leads to different phases. We further investigate the properties of edge modes under the influences of dissipation, which is represented by introducing a imaginary part in the chemical potential. Our work unveils the exotic properties induced by the staggered p-wave pairing and provides a new platform for further exploration of Majorana edge modes.
Abstract We introduce a new type of one-dimensional Kitaev chain with staggered p-wave superconducting pairing. We find three physical regimes in this model by tuning the p-wave pairing and the chemical potential of the system. In the topologically nontrivial phase, there are two Majorana zero modes localized at the opposite ends of the lattice, which are characterized and protected by nonzero topological invariants. More interestingly, we also find a regime where the system can hold four unprotected nonzero-energy edge modes in the trivial phase, which is analogous to a weak topological phase. The third regime is also trivial but holds no edge modes. The emergence of zero- and nonzero-energy edge modes in the system are analyzed by transforming the lattice model into a ladder consisting of Majorana fermions, where the competition between the intra- and inter-leg couplings leads to different phases. We further investigate the properties of edge modes under the influences of dissipation, which is represented by introducing a imaginary part in the chemical potential. Our work unveils the exotic properties induced by the staggered p-wave pairing and provides a new platform for further exploration of Majorana edge modes.
We investigate an inter-leg coupled hybrid Kitaev ladder composed of two parallel superconducting chains with distinct pairing interactions. The upper chain of the ladder hosts conventional p-wave pairing, while the lower chain exhibits long-range pairing that decays algebraically with distance. We demonstrate that the mutual influence of long-range pairing exponent, chemical potential, and inter-leg coupling strength gives rise to a rich topological phase diagram characterized by multiple Majorana zero modes and massive Dirac modes. In particular, we show that the inter-leg coupling renormalizes the effective energy scales, leading to a systematic shift of the topological phase boundaries and enabling controlled tuning of the Majorana modes. Furthermore, we identify a transition from a two Majorana zero mode phase to a phase encapsulating four Majorana zero modes, as the long-range pairing exponent is varied. This transition is accompanied by a crossover regime in which Majorana zero modes coexist with massive Dirac modes, reflecting hybridization between edge and bulk excitations. This ladder thus provides a minimal and attractive platform for realizing the impact of a long-range pairing on topological phases. Our results highlight the potential of long-range hybrid systems for engineering tunable topological states relevant for quantum information applications.
Evolution of Majorona zero-energy edge states in a T2=−1 symmetry protected 1D topological superconductor with dominant spin-orbit coupling
We consider a 1D topological superconductor (TSC) constructed by coupling a pair of Kitaev's Majorana chains with opposite spin configurations. Such a 1D lattice model is known to be protected by a T2=−1 time-reversal symmetry. Furthermore, we consider a modeled Rashba spin-orbit coupling on such a system of T2=−1 time-reversal symmetric TSC. The Rashba spin-orbit coupling together with the chemical potential engineered the phase transitions of the edge states in the system and consequently the number of Majorona's zero-energy edge modes (MZM's) emerging at the edge of the coupled chains. Correspondingly, the topological nature of the system is described by a phase diagram consisting of three different phases. The three phases are characterized by a topological winding number, W=1, 2 (with one and two MZM's: topological phases) and W=0 (devoid of any MZM: trivial insulating phase).
Proximity-induced superconductivity in low-dimensional systems offers a powerful pathway to engineer topological superconducting phases in, otherwise, non-superconducting systems. These exotic phases are of fundamental and technological interest due to the presence of robust zero-energy modes, the Majorana bound states. In this work, we propose a theoretical framework to realize Majorana bound states from the edge states of a two-dimensional semi-Dirac system. This anisotropic system, under specific conditions, can host non-chiral edge states that propagate only along particular edges, effectively forming separated one-dimensional channels. We show that the interplay between Rashba spin-orbit coupling and a Zeeman field on this setup provides the right conditions to get an effective p-wave pairing between the edge states by proximity with a s-wave superconductor. In finite geometries, each edge can independently undergo a topological phase transition into a one-dimensional topological superconductor and give rise to four zero-energy modes localized at the strip corners. At low energies, the edge states subspace admits a description in terms of coupled Kitaev chains, providing a clear picture of the origin, robustness, and tunability of the corner Majorana modes. Our results establish semi-Dirac materials as a natural platform for realizing Majorana modes in two dimensions without relying on engineered nanostructures, vortices, or crystalline higher-order topology.
We investigate the spectral and topological properties of a zigzag Kitaev chain constructed from two diagonally coupled one dimensional Kitaev chains with a zero and finite superconducting pairing phase difference. Using a Bogoliubov-de Gennes formulation, we analyze the energy spectrum, distribution of Majorana zero modes (MZMs), the quasi-particle dispersion, and the winding number, respectively. For a zero phase difference, the resulting energy spectrum shows topological phases with two, four MZMs, and trivial regions. The phases of gap closure determine the topological phase boundaries. In particular, for the phase difference between φ=π, the degeneracy of MZMs is partially lifted, leading to modified topological phases compared to the case φ=0. The topological and trivial phase boundaries are further confirmed by evaluating the quasi-particle dispersion and the topological invariant, namely the winding number. We show that the zigzag Kitaev chain contributes independently to the total winding number ν=1 and 2, giving rise to distinct topological phases that support two and four MZMs. The ν=0 gives rise to a trivial region. The energy spectrum of systems corroborates the analytical phase boundaries and reveals characteristics associated with hybridization, enabling us to obtain the complete phase diagram of the zigzag model. Our results establish the zigzag Kitaev chain as a minimal platform for engineering MZM quantum computations, with potential applications in the study of topological phases and Majorana based qubit physics.
We explore the salient features of the 'Kitaev ladder', a two-legged ladder version of the spin-1/2 Kitaev model on a honeycomb lattice, by mapping it to a one-dimensional fermionic p-wave superconducting system. We examine the connections between spin phases and topologically non-trivial phases of non-interacting fermionic systems, demonstrating the equivalence between the spontaneous breaking of global Z 2 symmetry in spin systems and the existence of isolated Majorana modes. In the Kitaev ladder, we investigate topological properties of the system in different sectors characterized by the presence or absence of a vortex in each plaquette of the ladder. We show that vortex patterns can yield a rich parameter space for tuning into topologically non-trivial phases. We introduce and employ a new topological invariant for explicitly determining the presence of zero energy Majorana modes at the boundaries of such phases. Finally, we discuss dynamic quenching between topologically non-trivial phases in the Kitaev ladder and, in particular, the post-quench dynamics governed by tuning through a quantum critical point.
We explore the salient features of the `Kitaev ladder', a two-legged ladder version of the spin-1/2 Kitaev model on a honeycomb lattice, by mapping it to a one-dimensional fermionic p-wave superconducting system. We examine the connections between spin phases and topologically non-trivial phases of non-interacting fermionic systems, demonstrating the equivalence between the spontaneous breaking of global Z(2) symmetry in spin systems and the existence of isolated Majorana modes. In the Kitaev ladder, we investigate topological properties of the system in different sectors characterized by the presence or absence of a vortex in each plaquette of the ladder. We show that vortex patterns can yield a rich parameter space for tuning into topologically non-trivial phases. We introduce and employ a new topological invariant for explicitly determining the presence of zero energy Majorana modes at the boundaries of such phases. Finally, we discuss dynamic quenching between topologically non-trivial phases in the Kitaev ladder and, in particular, the post-quench dynamics governed by tuning through a quantum critical point.