Majorana zero modes in Y-shape interacting Kitaev wires

Majorana zero modes (MZMs), charge-neutral non-Abelian quasiparticles, hold significant promise for fault-tolerant topological quantum computing. Experimental signatures, such as zero-bias peaks in tunneling spectroscopy, are crucial for their identification. Semiconductor nanowires proximitized to superconductors and magnet-superconductor hybrid systems offer promising platforms for MZM realization. However, achieving the ideal 'sweet spot' coupling conditions, central to Kitaev theory, poses challenges in these materials. Recent research has explored MZMs in quantum-dot-superconductor linear arrays, potentially mitigating the effects of disorder inherent in semiconductor nanowires. The experimental realization of minimal and three-site Kitaev chains using quantum dots has opened pathways to longer chains. Topological quantum computation requires manipulating and braiding MZMs; 1D geometries are insufficient as MZMs fuse during exchange. T- and Y-shaped wire geometries are proposed for realizing non-Abelian statistics through braiding operations. Existing studies on T-shaped and Y-shaped wires primarily focus on ground states in the non-interacting limit, neglecting the precise form of Majorana wave functions and the impact of Coulomb interactions. Coulomb repulsion can suppress the pairing-induced bulk gap and affect MZM stability. This research investigates MZMs in interacting Y-shaped quantum dots near the sweet spot, crucial for braiding and demonstrating non-Abelian statistics. The study aims to determine the exact form of Majorana wavefunctions near the junction and their dependence on superconducting phases.

The paper extensively reviews existing literature on Majorana zero modes (MZMs), their potential in topological quantum computing, and their experimental realization in various systems. It highlights the challenges associated with achieving the sweet spot in semiconductor nanowires and the advantages of using quantum dots to overcome disorder issues. The literature review also covers prior work on T- and Y-shaped wire geometries for braiding operations and non-Abelian statistics, emphasizing the gap in understanding the precise form of Majorana wave functions and the effect of Coulomb interactions in these systems. Previous research on three-terminal Josephson junctions and multi-legged star junctions is discussed, noting their limitations in addressing wave function details and interaction effects. This comprehensive review sets the stage for the current study, which addresses these gaps and provides new insights into the behavior of MZMs in Y-shaped interacting Kitaev wires.

The research employs both analytical and numerical methods to investigate Majorana zero modes (MZMs) in Y-shaped interacting Kitaev wires. Analytically, the study focuses on the sweet spot (t⊥ = Δ) where the Kitaev model can be solved exactly. The Y-shaped Kitaev model Hamiltonian is divided into four independent commuting Hamiltonians representing three arms and a central region. This decomposition, expressed in terms of Majorana operators, allows for exact diagonalization at the sweet spot for various superconducting phase values on each arm. This approach unveils the presence of both single-site and multi-site MZMs. The analytical solution provides the exact form of the Majorana wave functions, detailing their localization and distribution across multiple sites. For numerical analysis, density matrix renormalization group (DMRG) simulations using the DMRG++ program are performed. DMRG simulations account for the effects of nearest-neighbor Coulomb repulsion (V) on the stability of MZMs. The study calculates the local density of states (LDOS) to investigate the spatial distribution and stability of both single-site and multi-site MZMs. The electron and hole components of LDOS are examined separately to assess the robustness of MZMs against interaction strengths and deviations from the sweet spot. The methodology combines analytical solutions for precise wavefunction determination with numerical simulations to address interaction effects and confirm the analytical findings.

The study's key findings reveal the existence of exotic multi-site Majorana zero modes (MZMs) in Y-shaped Kitaev wires at the sweet spot (t⊥ = Δ). The analytical solution demonstrates that, depending on the superconducting phases on each arm of the Y-junction, there can be multi-site MZMs localized near the central region, in addition to the expected three single-site MZMs at the ends of the arms. The multi-site MZMs are not merely a consequence of being away from the sweet spot or the introduction of interactions, but are an intrinsic property of the Y-geometry at the sweet spot. For specific superconducting phase values (φ1 = π, φ2 = 0, φ3 = 0), six MZMs are predicted: three edge single-site MZMs, one central single-site MZM, and two multi-site MZMs distributed across two or three sites near the center. For φ1 = 0, φ2 = 0, φ3 = 0, four MZMs are found: three edge single-site MZMs and one multi-site MZM shared equally between two central sites. Finally, for φ1 = 0, φ2 = 0, φ3 = π/2, four MZMs are predicted: three edge single-site MZMs and one multi-site MZM distributed across three central sites with unequal weighting. Density matrix renormalization group (DMRG) simulations confirm these findings and further demonstrate the stability of both single-site and multi-site MZMs against moderate Coulomb repulsion. The LDOS calculations show that the multi-site MZMs are as robust as the single-site MZMs, even when deviating from the ideal sweet spot. The multi-site MZMs exhibit a specific weighting distribution: some sites host 1/2, 1/4, or 2/3 of a Majorana, a phenomenon that is not related to the exponential decay typically observed away from the sweet spot.

The discovery of multi-site Majorana zero modes (MZMs) in Y-shaped Kitaev wires has significant implications for topological quantum computing. The results demonstrate that these multi-site MZMs are not merely artifacts of deviations from ideal conditions but are an intrinsic feature of the Y-junction geometry at the sweet spot. This finding is important because the knowledge of the precise shape and distribution of the MZM wave functions is crucial for braiding operations, which require minimal overlap of the MZMs. The robustness of these multi-site MZMs against moderate Coulomb repulsion suggests that they are as topologically protected as their single-site counterparts, enhancing the feasibility of using Y-junctions in quantum dot arrays for braiding-based quantum gates. The observed unequal weighting of Majorana components on different sites in the multi-site MZMs is a noteworthy result, which needs to be further investigated in the context of braiding operations and quantum gate fidelity. The findings extend beyond the specific Y-junction considered here, as similar multi-site MZMs are predicted to occur in other geometries, such as triangular junctions of coupled quantum dots. The study's results provide valuable insights for designing and optimizing Y-junctions based on quantum dots arrays for experimental realizations of topological quantum computing.

This research unveils the existence and stability of exotic multi-site Majorana zero modes in Y-shaped interacting Kitaev wires. These multi-site MZMs, coexisting with single-site MZMs, are an intrinsic feature of the Y-geometry at the sweet spot and remain robust against moderate Coulomb repulsion. The precise knowledge of their spatial distribution is critical for braiding experiments in topological quantum computation. The findings suggest that Y-shaped quantum dot arrays provide a viable platform for realizing braiding-based quantum gates. Future research should explore the implications of these multi-site MZMs for topological quantum computation, including their impact on braiding fidelity and the development of fault-tolerant quantum gates. Investigating MZMs in X-shaped Kitaev wires and the effects of disorder and temperature would further advance this field.

The study primarily focuses on the idealized sweet spot (t⊥ = Δ) and moderate Coulomb repulsion. While the findings demonstrate robustness against deviations from the sweet spot and interaction effects, it is crucial to investigate the impact of stronger interactions and significant deviations from the sweet spot. The study does not explicitly address the effect of disorder, which is a significant factor in real experimental systems. Future work should explore the robustness of these multi-site MZMs in the presence of disorder and assess their experimental viability. Additionally, the analysis is limited to specific superconducting phase values; a more comprehensive study considering a broader range of phases might reveal further insights into the behavior of multi-site MZMs.