Majorana zero modes in Y-shape interacting Kitaev wires

The study addresses how Majorana zero modes manifest and localize in Y-shaped Kitaev wire geometries, particularly near junctions, and how their properties depend on superconducting phase differences between arms. MZMs are sought for topological quantum computing due to their non-Abelian statistics, often identified by zero-bias peaks in tunneling. Conventional semiconductor-superconductor platforms struggle to realize the Kitaev sweet spot and are sensitive to disorder. Quantum-dot-based Kitaev chains mitigate disorder and have recently realized minimal chains with observed MZMs at the sweet spot. Braiding requires non-1D geometries such as T- or Y-junctions. Prior Y/T-junction studies in proximitized nanowires (Δ < t) have explored subgap states and parafermions but not the exact Majorana wavefunctions nor interaction effects. This work focuses on interacting Y-shaped quantum-dot-based Kitaev wires at the sweet spot to find exact junction MZM wavefunctions, determine their dependence on superconducting phases, and assess stability under Coulomb repulsion.

The introduction situates the work within: (i) MZM realizations and signatures in proximitized semiconductor nanowires and magnet–superconductor hybrids; (ii) quantum-dot-superconductor arrays as an alternative platform reducing disorder effects, including experimental realization of two- and three-site Kitaev chains; (iii) the need for T/Y geometries to braid MZMs; (iv) prior analyses of T/Y junctions using scattering matrix and proposals of parafermionic modes, which did not provide exact Majorana wavefunctions nor address Coulomb interactions. Interaction effects are expected to reduce pairing gaps and potentially destabilize MZMs, motivating an interacting study with unbiased numerics.

Analytical approach: The Y-shaped spinless Kitaev model with superconducting phases ϕ1, ϕ2, ϕ3 on each arm is studied at the sweet spot t⊥ = Δ and V = 0. The full Hamiltonian is decomposed into four commuting sectors when written in Majorana operators: three independent 1D legs (I, II, III) and a central junction region (IV). Each leg Hamiltonian, at t = Δ, leaves one Majorana operator at the outer end unpaired, yielding three edge MZMs independent of phase. The central sector depends on the phases; exact diagonalization in the Majorana basis yields closed-form multi-site MZM wavefunctions for specific phase configurations: (i) ϕ1 = π, ϕ2 = 0, ϕ3 = 0; (ii) ϕ1 = 0, ϕ2 = 0, ϕ3 = 0; (iii) ϕ1 = 0, ϕ2 = 0, ϕ3 = π/2. Analytical diagonalizations (4×4 or 5×5 Majorana matrices) produce explicit zero modes and their site distributions near the junction, as well as ordinary fermionic modes and spectra. Ground-state degeneracies follow from the number of MZMs and fusion rules (eightfold for six MZMs, fourfold for four MZMs). Numerical methods: Unbiased DMRG (DMRG++) in the two-site algorithm is used on Y-junction geometries with total size L = 46, keeping m = 1500 states with truncation error ≤ 1e−10. Nearest-neighbor repulsive interaction V is included via Hint = ∑j V njnj+1. Local density of states LDOS(ω, j) is computed using the Krylov-space correction-vector method with broadening η = 0.1, separating electron and hole components to assess Majorana character (equal spectral weights at ω ≈ 0). Bogoliubov–de Gennes (BdG) calculations at V = 0 explore deviations from the sweet spot by varying Δ/t⊥ (e.g., 1.0, 0.6, 0.2).

• Exact analytical solution at the sweet spot reveals three universal single-site edge MZMs (one per arm) plus additional junction MZMs whose number and spatial distribution depend on arm phases. Case-dependent results: • ϕ1 = π, ϕ2 = 0, ϕ3 = 0: total of six MZMs: three single-site edge MZMs (j = 1, 31, 46), one single-site MZM at the central site (j = 16), and two multi-site MZMs: x3 on j = 15 and 32 (equal amplitude), and x4 on j = 15, 17, 32 (unequal amplitudes). Corresponding LDOS(ω = 0, j) shows three central-region peaks with spectral weight 2/3 relative to edge peaks. Ground state is eightfold degenerate. • ϕ1 = 0, ϕ2 = 0, ϕ3 = 0: total of four MZMs: three single-site edge MZMs and one multi-site MZM R2 equally distributed on j = 17 and 32 (LDOS weight 1/2 vs. edge). Ground state is fourfold degenerate. • ϕ1 = 0, ϕ2 = 0, ϕ3 = π/2: total of four MZMs: three single-site edge MZMs and one multi-site MZM X5 distributed on j = 15, 17, 32 with weights 1/4, 1/4, and 1/2, respectively. Fourfold ground-state degeneracy. Although energy spectra for ϕ3 = 0 and ϕ3 = π/2 cases are identical, the multi-site Majorana wavefunctions differ. - Multi-site MZMs are intrinsic to the Y-junction at the sweet spot and are not exponential tails; instead their “core” is spread over a small discrete set of sites with exact fractional spectral weights (e.g., 2/3, 1/2, 1/4). - DMRG LDOS confirms analytical predictions: sharp ω = 0 peaks at predicted sites and relative heights (2/3, 1/2, 1/4 patterns). Under repulsive interactions up to V ≈ 2, both single-site and multi-site MZMs remain localized over a few sites and exhibit equal electron/hole LDOS at ω = 0, indicating preserved Majorana character. The decay of LDOS peak heights vs V is comparable for edge and multi-site junction MZMs. - Away from the sweet spot (BdG, V = 0): At Δ/t⊥ = 0.6, MZMs remain robust with short exponential spreads. At Δ/t⊥ = 0.2, both edge and junction MZMs spread over many sites and the central multi-site MZM overlaps with edge MZMs on two arms. - The explicit Majorana wavefunctions at the junction enable counting and localization crucial for braiding design in quantum-dot Y-junctions.

The work addresses whether and where Majorana modes arise in Y-shaped Kitaev wire junctions and how they depend on superconducting phase differences and interactions. By decomposing the Hamiltonian into independent leg and central sectors in the Majorana basis at the sweet spot, the authors exactly identify not only the known single-site edge MZMs but also intrinsic multi-site MZMs localized over several junction-adjacent sites, with phase-tunable spatial distributions. These analytic results, corroborated by DMRG LDOS, demonstrate that multi-site junction MZMs are as robust to moderate Coulomb repulsion (V ≤ 2) as single-site edge MZMs, with electron and hole LDOS components remaining equal at ω ≈ 0. Away from the sweet spot, MZMs gradually become extended and can overlap, which is relevant for device design to avoid unwanted fusion during braiding. The findings are directly relevant to quantum-dot arrays, where sweet-spot operation is experimentally accessible, suggesting Y-junctions can host and manipulate MZMs over just a few sites. Knowledge of the precise multi-site wavefunction shapes is essential for braiding protocols to minimize overlap and optimize gate operations; alternative junction implementations (e.g., triangular triple-dot junctions) are predicted to also host multi-site MZMs with modified distributions.

The paper provides exact analytical solutions for Majorana zero modes in Y-shaped Kitaev wires at the sweet spot, revealing phase-controlled multi-site MZMs at the junction in addition to universal single-site edge MZMs. Unbiased DMRG confirms the predicted spatial distributions and spectral weights, and shows that both single-site and multi-site MZMs remain robust under moderate nearest-neighbor repulsion and deviations from the sweet spot. These insights enable the design of braiding-capable Y-junctions from quantum-dot arrays with only a few sites. Future directions include studying X-shaped junctions, the impact of disorder and temperature, and optimizing junction geometries (e.g., triangular triple-dot couplings) for braiding operations.

• The exact analytical treatment focuses on the sweet spot t = Δ and V = 0; interaction effects are primarily assessed numerically and up to moderate repulsion (V ≤ 2). - Idealized spinless Kitaev-chain modeling neglects spin, disorder, and realistic multi-orbital effects; temperature and disorder are not included. - Finite-size numerics use L = 46 and specific phase configurations; general phase landscapes beyond the three representative cases are not exhaustively mapped. - Away-from-sweet-spot behavior is explored within non-interacting BdG; combined strong interactions and deviations from the sweet spot are not fully addressed.