Non-Hermitian topological superconductivity with symmetry-enriched spectral and eigenstate features

We investigate a one-dimensional superconducting lattice that realizes all internal symmetries permitted in non-Hermitian systems, characterized by nonreciprocal hopping, onsite dissipation, and $s$-wave singlet pairing in a Su-Schrieffer-Heeger-type structure. The combined presence of pseudo-Hermiticity and sublattice symmetry imposes constraints on the energy spectra. We identify parameter regimes featuring real spectra, purely imaginary spectra, complex flat bands, and Majorana zero modes, the latter emerging when a uniform transverse magnetic field suppresses the non-Hermitian skin effect. We show that a uniform onsite dissipation is essential for stabilizing the zero modes, whereas a purely staggered dissipation destroys the topological superconductivity. Through Hermitianization, we construct a spectral winding number as a topological invariant and demonstrate its correspondence with the gap closing conditions and appearance of the Majorana zero modes, allowing us to establish topological phase diagrams. Moreover, we reveal nontrivial correlations between the particle-hole and spin components of left and right eigenstates, enforced by chiral symmetry, pseudo-Hermiticity, and their combination. Our results highlight how non-Hermiticity, sublattice structure, and superconductivity together enrich symmetry properties and give rise to novel topological phenomena.

💡 Research Summary

In this work the authors introduce and thoroughly analyze a one‑dimensional non‑Hermitian superconducting lattice that incorporates all internal symmetries allowed in non‑Hermitian systems. The model consists of a Su‑Schrieffer‑Heeger (SSH)‑type chain with two sublattices (A and B), spin‑dependent non‑reciprocal hopping, onsite dissipation on each sublattice, and an s‑wave spin‑singlet pairing Δ₀. The hopping amplitudes are t ± σ g/4 for spin σ = ↑,↓, giving rise to asymmetric (non‑reciprocal) particle motion, while the dissipation terms Γ_A and Γ_B introduce complex on‑site potentials.

A key theoretical step is the identification of two simultaneous internal symmetries: pseudo‑Hermiticity (pH) and sublattice symmetry (SLS). Pseudo‑Hermiticity (implemented by ηₓτ_zσ_z) guarantees that the complex conjugate of any eigenvalue is also an eigenvalue, reflecting the spectrum across the real axis. SLS (η_yτ₀σ_x) forces the Hamiltonian to change sign under a sublattice‑exchange operation, reflecting the spectrum across the imaginary axis. When both are present, eigenvalues appear in quartets (E, E*, −E, −E*) and the spectrum acquires a highly constrained structure.

Including a uniform transverse magnetic field term δhₓ η_zτ₀σₓ (which mixes the two spin components) breaks the block‑diagonal structure but preserves a set of anti‑unitary symmetries. The full Hamiltonian therefore belongs to the non‑Hermitian BDI† class (real AZ † classification) characterized by time‑reversal symmetry (T⁺, T² = +1) and particle‑hole symmetry (C⁻, C² = +1). Because SLS is also present, the system simultaneously satisfies the daggered versions TRS† and PHS†. In one dimension this class supports three independent topological invariants: a Z invariant for point gaps and two Z⊕Z invariants for real‑line and imaginary‑line gaps.

The authors solve the Bloch Hamiltonian under periodic boundary conditions analytically, obtaining eight bands E_{±}^{λ,ε}(k) = ±¼ F_r(k) + i λ F_i(k) + 2i ε D + (i/2)