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

TL;DR

This work analyzes a one-dimensional non-Hermitian superconducting lattice with sublattice symmetry, onsite dissipation, and spin-dependent nonreciprocal hopping, showing how pseudo-Hermiticity and SLS constrain the spectrum and support rich topological physics. By Hermitianizing the Hamiltonian, the authors define a set of winding numbers $W_{\delta,\pm}$ that track complex-gap closures and predict Majorana zero modes under a uniform onsite dissipation when a transverse field suppresses the non-Hermitian skin effect. They demonstrate real, imaginary, and complex spectra, gapless superconducting phases, and symmetry-enriched correlations between left/right eigenstates and particle/hole and spin sectors, all tied to topological invariants and gap closures. The results extend non-Hermitian topology to symmetry-enriched superconductivity, offering experimental routes in cold-atom, photonic, circuit, and metamaterial platforms for realizing and detecting symmetry-protected Majorana modes in non-Hermitian settings.

Abstract

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.

Figures (22)

• Figure 1: Illustration of the setup described by Eq. \ref{['Eq:H_nHSC']}: a one-dimensional lattice with sublattice sites (labeled by A and B) and onsite $s$-wave singlet pairing (ellipse) with pairing strength $\Delta_0$. The curly arrows indicate the spin-dependent nonreciprocal hopping strengths for particles moving to the right ($t+\sigma g /4$) and to the left ($t-\sigma g /4$) for spin $\sigma \in \{\uparrow \, {\rm (blue \, arrow)}, \, \downarrow \, {\rm (red \, arrow)} \}$. The subscripts ${ \space {\mathrel{{\ooalign{\raisebox{1.2ex}{$\circlearrowright$}\cr\raisebox{1.2ex}{$\circlearrowleft$}}}} \space } }$ and $\curvearrowleftright$ denote intra- and inter-unit cell processes, respectively. The wavy curves represent onsite dissipation terms, with $\Gamma_{a}/2$ and $\Gamma_{b}/2$ corresponding to sublattices A and B, respectively.
• Figure 2: Gap closing curves for Case (i) in the $t_{+}$--$g_{-}$ (left) and $t_{-}$--$g_{+}$ (right) planes for a general parameter set. The solid curves are derived from Eq. \ref{['Eq:Transition_H_nHSC(i)']} and can be used to deduce phase diagrams. The red dashed lines, $|g_{\pm }| = 4 |t_{\mp}|$, correspond to the asymptotic limit of $D_{\pm} = 0$.
• Figure 3: (a,b) Phase diagrams and (c--h) energy spectra for Case (iib), where Eqs. \ref{['eq:constraintconditions1']}--\ref{['Eq:GaplessSC_condition']} hold with $C_0=0.5$; in this case, $g_{\pm}$ and $t_{\pm}$ are mutually dependent. (a) Phase diagram with gapless superconducting phase in the shaded regions. (b) The first quadrant of Panel (a), with dots indicating parameter sets of $t_+ = 1$ and $g_{-} = 3$, 4, 6, 7.6, 9, and $15$ at $\Gamma_0 = 5.0$ and $\Delta_0 = 1.0$, which are marked by the corresponding colors to the energy spectra in Panels (c--h). See Table \ref{['Table:Parameters']} for the adopted values of the full parameter set. Here, the PBC spectra coincide with the OBC ones.
• Figure 4: Energy spectra of the system when Eq. \ref{['Eq:Case(ii)_criterion']} and Eq. \ref{['Eq:complex_flex_band_condition(iso-b)']} are fulfilled with $t_{+} = t_{-} = 1$. (a) $g_{+}=g_{-}=3$, $\Gamma_{0}=4$, and $\Delta_{0}=3$. (b) $g_{+}=g_{-}=9$, $\Gamma_{0}=5$, and $\Delta_{0}=1$. (c) $g_{+}=g_{-}=4.5$, $\Gamma_{0}=4$, and $\Delta_{0}=3$. (d) $g_{+}=g_{-}=5$, $\Gamma_{0}=5$, and $\Delta_{0}=5$. See Table \ref{['Table:Parameters']} for the adopted values of the full parameter set. Here, the PBC spectra coincide with the OBC ones.
• Figure 5: Energy spectra under the PBC (gray curves) and OBC (blue dots) for $\Gamma_{0}=5$, $\Delta_{0}=1$, $\delta h_x = 0.01$ and $N = 200$ (corresponding to 100 unit cells). The range of the parameter diagrams corresponds to the first quadrant of Fig. \ref{['Fig:GapClosing_H_nHSC']}. In each row, we adopt the values for the parameters $(t_{-},g_{+})$ as labeled by the dots I, II and III in the leftmost panels, whereas in each column, we adopt the values for $(t_{+},g_{-})$ as labeled by the dots A, B and C in the topmost panels. In the leftmost panels, the relations in Eq. \ref{['Eq:Case(ii)_criterion']} are shown as dashed lines, with colors corresponding to sets A, B and C. Along these lines, the solid segments mark the additional gap closing curves satisfying Eq. \ref{['Eq:GaplessSC_condition']}. Similarly, in the topmost panels, the corresponding dashed lines and solid segments are shown for sets I, II and III. See Table \ref{['Table:Parameters']} for the adopted values of the full parameter sets.