Reconstructed conventional bulk-boundary correspondence in the non-Hermitian $s$-wave nodal-ring superconductor

Recently, the interplay between the non-Hermitian skin effect (NHSE) and topological phases has attracted a lot of attention. When the NHSE occurs in a topological system, the conventional bulk-boundary correspondence (BBC) based on the Bloch band theory is destroyed. In this work, we investigate a non-Hermitian nodal-ring semimetal with the $s$-wave superconducting (SC) term. We find that the SC term introduces the intrinsic coupling between the different general Brillouin zones (GBZs) of the particle and hole subbands in our model, which drives the total GBZ to the BZ. Furthermore, when the SC term is finite, we show a possibility that the GBZ is the same as the BZ. We also find that the Bloch winding number of the quasi-one-dimensional model can predict the number of Majorana zero states faithfully in this case, which implies that the conventional BBC is reconstructed in our model. The localization strengths of the Majorana zero states are also calculated to show the reconstruction of conventional BBC. The finite-size effect, related experimental regimes in the electric circuit, and the fragility of this reconstructed conventional BBC are also discussed in this work.

Figure 1

Schematic for the effectively removed NHSE and the reconstructed conventional BBC. The blue curves and lines denote an eigenstate. Since our quasi-1D model possesses the NHSE, after adding the finite SC pairing term, an eigenstate of the particle part tends to be localized at one edge of the system, and the corresponding eigenstate of the hole part tends to be localized at another edge of the system. i.e., the particle and hole parts have different localized eigenstates due to the NHSE. Meanwhile, the two parts are coupled by the SC pairing term $\mathrm{\Delta }$. If such two NHSE localizations cancel each other, the NHSE can be effectively removed in our system. This makes the Bloch topological invariant give the correct phase boundary again, which indicates that the conventional BBC is reconstructed.

Figure 2

(a1)–(a4) The sub-GBZs corresponding to the particle block $h_{\mathrm{P}}$ (green solid circle) and the hole block $h_{\mathrm{H}}$ (green dash circle). Red points denote total GBZs with different $\mathrm{\Delta }\mathrm{s}$. The blue solid circle denotes BZ. (b1)–(b4) Complex eigenenergies under the PBC and OBC, which correspond to (a1)–(a4). Other parameters are $L=80, \gamma =0.3, v_z=2B=1, m=0$.

Figure 3

(a) The OBC spectrum for our model with $\mathrm{\Delta }=0.2$. The red dashed lines are phase boundaries $k_x=\pm k_{+}$ and $k_x=\pm k_{-}$ given by the Bloch winding number. The numbers denote the number of the zero modes. (b) The real-space winding number as a function of $k_x$ and $k_y$. The black solid lines are the phase boundaries given by the Bloch winding number. (c) Density distribution of zero modes for $k_x=0.44\pi$ and $k_y=\pi$. (d) ${\beta }_{i,0}-k_x$ curves from Eq. (21), where each ${\beta }_{i,0}$ ($i=1,2,3,4$) is the solution of the characteristic equation Eq. (20) for $E=0$. The two red (blue) lines denote ${\beta }_{1,0}$ and ${\beta }_{4,0}({\beta }_{2,0}$ and ${\beta }_{3,0}$). The four dashed perpendicular lines are $k_x=\pm k_{+}$ and $k_x=\pm k_{-}$. The horizontal line is $\left|\beta \right|=1$. Other parameters are $L=80, \gamma =0.3, v_z=2B=1$, and $m=0$.

Figure 4

(a1)–(a3) Complex eigenenergies for $\mathrm{\Delta }=0.01$. (b1)–(b3) Complex eigenenergies for $\mathrm{\Delta }=0.2$. The blue (red) points denote eigenenergies under PBC (OBC). The system length $L=10$, 15, and 60. Other parameters are $L=80, \gamma =0.3, v_z=2B=1$, and $m=0$.

Figure 5

Electric circuit simulating the non-Hermitian nodal-ring semimetal with the $s$-wave pairing. The circuit consists of two parts. In this electron circuit, each unit cell has four nodes and these nodes are labeled by 1–4. The part containing 1 and 2 (3 and 4) nodes simulates the particle (hole) band. (a) The circuit realization of the terms in the $z$ direction. The operational amplifier is colored red. (b) The circuit realization of the terms in the $x\left(y\right)$ direction. (c) Circuit structure of an operational amplifier shown in (a), where $R_1=R_2$ and the resistive parameter is $R$.