Defective Majorana zero modes in a non-Hermitian Kitaev chain

Topological stability is an important property for topological materials. However, the non-Hermitian effects may change this situation. Here, we investigate the robustness of edge states in the non-Hermitian Kitaev chain with an imbalanced tunneling term and a superconducting pairing term. By defining the similarity of Majorana zero modes (MZMs) and magnetic factor, the coalescing phase diagram of the MZMs and corresponding spin polarization phase diagram are provided. Because of the non-Hermitian coalescence effect and non-Hermitian suppression effect induced by the breakdown of sublattice symmetry and particle-hole symmetry, the system emerges with very interesting phenomena, such as defective MZMs, number-anomalous bulk-boundary correspondence, coalescence of many-body ground states, and the magnetic phase crossover without gap closing. Those non-Hermitian effects offer fresh insights into MZMs and topological physics.

Figure 1

Schematic illustration of the non-Hermitian Kitaev chain in different representation; the dotted box indicates the unit cell. (a) The lattice structure in Dirac fermion representation, which is described by $H_{\mathrm{NH}}$. The imbalanced hopping amplitudes from $A$ to $B$ (orange arrow) and from $B$ to $A$ (blue arrow) are described by ${\beta }_1$. The imbalanced pairing amplitudes of particles and holes are described by ${\beta }_2$. (b) Kitaev chain viewed as two coupled SSH chains in Majorana representation, which is described by $H_{\mathrm{NH}}^{\mathrm{M}}$. The Majorana fermion $a_{j,A/B}\left(b_{j,A/B}\right)$ are marked by blue (red) filled circles. The solid lines indicate the couplings $m_1,m_2,i{m}_3,i{m}_4$ between nearest neighbor sites and the dashed lines indicate the couplings $i\mu$ intrasite. (c) The lattice schematic diagram of the Hermitian counterpart systems in Majorana representation, which is described by $H_{\mathrm{cp}}^{\mathrm{M}}$.

Figure 2

Similarity ${\gamma }_{\mathrm{M}}$ and localized properties of the two defective MZMs with $L=24, \mu =0.1, t=1$, and ${\mathrm{\Delta }}_0=0.8$. (a) The blue area represents ${\gamma }_{\mathrm{M}}=0$, which means two MZMs are local at two ends of the chain independently; the yellow areas represent ${\gamma }_{\mathrm{M}}=1$, which means the two MZMs coalesce to one and are local at either the left or right end of the chain. (b1)–(b9) The distributions of the MZMs for different cases marked by the points P1–P9 in (a), respectively. For the cases in P1, P5, P9, which are away from ${\beta }_1=0$ and ${\beta }_2=0$, the two MZMs coalesce to one

Figure 3

Breakdown of symmetry induced by imbalanced hopping amplitude (a) and imbalanced pairing amplitude (b).

Figure 4

Similarity ${\gamma }_{\mathrm{spin}}$ of the two ground states in spin representation with $L=4, t={\mathrm{\Delta }}_0=1$, and $\mu =0.1$. The blue area I represents ${\gamma }_{\mathrm{spin}}=0$, which means the two ground states are orthogonal. The yellow areas II–V represent ${\gamma }_{\mathrm{spin}}=1$, which means the two ground states coalesce to one and act as exceptional points (EPs).

Figure 5

Spin structure of the many-body ground state ${|G\rangle }_{\mathrm{NH}}^1$ with (a) ${\beta }_2=\pm 4$ and (b) ${\beta }_1=\pm 4$. Here, we set $L=4, t={\mathrm{\Delta }}_0=1$, and $\mu =0.1$. The non-Hermitian suppression effect, induced by the competition of ${\beta }_1$ and ${\beta }_2$, drives the spin flipping to opposite direction continuously without gap closing.

Figure 6

Global magnetic phase diagram of the many-body ground states. The variations of ${\chi }_{\mathrm{M}}$ induced by the non-Hermitian suppression effect are marked by different colors: FM phase (blue and red area), AFM phase (green and yellow area), and so on. Here, we set $L=4, t={\mathrm{\Delta }}_0=1$, and $\mu =0.1$.