Parity-time-symmetric topological superconductor

We investigate a topological superconducting wire with balanced gain and loss that is effectively described by the non-Hermitian Kitaev/Majorana chain with parity-time symmetry. This system is shown to possess two distinct types of unconventional edge modes, those with complex energies and nonorthogonal Majorana zero modes. The latter edge modes cause nonlocal particle transport with currents that are localized at the edges and absent in the bulk. This anomalous particle transport results from the interplay between parity-time symmetry (non-Hermiticity) and topological superconductivity.

Figure 1

(a) Phase diagram of the PT-symmetric Majorana chain with $J=\mathrm{\Delta }$. The topological (trivial) phase lies in the region $\left|\mu /2J\right|<\left(>\right)1$, while the PT-unbroken (-broken) phase lies in the region $\gamma <\left(>\right)\sqrt{2\left|\mu \right|(\sqrt{J^2+{\mu }^2}-|\mu \left|\right)}$. The sweet spot $\left(J=\mathrm{\Delta },\mu =0\right)$ is singular, where the PT phase boundary lies at $\gamma =2J$. (b)–(e) Complex spectra of the PT-symmetric Majorana chain $\left(L=200;J=\mathrm{\Delta }=0.5\right)$. (b) Topological and PT-unbroken phase $\left(\mu =0.2,\gamma =0.2\right)$; Majorana zero edge modes (blue dots) emerge. (c) Topological and PT-broken phase $\left(\mu =0.2,\gamma =1.0\right)$; both Majorana zero edge modes (blue dots) and complex edge modes (red dots) emerge. (d) Trivial and PT-unbroken phase $\left(\mu =1.5,\gamma =0.2\right)$; no edge modes emerge. (e) Trivial and PT-broken phase $\left(\mu =1.5,\gamma =1.0\right)$; complex edge modes (red dots) emerge.

Figure 2

(a) Real and (b) imaginary parts of the single-particle spectrum as a function of the disorder strength $d$. Black, red, and blue dots represent the bulk modes, complex edge modes, and Majorana edge modes, respectively. The Majorana chain of $L=200$ sites is characterized by the parameters $J_j=0.5+0.2{\epsilon }_j,\mathrm{\Delta }=1.0,\gamma =1.0$, and ${\mu }_j=0.5+d{\epsilon }_j^{\prime }$, where $J_j$ is the disordered hopping amplitude between sites $j-1$ and $j,{\mu }_j$ is the disordered chemical potential on site $j$, and ${\epsilon }_j$ and ${\epsilon }_j^{\prime }$ are uniform random variables over $\left[-0.5,0.5\right]$.

Figure 3

Schematic representations of (a) the original non-Hermitian Majorana chain ${\widehat{H}}_{\mathrm{PT}}$ and (b) the accompanying Hermitian Majorana chain ${\widehat{H}}_{\mathrm{H}}$ constructed with $\widehat{\eta }$. Here ${\widehat{a}}_j$ and ${\widehat{b}}_j$ represent Majorana operators, and the lines connecting them show their couplings. As we approach the PT-transition point $\left(\gamma =J+\mathrm{\Delta }\right)$, the complex edge modes appear in the original representation (a), whereas the additional unpaired Majorana fermions ${\widehat{b}}_1$ and ${\widehat{a}}_L$ emerge in the transformed representation (b).

Figure 4

Current-gain/loss characteristics for the PT-symmetric fermionic system with two sites $\left(J=1.0;{\lambda }_1=1/\sqrt{5},{\lambda }_2=2/\sqrt{5}\right)$. The current is the long-time average. The PT-transition point lies at $\gamma =J=1.0$, at which the current reaches the maximum.

Figure 5

Particle currents through the PT-symmetric Majorana chains of $L=200$ sites with $J=\mathrm{\Delta }=0.5$. The red/blue curves represent the currents at the right/left edge $\left(j=2/200\right)$, while the green curves represent those at the center of the bulk $\left(j=100\right)$. (a)–(c) Time evolution of the currents in the topological phase $\left(\mu =0.2\right)$ for the Hermitian case $\mathrm{[}\gamma =0.0$, (a)], the PT-unbroken phase $\mathrm{[}\gamma =0.3$, (b)], and the PT-broken phase $\mathrm{[}\gamma =0.4$, (c)]. (d)–(f) Time evolution of the currents in the trivial phase $\left(\mu =1.5\right)$ for the Hermitian case $\mathrm{[}\gamma =0.0$, (d)], the PT-unbroken phase $\mathrm{[}\gamma =0.4$, (e)], and the PT-broken phase $\mathrm{[}\gamma =0.6$, (f)]. (g) Current-gain/loss characteristics for the topological phase $\left(\mu =0.2\right)$. The current is the long-time average. The PT-transition point lies at $\gamma =0.37$, at which the edge current reaches the maximum.

Figure 6

Finite-size scaling of the particle currents at the PT-transition point. (a) Topological phase $\left(J=\mathrm{\Delta }=0.5,\mu =0.2,\gamma =0.365\right)$. The current in the bulk decreases according to $1/L$, whereas the current at the edge does not. (b) Trivial phase $\left(J=\mathrm{\Delta }=0.5,\mu =1.5,\gamma =0.490\right)$. Both currents in the bulk and at the edge decrease according to $1/L$.