Topological quantum wires with balanced gain and loss

We study a one-dimensional topological superconductor, the Kitaev chain, under the influence of a non-Hermitian but $\mathcal{PT}$-symmetric potential. This potential introduces gain and loss in the system in equal parts. We show that the stability of the topological phase is influenced by the gain/loss strength and explicitly derive the bulk topological invariant in a bipartite lattice as well as compute the corresponding phase diagram using analytical and numerical methods. Furthermore, we find that the edge state is exponentially localized near the ends of the wire despite the presence of gain and loss of probability amplitude in that region.

Figure 1

Topological phase diagram for the Kitaev chain with $N=100$ sites and the alternating potential (5) for $t=1$ and $\mathrm{\Delta }=1$. The color map indicates the sign of the Pfaffian where $-1$ (black) means topological and $+1$ (white) means trivial. The dashed line corresponds to the analytically calculated phase boundary.

Figure 2

We show spectra for a chain with $N=100$ sites and the alternating potential (5) at different chemical potential over the gain/loss strength $\gamma$. The dashed line indicates the sign of the Pfaffian invariant. Zero modes [$Re\left(E\right)=Im\left(E\right)=0$] are highlighted with green dots. Evidently, the potential can close the gap and lead to a topological phase transition. In the topological regime, we always have a state at zero energy.

Figure 3

We show bulk spectra for a chain with $N=100$ sites and the alternating potential (5) at different chemical potential over the gain/loss strength $\gamma$. The dashed line indicates the sign of the Pfaffian invariant. For periodic boundary conditions, the spectra differ qualitatively from open boundary conditions.

Figure 4

(a) Modulus of the poles of the generating function for $t=\mathrm{\Delta }$. The modulus of the pole gives the decay constant of the edge state. Their intersection at $\left|z\right|=1$ corresponds to the topological phase transition which is also found in the numerical calculation of the Pfaffian (filled dots). In (b) and (c), the dots denote the unnormalized edge state from a numerical calculation with $N=100$ sites overlayed with lines for the analytical solution.

Figure 5

Topological phase diagram for the partitioning potential (C1) in a chain with $N=100$ sites and a partitioning fraction of $f=0$. The color map indicates the sign of the Pfaffian, the dashed line corresponds to the analytical phase boundary as extracted from the generating function approach.

Figure 6

Topological phase diagram for the partitioning potential (C1) in a chain with $N=100$ sites. A partitioning fraction of $f/N=0$ corresponds to half loss/half gain, whereas $f/N=0.5$ corresponds to the Hermitian model without any potential. The color map indicates the sign of the Pfaffian. The left edge of the phase diagram corresponds to the Hermitian Kitaev chain. Clearly, the phase diagram changes with the partitioning.

Figure 7

Numerically determined phase boundary for the potential (C1) at different partitioning fractions $f/N$ in a chain with $N=100$ sites. With increasing partitioning fraction we approach the phase boundary of the Hermitian case ($f/N=0.5$).

Figure 8

We show spectra for the non-$\mathcal{PT}$-symmetric potential in Eq. (D1) in a chain with $N=100$ sites at different chemical potential over the gain/loss strength $\gamma$. The dashed line corresponds to the sign of the real part of the Pfaffian, the solid line to the value of the imaginary part.

Figure 9

We show bulk spectra for the non-$\mathcal{PT}$-symmetric potential in Eq. (D1) in a chain with $N=100$ sites at different chemical potential over the gain/loss strength $\gamma$.