Dissipation induced Majarona $0$- and $π$-modes in a driven Rashba nanowire

TL;DR

The paper analyzes a periodically driven Rashba nanowire proximitized by an s-wave superconductor, embedded in a Markovian environment. By mapping the driven Liouvillian to a Floquet problem via third-quantization, it identifies edge Majorana zero modes (MZMs) at quasienergy $E=0$ and Majorana π-modes (MPMs) at $E=\pi$, protected by bulk winding numbers $\nu_0$ and $\nu_\pi$. It also uncovers trivial edge modes TZMs and TPMs arising from exceptional points, which are not bulk-topology protected. The study shows that dissipation can modify the topological phase diagram, even creating topological phases absent in the closed-drive case, and that these boundary modes are robust against onsite disorder. The results extend driven topological superconductivity to open, dissipative systems and outline experimental parameters for realizing such phases.

Abstract

Periodic drive is an intriguing way of creating topological phases in a non-topological setup. However, most systems are often studied as a closed system, despite being always in contact with the environment, which induces dissipation. Here, we investigate a periodically driven Rashba nanowire in proximity to an $s$-wave superconductor in a dissipative background. The system's dynamics is governed by a periodic Liouvillian operator, from which we construct the Liouvillian time-evolution operator and use the third-quantization method to obtain the `Floquet damping matrix', which captures the spectral and topological properties of the system. We show that the system exhibits edge-localized topological Majorana $0$-modes (MZMs) and $π$-modes (MPMs). Additionally, the system also supports a trivial $0$-modes (TZMs) and $π$-modes (TPMs), which are also localized at the edges of the system. The MZMs and the MPMs are connected to the bulk topology and carry a bulk topological invariant, while the emergence of TZMs and TPMs is primarily tied to exceptional points and is topologically trivial. We show that both the topological (MZMs and MPMs) and trivial (TZMs and TPMs) edge modes are robust against onsite disorder. We study the topological phase diagrams in terms of the topological invariants and show that the dissipation can modify the topological phase diagram substantially and even induce topological phases in the system. Our work extends the understanding of a driven-dissipative topological superconductor.

Figures (5)

• Figure 1: (a) Real part of the quasienergy ${\rm Re}~E$ as a function of the magnetic field $B$ of the Floquet damping matrix, considering OBC. The color represents the imaginary part of the quasienergy ${\rm Im}~E$ and the amplitude is represented by the colorbar. The red, black, and green dashed lines represent parameters used in Fig. \ref{['fig:Eigenvalue_chain_probability_plot']}. The yellow shade represents regions of TZMs and TPMs. We use $100$ lattice sites. (b) Quasienergy spectra close to ${\rm Re}~E=0$ for a system obeying PBC (orange) and OBC (gray). The right axis represents the winding number $\nu_0$. We demarcate the equilibrium critical point $B_c=\sqrt{2}$ to highlight the emergence of new topological phases out of a driven dissipative quantum system. We use $400$ lattice sites. (c) We repeat (b) but close to ${\rm Re}~E=\pi$ and the right axis represents the winding number $\nu_\pi$. (d) The absolute quasienergy $\lvert E \rvert$ as a function of $B$ is shown close to the black dashed line in (b). The right axis represents the scalar product $\lvert \langle \psi_i | \psi_j \rangle \rvert$. The inset shows $\lvert E \rvert$ and $\lvert \langle \psi_i | \psi_j \rangle \rvert$ close to the green dashed line in (b). (e) We repeat (d) but for the black and green dashed lines in (c). Here, $2 t_h = \lambda_{R} = \Delta=\gamma=1$, $\mu=0$, $\mu_1=0.3$, and $\omega=2.5$.
• Figure 2: (a) Real part of the quasienergy spectrum ${\rm Re}~E$ as a function of $m$ for $B=1.2$ (red line in Fig. \ref{['Fig:PBC-OBC']}(a)). The insets represent zoomed-in spectra close to ${\rm Re}~E=0$ and $\pi$. The color bar indicates the imaginary part of the quasienergy. (b) LDOS associated with the MZMs, while the inset shows the LDOS for the MPMs. (c) Real part of the quasienergy spectrum close to ${\rm Re}~E=0.0$ showing four TZMs for $B=4.3$ (green line in Fig. \ref{['Fig:PBC-OBC']}(a)). The inset shows real part of the quasienergy spectrum close to ${\rm Re}~E=\pi$, showing two TPMs (the other two appear at ${\rm Re}~E=-\pi$) for $B=2.3$ (black line in Fig. \ref{['Fig:PBC-OBC']}(a)). (d) LDOS associated with the TZMs. The inset shows the LDOS for the TPMs. We consider $N=100$ lattice sites and the rest of the parameters remain the same as in Fig. \ref{['Fig:PBC-OBC']}.
• Figure 3: The Phase diagram in terms of (a) $\nu_0$ and (b) $\nu_\pi$ in the $B - \gamma$ plane. The $\gamma=0$ line represents the Hermitian driven system. Here, the colorbar represents the values of $\nu_{0,\pi}$. We consider $200$ lattice sites. Rest of the parameters remain the same as in Fig. \ref{['Fig:PBC-OBC']}.
• Figure 4: Disorder-averaged LDOS for (a) MZMs, (b) MPMs, (c) TZMs, and (d) TPMs for $W=0.25$. The inset shows LDOS associated with $W=0.5$. We consider $100$ independent disorder realizations for computing disorder averaging. The parameters remain the same as Fig. \ref{['fig:Eigenvalue_chain_probability_plot']}.
• Figure 5: Real part of the quasienergy spectrum as a function of the state index showing (a) $\nu_0=2$ phase ($B=0$ and $\gamma=5$), (b) $\nu_\pi=2$ phase ($B=3.4$ and $\gamma=0.3$). The insets show zoomed-in spectra around ${\rm Re}~E = 0$ in (a) and around ${\rm Re}~E = \pi$ in (b). The colorbar represents the imaginary part of the eigenvalue. The rest of the parameters take the same value as Fig. \ref{['Fig:PBC-OBC']}.