Non-Hermitian Josephson junctions with four Majorana zero modes

TL;DR

The paper investigates non-Hermitian phase-biased Josephson junctions hosting four Majorana zero modes, modeled by an effective NH Hamiltonian that includes lead-induced self-energies. It establishes that Andreev exceptional points (EPs) emerge in the Andreev spectrum and are organized by the superconducting phase and asymmetric lead couplings, producing zero real-energy lines akin to NH bulk Fermi arcs. The work provides analytical and numerical evidence for EP formation between the lowest (and, at stronger NH, between higher) ABSs and shows that EPs yield strong local and nonlocal spectral signatures accessible via conductance measurements. These findings demonstrate a pathway to design NH topological phases and to operate Andreev bound states in Majorana-based Josephson junctions, with robust markers in the Green’s-function spectrum. The results have relevance for NH topological physics and Majorana-based quantum devices where dissipation is intrinsic.

Abstract

Josephson junctions formed by finite-length topological superconductors host four Majorana zero modes when the phase difference between the superconductors is $\varphi=π$ and their length is larger than the Majorana localization length. While this picture is understood in terms of a Hermitian description of isolated junctions, unavoidable transport conditions due to coupling to reservoirs make them open and ground for non-Hermitian effects that still remain largely unexplored. In this work, we investigate the impact of non-Hermiticity on Josephson junctions hosting four Majorana zero modes when they are coupled to normal leads. We demonstrate that, depending on whether inner or outer Majorana zero modes are subjected to non-Hermiticity, Andreev exceptional points can form between lowest (higher energy) Andreev bound states connected by stable zero real energy lines. We further find that the Andreev exceptional points give rise to strong local and nonlocal spectral weights, thus providing a way for their identification via, e.g., conductance measurements. Our findings unveil non-Hermiticity for designing non-Hermitian topological phases and for operating Andreev bound states in Josephson junctions hosting Majorana zero modes.

Figures (7)

• Figure 1: A NH Josephson junction with four MZMs ($\gamma_{i}$), where MZMs emerging at the end of each topological superconductor (green) is coupled to normal N reservoirs (blue and red).
• Figure 2: Re (blue) and Im (red) parts of the eigenvalues as a function of the superconducting phase difference $\phi$ at finite but equal non-Hermiticity $\Gamma_{i}=0.5$ for (a) $t'=0.3$, $t=0.5$ and (b) $t'=0.01$, $t=0.01$. The gray curves below the blue curves correspond to the eigenvalues without non-Hermiticity $\Gamma_{i}=0$. Parameters: $\Gamma_{i}=0.5$, $\bar{\tau}=2$.
• Figure 3: (a,b) Re (blue) and Im (red) parts of the eigenvalues as a function of the superconducting phase difference $\phi$ when $\Gamma_{2,3}=0$ for (a) $\Gamma_{1,4}=0.5$ and (b) $\Gamma_{1,4}=2$, in both cases with $t'=0.3$, $t=0.5$. The inset shows the Re and Im energies obtained from Eq. (\ref{['ABSEQ7']}) for the same regime of (a). The gray curves correspond to the eigenvalues without non-Hermiticity $\Gamma_{i}=0$. The ends of the shaded magenta regions mark the position of EPs connecting zero Re energy lines. (c) Re part of the energy difference between the positive and negative levels closest to zero real energy as a function of $\phi$ and $\Gamma_{1}=\Gamma_{4}$, with the magenta regions indicating the zero Re energy and their ends marking the EPs. The inset in (c) shows the same as in (c) but at $\Gamma_{4}=0$. (d) Same as in (c) but for the Re part of the energy difference between first excited positive and negative levels. Parameters: $\bar{\tau}=2$.
• Figure 4: (a,b) Re (blue) and Im (red) parts of the eigenvalues as a function of $\phi$ when $\Gamma_{1,4}=0$ for (a) $\Gamma_{2,3}=0.5$ and (b) $\Gamma_{2,3}=2$, in both cases with $t'=0.3$, $t=0.5$. The gray curves correspond to the eigenvalues without non-Hermiticity $\Gamma_{i}=0$. The inset shows the Re and Im energies obtained from Eq. (\ref{['EqInnerMZMsG']}) for the same regime of (a). The ends of the shaded magenta regions mark the position of EPs connecting zero Re energy lines. (c) Re part of the energy difference between the positive and negative levels closes to zero as a function of $\phi$ and $\Gamma_{2}=\Gamma_{3}$, with the magenta regions indicating the zero Re energy and their ends marking the EPs. (d) Same as in (c) but for the Re part of the energy difference between first excited positive and negative levels. Parameters: $\bar{\tau}=2$.
• Figure 5: (a,b) Re (blue) and Im (red) parts of the eigenvalues as a function of $\phi$ for $\Gamma_{2}=1$ and $\Gamma_{3}=2$ at $\bar{\tau}=2$, $t'=0$, $t=0$ and $\Gamma_{1,4}=0$. The gray curves correspond to the eigenvalues at $\Gamma_{i}=0$. (b) Re part of the energy difference between the first excited positive and negative levels of (a) as a function of $\phi$ and $\Gamma_{3}$ at $\Gamma_{2}=1$ marked by the horizontal dashed line. (c) The same quantity as in (b) but as a function of $\phi$ and the tunneling coupling $\bar{\tau}$ between superconductors forming the Josephson junction. The ends of the shaded magenta regions in (b,c) mark the position of EPs connecting zero Re energy lines indicated by magenta color. The horizontal dotted line in (c) indicates that below it the Re energies are zero for any $\phi$ but no EPs appear.