Signatures of the long-range phase transition in topological Josephson junctions

The extended Kitaev models spark a recent surge of interest due to its unprecedented topological phase with fractional invariants. In this extended model of topological superconductors, long-range pairing interactions can induce a pair of subgap massive edge modes (MEMs), posing novel challenges to the unambiguous detection of Majorana bound states (MBSs) sustained by the same model in the short-range limit. Here we investigate the effect of a power-law decayed long-range pairing on the dc and ac Josephson currents between two Majorana nanowires. By varying the magnetic field and long-range pairing, the nanowires can be driven into different phases hosting MEMs, MBSs, or trivial Andreev bound states (ABSs), which considerably modulate the Josephson current. For weakly linked Josephson junctions, we show that the MEMs emerging in the long-range phases can induce a zigzag profile in the dc current-phase relation featured by two sign reversals deviating from the point where the superconducting phase difference $\varphi =\pi$. In contrast, the Josephson current indicates a sharp sign reversal at $\varphi =\pi$ in the short-range phase that hosts MBSs, while a smooth sinelike dc Josephson current appears in the presence of ABSs. In the nonequilibrium junction biased by a voltage, we identify a $4\pi$ ac Josephson current in the short-range phases, indicating a MBSs-mediated tunneling process dominating the supercurrents. Differently, when the system enters MEMs-hosted long-range phases, the ordinary ac Josephson current with $2\pi$ periodicity can be restored. In addition, we find the ac Josephson current induced by MEMs is significantly suppressed under the strong long-range pairing interactions. These signatures can serve as probes to discriminate MBSs from MEMs and ABSs.

Figure 1

(a) Schematic view and (b) the corresponding lattice model of the one-dimensional Josephson junction consisting of two superconducting regions and a central normal region with length $L_S$ and $L_N$. The proximity-induced long-range pairing interactions take a power-law dependence on the spatial distance $r$ and acquire a phase difference $\varphi ={\varphi }_1-{\varphi }_2$ across the junction. The tunneling couplings at the junction interfaces are described by $t_L$ and $t_R$. The central region can sustain static or nonequilibrium supercurrents when the applied voltage is zero or finitely biased.

Figure 2

(a) Phase diagram on the $\varepsilon -\alpha$ plane of a single long-range nanowire with power-law decaying pairing exhibits a dependence on the on-site pairing potential. [(b)–(d)] Trajectories of the winding vector on the complex plane at $\varepsilon =2t$ for (b) MBSs in the topological short-range regime, (c) trivial states in the nontopological short-range regime, and (d) MEMs in the long-range regime. (e) Mass gap as a function of $\alpha$ in the long-range superconducting region with open boundary condition and $L_S=1000a$. The other parameters are taken as $V_Z=t_{\lambda }=t$ and $\mathrm{\Delta }=t/4$.

Figure 3

(a) Phase diagram in the $\mu -\alpha$ plane of a long-range Kitaev chain with its pairing interactions quantified by power-law decay rate $\alpha$, adapted from Ref. [52]. The winding number $W=0, W=1$ are associated with the trivial and topological phases in the short-range sector ($\alpha >1$), while $W=+1/2, W=-1/2$ corresponds to the unconventional trivial and topological phases in the long-range sector ($\alpha <1$). [(b) and (c)] Phase-dependent dc Josephson currents of a LRK-based Josephson junction involving (b) MBSs in the short-range sector ($\alpha \gg 1$), or (c) MEMs in the long-range sector ($\alpha =0.25$). The chemical potentials are fix at $\mu =0$.

Figure 4

[(a) and (b)] Low-lying eigenenergies in a long tunneling junction as a function of Zeeman field $V_z$ in the (a) short-range regime $\alpha =10$ and (b) long-range regime $\alpha =0.2$. [(c) and (d)] Spatial distributions of the wave functions for the lowest four eigenstates in the short-range regime marked in (a) by the red star and black annotation. [(e) and (f)] The same quantities while in long-range regime indicated in (b) by the blue triangle and black annotation. The other parameters are taken as ${\mu }_S=\mathrm{\Delta }=0.5\mathrm{meV}, t=50\mathrm{\Delta }$, and $t_{\lambda }=4\mathrm{\Delta }$. The chemical potential in the normal region is set as ${\mu }_N={\mu }_S/2$ to induce zero-energy ABSs via the confinement effect [82].

Figure 5

[(a) and (c)] Phase-dependent low-lying eigenenergies in the (a) short-range $\alpha =10$, (b) long-range regime $\alpha =0.2$ at $V_Z=1.5V_{ZC}$, and (c) a nontopological case at $V_Z=0.5V_{ZC},\alpha =10$ for comparison. [(d) and (f)] Phase-dependent dc Josephson currents for different $V_Z$ in the (d) short-range $\alpha =10$, (e) long-range regime $\alpha =0.2$, and (f) nontopological cases with $V_Z⩽0.6V_{ZC},\alpha =10$ accompanied by ABSs.

Figure 6

[(a) and (b)] Phase-dependent dc Josephson current $I_{dc}$ on the $\alpha -\varphi$ plane at $V_Z=1.5V_{ZC}$ in the (a) short junction case with $L_N=10a$ and (b) long junction case with $L_N=100a$. Other parameters are taken the same as in Fig. 5.

Figure 7

Critical current $I_c$ as a function of the Zeeman splitting $V_Z$ with the long-range exponent taking values in $0.2⩽\alpha ⩽10$ for (a) the short junction case with $L_N=2a$ and (b) long junction case with $L_N=100a$. The rest of parameters are the same as in Fig. 5.

Figure 8

(a) Time-dependent ac Josephson currents as a function of phase difference $\varphi$ and Zeeman field $V_Z$ in the short-range regime $\alpha =10$. (b) The same quantity but in the long-range regime $\alpha =0.2$. Color symbols mark the cut-lines where the CPR are extracted accompanied by MBSs (red star), trivial states (black rhombus), and MEMs (blue triangle), respectively. The coherence length and wire length of the superconducting region are reduced to $\xi \sim t/\mathrm{\Delta }=5$ and $L_S=20a$ to alleviate the computation of Eq. (26) in small time steps. Other parameters are the same as in Fig. 4.

Figure 9

[(a)–(e)] Time-dependent ac Josephson currents at $V=2{\delta }_m$ over a wide range of long-range decay rates: (a) $\alpha =10$ marked by the red star in Fig. 8, (b) $\alpha =4.0$, (c) $\alpha =2.2$, (d) $\alpha =0.8$, and (e) $\alpha =0.2$ marked by the blue triangle in Fig. 8. (f) Same current measurement marked by the black rhombus in Fig. 8 in the nontopological regime.