Josephson current signature of Floquet Majorana and topological accidental zero modes in altermagnet heterostructures

TL;DR

The paper develops Floquet engineering of Majorana end modes in a 1D Rashba nanowire proximitized by an $s$-wave superconductor and a $d$-wave altermagnet, using real-space dynamical winding numbers to classify zero and $\pi$ Floquet Majorana end modes ($0$-FMEMs and $\pi$-FMEMs) and showing that periodic driving can gap out static accidental zero modes to yield topological AZMs (TAZMs) with $\pi$-FMEMs. A Floquet Josephson current framework with energy-resolved occupations is introduced to distinguish $0$- and $\pi$-FMEMs in static and driven Josephson junctions, revealing robust $4\pi$ periodic signatures that persist under disorder. The work demonstrates how AM-based platforms enable and identify FMEMs through Floquet protocols, with AM offering larger bulk gaps and expanded topological regions relative to Zeeman-field approaches, thereby broadening the feasible parameter space and enhancing experimental prospects. Overall, the study provides a concrete route to realize, detect, and differentiate Floquet Majorana modes via Josephson response in altermagnet heterostructures.

Abstract

We theoretically investigate the generation and Josephson current signatures of Floquet Majorana end modes (FMEMs) in a periodically driven altermagnet (AM) heterostructure. Considering a one-dimensional (1D) Rashba nanowire (RNW) proximitized to a regular $s$-wave superconductor and a $d$-wave AM, we generate both $0$- and $π$-FMEMs by driving the nontopological phase of the static system. While the static counterpart hosts both topological Majorana zero modes (MZMs) and nontopological accidental zero modes (AZMs), the drive can gap out the static AZMs and generate robust $π$-FMEMs, termed as topological AZMs (TAZMs). We topologically characterize the emergent FMEMs via dynamical winding numbers exploiting chiral symmetry of the system. Moreover, we consider a periodically driven Josephson junction comprising of RNW/AM-based 1D topological superconduting setup. We identify the signature of MZMs and FMEMs utilizing $4π$-periodic Josephson effect, distinguishing them from trivial AZMs exhibiting $2π$-periodicty, in both static and driven platforms. This Josephson current signal due to Majorana modes survives even in presence of finite disorder. Our work establishes a route to realize and identify FMEMs in AM-based platforms through Floquet engineering and Josephson current response.

Figures (12)

• Figure 1: Schematic illustration of our AM-SC heterostructure: (a) A 1D RNW (blue) is placed in close proximity to a $s$-wave SC (yellow) and a $d$-wave AM (green) in presence of an external sinusoidal Floquet drive, $V(t)$. This setup hosts FMEMs localized at the ends of the RNW (red spikes). Panel (b) depicts the JJ setup comprised of two weakly coupled TSCs (as shown in panel (a)), with coupling strength $\tilde{t}$ and superconducting phase bias, $\phi=(\phi_L-\phi_R)$. This setup is periodically driven by $V(t)$.
• Figure 2: Illustration of DWNs and quasienergy spectrum: In panels (a)-(b), we depict the DWNs, $\mathcal{W}_0$ and $\mathcal{W}_{\pi}$ in the $V_0-\Omega$ plane when the static model is in the trivial phase and does not host any zero energy modes, while panels (c)-(d) showcase the same when the static model hosts AZMs (see SM supp for static winding number of the undriven system). Inset of each panel displays the quasienergy spectrum, $E_\alpha/\Omega$ as a function of $\Omega$ (along the vertical dashed lines of each panel) for $V_0=2$ employing both OBC and PBC, highlighting the emergence of $0$-FMEMs (inset of (a),(c)) and $\pi$-FMEMs (inset of (b),(d)). We choose the model parameters as $(\mu/t,J_A/t=1.5,0.4)$ in panels (a)-(b) and $(\mu/t,J_A/t=0,1.2)$ in panels (c)-(d) while consider a finite size system with $N_x=100$ lattice sites across all the panels.
• Figure 3: Behaviour of JC flowing across the static JJ: In panel (a), we present the ${\epsilon}-\phi$ relation due to MZMs and AZMs localized near (${\rm AZM_{jun}, MZM_{jun}}$) and far from the junction (${\rm AZM_{end}, MZM_{end}}$). In panel (b), we depict the variation of JC, $I(\phi)$ as a function of $\phi$ for MZMs and AZMs. Panels (c) and (d) display the effect of disorder on $I(\phi)$ for different disorder strengths, $V_{dis}$ corresponding to MZMs and AZMs, respectively. For MZMs, we choose $(\mu, J_A) = (1.1t, 0.4t)$, while for AZMs $(\mu, J_A) = (0, 1.2t)$. We fix the system size $N_x = 340$ lattice sites and $\tilde{t} = 0.01t$ in all panels. We consider 50 disorder configurations in panels (c) and (d).
• Figure 4: Variation of Floquet JC across the driven JJ: In panels (a) and (b), we present the variation of JC when the driven system hosts only $0$-FMEMs and only $\pi$-FMEMs, respectively. On the other hand, panel (c) [inset] illustrates the JC for TAZMs [zero energy Floquet AZMs] for various system sizes. Panel (d) highlights the discontinuous jump in JC at $\phi=\pi$ as a function of $N_x$ corresponding to panels (a) and (b) while the inset represents the discontinuity for TAZMs as presented in panel (c). The model parameters in $(\mu,J_A,\Omega,V_0)$ space is chosen as, (a) $(1.5t,0.4t,2.4,4.2t)$,(b) $(1.5t,0.4t,4.8,4.5t)$, (c) $(0,1.2t,3,3.1t)$ and $\tilde{t}=0.01t$ in all the panels.
• Figure 5: Variation of winding number $\mathcal{W}$ and number of zero modes $N_{0}$ for the static case: In panel (a) we depict the static winding number $\mathcal{W}$ in the $\mu/t - J_A/t$ plane for the undriven system (see Eq.(1) of the main text). In panel (b), we highlight the number of zero energy modes, $N_0$, localized at the ends of the AM based 1D TSC. Note that in panel (b) $N_0=4$ for $\mu/t=0$ and $|J_A/t|\ge 1$ signifying the presence of four end localized zero energy modes which we refer as AZMs. However, for the same parameter values $\mathcal{W}$ is zero in panel (a) which establishes their nontopological nature. Other model parameters of the static Hamiltonian are chosen as $\lambda_R=0.5t, \Delta_0=0.3t$ in both the panels.