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  • Letter

Engineering anomalous Floquet Majorana modes and their time evolution in a helical Shiba chain

Debashish Mondal, Arnob Kumar Ghosh, Tanay Nag, and Arijit Saha
Phys. Rev. B 108, L081403 – Published 4 August 2023
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Abstract

We theoretically explore the Floquet generation of Majorana end modes (MEMs; both regular 0 modes and anomalous π modes) implementing a periodic sinusoidal modulation in the chemical potential in an experimentally feasible setup based on a one-dimensional chain of magnetic impurity atoms having spin spiral configuration (out-of-plane Néel type) fabricated on the surface of a common bulk s-wave superconductor. We obtain a rich phase diagram in the parameter space, highlighting the possibility of generating multiple 0- and π-MEMs localized at the end of the chain. We also study the real-time evolution of these emergent MEMs, especially when they start to appear in the time domain. These MEMs are topologically characterized by employing the dynamical winding number. We observe that the existing perturbative analysis is unable to explain the numerical findings, indicating the complex mechanism behind the formation of the Floquet Shiba minigap, which is characteristically distinct from other setups, e.g., the Rashba nanowire model. We also discuss the possible experimental parameters in connection to our model. Our work paves the way to realize Floquet MEMs in a magnet-superconductor heterostructure.

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  • Received 12 April 2023
  • Revised 4 July 2023
  • Accepted 20 July 2023

DOI:https://doi.org/10.1103/PhysRevB.108.L081403

©2023 American Physical Society

Physics Subject Headings (PhySH)

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Debashish Mondal1,2,*, Arnob Kumar Ghosh1,2,†, Tanay Nag3,4,‡, and Arijit Saha1,2,§

  • 1Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, India
  • 2Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India
  • 3Department of Physics and Astronomy, Uppsala University, Box 516, 75120 Uppsala, Sweden
  • 4Department of Physics, BITS Pilani–Hyderabad Campus, Telangana 500078, India

  • *debashish.m@iopb.res.in
  • arnob@iopb.res.in
  • tanay.nag@physics.uu.se
  • §arijit@iopb.res.in

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Issue

Vol. 108, Iss. 8 — 15 August 2023

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Images

  • Figure 1
    Figure 1

    Schematic representation of our setup. A 1D chain of magnetic adatoms with their spins (red arrows) confined in the xz plane (out-of-plane Néel-type spin spiral configuration) is placed on top of a bulk s-wave superconductor (green).

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  • Figure 2
    Figure 2

    (a) and (b) The quasienergy spectra of the Floquet operator for B/Δ=2 and B/Δ=3, respectively. Floquet 0- and π-MEMs are highlighted in the top-left and bottom-right insets, respectively. (c) The energy-resolved normalized LDOS computed at the end (blue curve) and middle (green curve) of the chain for B/Δ=2. The green peaks represent the emergent Shiba modes within the Floquet quasienergy spectrum. Here, we consider a 1D chain of 600 lattice sites, and we choose the model parameters (μ/Δ,th/Δ,θ)=(4,1,2π/3), Ω/Δ=1.5, and V0/Δ=5.

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  • Figure 3
    Figure 3

    (a) The time-dependent eigenvalue spectra of the time-evolution operator U(t,0) as a function of time t. Both the 0 and π modes appear and disappear with time during the time interval t[0,T]. We consider four time points: t=0.28T with only 0 modes (time point 1), t=0.63T with only π modes (time point 2), t=0.9T with both 0 and π modes (time point 3), and t=T with both 0 and π modes (time point 4). (b), (c), (d) and (e) denote the time-dependent site resolved LDOSs for 0 and/or π modes corresponding to time points 1, 2, 3 and 4, respectively. Clearly, at t/T=0.28 (t/T=0.63) only 0 modes (π modes) are present, while at t/T=0.9 and t/T=1 both the 0- and π-MEMs appear. Here, we consider a 1D chain of 500 lattice sites, and all the model parameters take the values as in Fig. 2.

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  • Figure 4
    Figure 4

    (a) and (b) The dynamical winding numbers W0 and Wπ, respectively, in the V0Ω plane. Here, Wε characterizes the Floquet MEMs residing at the quasienergy gap ε. (c) and (d) W0 and Wπ, respectively, as a function of Ω for three fixed values, V0/Δ= 5 (solid blue curve), 8 (dashed red curve), and 10 (dotted green curve), for better clarity. (e) and (f) The quasienergy gap around quasienergies 0 and π, respectively. We choose B/Δ=5 (topological regime of the static model), and the rest of the model parameters take the same values as in Fig. 2.

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  • Figure 5
    Figure 5

    Comparison of the eigenvalue spectra Em as a function of m, obtained from the FPT (blue dots), the BW perturbation theory (red dots), and the exact Floquet operator (green dots). In (a), we depict the eigenvalue spectra for (V0/Δ,Ω/Δ)=(10,3), while in the inset we show the modes near quasienergy zero. We choose (V0/Δ,Ω/Δ)=(5,10) for (b), while the zoomed-in spectra near zero quasienergy are shown in the top-left inset. In the bottom-right inset, we demonstrate the eigenvalue spectra for (V0/Δ,Ω/Δ)=(5,6.5). We choose a 1D chain of 300 lattice sites, and the rest of the model parameters are the same as in Fig. 2.

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