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Unconventional topological phase transitions in helical Shiba chains

Falko Pientka, Leonid I. Glazman, and Felix von Oppen
Phys. Rev. B 89, 180505(R) – Published 20 May 2014
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Abstract

Chains of magnetic impurities placed on a superconducting substrate and forming helical spin order provide a promising venue for realizing a topological superconducting phase. An effective tight-binding description of such helical Shiba chains involves long-range (power-law) hopping and pairing amplitudes which induce an unconventional topological critical point. At the critical point, we find exponentially localized Majorana bound states with a short localization length unrelated to a topological gap. Away from the critical point, this exponential decay develops a power-law tail. Our analytical results have encouraging implications for experiment.

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  • Received 13 January 2014

DOI:https://doi.org/10.1103/PhysRevB.89.180505

©2014 American Physical Society

Authors & Affiliations

Falko Pientka1, Leonid I. Glazman2, and Felix von Oppen1

  • 1Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany
  • 2Department of Physics, Yale University, New Haven, Connecticut 06520, USA

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Vol. 89, Iss. 18 — 1 May 2014

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  • Figure 1
    Figure 1

    (a) Phase diagram for kh=0.1π/a and ξ0=∞ (black phase boundaries) or ξ0=15a (gray lines), with topological (T) and nontopological (N) phases. (Energies are given in units of Δ0.) The yellow dashed line indicates the Bragg point kF=kh with exponentially localized Majorana states [|β|<1 in Eq. (6)]. For ξ0=∞, this coincides with the phase transition between the two- and single-channel phases. (b), (c) Winding of the unit vector B̂k=Bk/Bk as k is tuned across the Brillouin zone for (b) the single-channel and (c) the two-channel phase (partially shifted radially for visibility). While in the single-channel phase B̂k winds once around the origin; the winding is trivial in the two-channel phase, reflecting the topological phase transition. Insets: Dispersion hk and pairing Δk in the two phases. The two-channel dispersion has a second pair of Fermi points.

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  • Figure 2
    Figure 2

    (a) Energy of the two positive-energy subgap states (in units of Δ0) in the nontopological two-channel phase (δk<0) near the Bragg point, for ξ0=∞ and various chain lengths L. As L→∞, the two states become degenerate with energy ∼|δk| near the phase transition. At the critical point, one subgap state merges discontinuously with the quasiparticle continuum. For finite L, the discontinuity is smeared and the degeneracy is lifted on the scale 1/L. (b) Majorana wave function vj at the Bragg point kF=kh. The exact numerical solution of the BdG Hamiltonian (green crosses) agrees with the analytical solution (black line) in Eq. (6). Inset: Localization length ξeff=a/ln|β−1| along the yellow line in the phase diagram in Fig. 1. The localization length is of order a and decreases with increasing coherence length ξ0. (c) Majorana wave function vj for kF=kh+δk with δk=0.003/a. The numerical solution of the BdG Hamiltonian (orange crosses) agrees with the analytical solution (blue line) as obtained by numerical evaluation of the inverse Laplace transform in Eq. (9). Inset: Blowup near the end of the chain emphasizing the initial exponential decay. Parameters: ε0=0.03Δ0, kh=0.1π/a, and kF=4.1π/a+δk.

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