Non-Kondo many-body physics in a Majorana-based Kondo type system

Ian J. van Beek and Bernd Braunecker

Phys. Rev. B 94, 115416 – Published 12 September 2016

Abstract

We carry out a theoretical analysis of a prototypical Majorana system, which demonstrates the existence of a Majorana-mediated many-body state and an associated intermediate low-energy fixed point. Starting from two Majorana bound states, hosted by a Coulomb-blockaded topological superconductor and each coupled to a separate lead, we derive an effective low-energy Hamiltonian, which displays a Kondo-like character. However, in contrast to the Kondo model which tends to a strong- or weak-coupling limit under renormalization, we show that this effective Hamiltonian scales to an intermediate fixed point, whose existence is contingent upon teleportation via the Majorana modes. We conclude by determining experimental signatures of this fixed point, as well as the exotic many-body state associated with it.

Received 28 June 2016

DOI:https://doi.org/10.1103/PhysRevB.94.115416

Physics Subject Headings (PhySH)

Coulomb blockade Kondo effect Majorana fermions Topological superconductors

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Ian J. van Beek and Bernd Braunecker

SUPA, School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews KY16 9SS, United Kingdom

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Issue

Vol. 94, Iss. 11 — 15 September 2016

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Images

Figure 1
A Majorana system with a floating superconductor. An $s$ -wave superconductor (red) is grown on a nanowire with strong spin-orbit coupling (blue). For sufficiently large applied magnetic field and appropriate chemical potential the nanowire becomes a topological superconductor (TSC) with Majorana bound states $γ_{1}$ and $γ_{2}$ at each end. A gate at voltage $V_{g}$ is used to tune the ground-state occupation number, which is dictated by the capacitive charging energy $E_{C}$ . The nanowire couples to leads at either end with tunneling coefficients $t_{1}$ and $t_{2}$ between lead electrons and $γ_{1}$ and $γ_{2}$ .
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Figure 2
Charging energy against total number of electrons, $N_{e}$ , in the nanowire for $n_{g} = 2 n - \frac{1}{2}$ . Filled (empty) circles represent states with $n_{d} = 1$ ( $n_{d} = 0$ ). Both the ground and excited states are degenerate, with transitions between them via (a) normal tunneling and (b) anomalous tunneling. Note that there is no process mediating transitions between the two excited states.
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Figure 3
Scattering channels which contribute to renormalization of $J_{\pm}^{(1)}$ . Diagrams (a) and (b) show particle-mediated scattering via the left and right leads, respectively. Similarly, (c) and (d) depict hole-mediated scattering. Curved lines represent lead electrons, while the straight lines correspond to the nanowire with dashed and solid lines denoting $S_{z} = - 1$ and $S_{z} = + 1$ , respectively.
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Figure 4
Change in anomalous tunnel couplings, $J_{\pm}^{(j)}$ , with scaling parameter $ℓ$ , as given by Eq. (3) for $t_{1} = 0.1$ meV, $t_{2} = 0.01$ meV, and further parameters [65] based on a recent experiment [45]. The solid purple and dashed red lines show $J_{\pm}^{(1)}$ and $J_{\pm}^{(2)}$ , respectively. The vertical dashed black line is the value of $ℓ$ corresponding to the crossover temperature $T_{c}$ . The couplings display a rapid, exponential change from their initial values of $t_{1}$ and $t_{2}$ as $ℓ$ increases. This high temperature sensitivity even far above $T_{c}$ is due to the large ratio $t_{1} / t_{2} = 10$ .
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Figure 5
Variation of conductance amplitude with temperature, adjusted to account for generic $1 / T$ dependence, as given by Eqs. (3) and (5). The solid blue line depicts the result for a many-body state while the dashed orange line corresponds to the bare tunnel couplings. The dot-dashed magenta line is a high-temperature ( $T ≫ T_{c}$ ), high-asymmetry ( $t_{1} ≫ t_{2}$ ) approximation, $G = K t_{2}^{2} (e^{- 2 α} - 4 e^{- α} + 4)$ , where $α = ln (k_{B} T / D_{0}) / ln (k_{B} T_{c} / D_{0})$ . The parameters used [65] imply $T_{c} = 2$ mK.
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Figure 6
The lowest-order particle mediated process contributing to the scaling of $J_{z}^{(11)}$ . The line between the two vertices denotes a particle excited to the high-energy shell. Note that the two scattering events commute, so the pathway shown here is exactly canceled by a corresponding hole-mediated process and does not contribute to scaling.
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