Long-range Kitaev chains via planar Josephson junctions

Dillon T. Liu, Javad Shabani, and Aditi Mitra

Phys. Rev. B 97, 235114 – Published 11 June 2018

Abstract

We show how a recently proposed solid-state Majorana platform comprising a planar Josephson junction proximitized to a 2D electron gas (2DEG) with Rashba spin-orbit coupling and Zeeman field can be viewed as an effectively one-dimensional (1D) Kitaev chain with long-range pairing and hopping terms. We highlight how the couplings of the 1D system may be tuned by changing experimentally realistic parameters. We also show that the mapping is robust to disorder by computing the Clifford pseudospectrum index in real space for the long-range Kitaev chain across several topological phases. This mapping opens up the possibility of using current experimental setups to explore 1D topological superconductors with nonstandard and tunable couplings.

Received 23 March 2018
Revised 29 May 2018

DOI:https://doi.org/10.1103/PhysRevB.97.235114

Physics Subject Headings (PhySH)

Topological superconductors

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Dillon T. Liu, Javad Shabani, and Aditi Mitra

Center for Quantum Phenomena, Department of Physics, New York University, New York, New York 10003, USA

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Issue

Vol. 97, Iss. 23 — 15 June 2018

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Images

Figure 1
Schematic of 2D Majorana platform proposed in Refs. [24, 25]. A 2DEG with strong spin-orbit coupling is contacted with $s$ -wave superconducting leads that have a phase difference $ϕ$ and an in-plane longitudinal magnetic field, $\vec{B}$ . The top gate can be used to tune the chemical potential. Majorana edge modes appear at either end of the 1D normal channel.
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Figure 2
(a) Comparison of the topological invariant computed via a winding number in 2D system (dashed line) and via the real-space invariant in the 1D model (circles) defined in Eq. (15). The fluctuations across disorder realizations are also plotted (boxes), but are very close to zero. (b)–(d) The amplitude of four positive energy modes closest to zero for a finite 1D system in the three topological phases shown in the top panel. The amplitudes are averaged over disorder realizations ( $N_{R} = 100$ ) and show that there are as many edge modes as the invariant predicts.
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Figure 3
(a) Winding angle $θ (k)$ extracted from a 2D system. This angle winds around the unit circle three times to give an invariant $W = 3$ . Several points are extracted and labeled with roman numerals (i)–(viii) illustrating the winding angle. (b) Components $d_{z} (k) and d_{y} (k)$ , defining a 1D Hamiltonian, as been derived from a 2D system. These differ dramatically from simple sinusoidal functions, which implies that the 1D system is not a nearest-neighbor model, but instead includes long-range hopping and pairing terms.
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Figure 4
(Left) Hopping and (right) pairing parameters ( $t_{i} and Δ_{i}$ , respectively) are plotted as functions of the separation $i$ between sites involved. These parameters are shown for three different values of 2D chemical potential, which correspond to three distinct topological regimes. The presence of significant terms beyond nearest neighbors is evident and there is a noticeable difference in the decay at large separations for the various topological regimes.
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Figure 5
(a) Hopping and (b) pairing parameters for the 1D model vary as the underlying 2D system is tuned through several topological phase transitions, as indicated in Fig. 2. In these panels, we have included the first four nearest neighbor coupling terms.
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Figure 6
(a) Topological invariant, $W$ defined as half the signature of $N = X Γ + H$ , plotted for varying $μ$ . (b) (solid blue) Energy spectrum and (dashed red) pseudospectrum of Kitaev chain for varying $μ$ . One can clearly see that the pseudospectrum crosses zero around $μ = 2$ . This is the usual topological phase transition that corresponds to the appearance of edge modes at zero energy. (right) Probability density of eigenstates with eigenvalue closest to zero of $N and H$ for two values of $μ$ corresponding to being deep in the trivial ( $μ = 3.0$ ) and topological ( $μ = 1.0$ ) regimes. The probability densities computed from $N, H$ are almost indistinguishable, with the topological edge-mode clearly visible.
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