Non-Abelian topological insulators from an array of quantum wires

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Non-Abelian topological insulators from an array of quantum wires

Eran Sagi and Yuval Oreg

Phys. Rev. B 90, 201102(R) – Published 4 November 2014

Abstract

We suggest a construction of a large class of topological states using an array of quantum wires. First, we show how to construct a Chern insulator using an array of alternating wires that contain electrons and holes, correlated with an alternating magnetic field. This is supported by semiclassical arguments and a full quantum-mechanical treatment of an analogous tight-binding model. We then show how electron-electron interactions can stabilize fractional Chern insulators (Abelian and non-Abelian). In particular, we construct a non-Abelian $Z_{3}$ parafermion state. Our construction is generalized to wires with alternating spin-orbit couplings, which give rise to integer and fractional (Abelian and non-Abelian) topological insulators. The states we construct are effectively two dimensional, and are therefore less sensitive to disorder than one-dimensional systems. The possibility of experimental realization of our construction is addressed.

Received 20 March 2014
Revised 18 July 2014

DOI:https://doi.org/10.1103/PhysRevB.90.201102

Authors & Affiliations

Eran Sagi and Yuval Oreg

Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel

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Issue

Vol. 90, Iss. 20 — 15 November 2014

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Images

Figure 1
(a) Physical scheme of the Q1D model we study. Blue wires contain electrons, and red wires contain holes. The black arrows represent the magnetic field through the system. The circles represent the sites of the corresponding tight-binding model [30], and the tunneling amplitudes of the tight-binding model are represented by colored arrows. (b) The energy spectrum of the wires (as a function of $k_{x}$ ) near zero energy without tunneling between the wires ( $t, t^{'}, t^{''} = 0$ ). The wires are tuned such that the four parabolas cross each other at zero energy, and the chemical potential is set to be zero. The spectra in blue, dashed blue, dashed red, and red correspond to wires 1, 2, 3, and 4 in a unit cell, respectively. (c) The energy spectrum when $t$ is switched on. A gap opens near $k_{x} = 0$ . (d) The spectrum when $t^{'}$ is switched on as well. This gives an additional gap at $k_{x} > 0$ . Free chiral modes are left on wires 1 and 4. Finally, if one switches on $t^{''}$ , there are free chiral modes at the edge of the system, which suggests that there is a nonzero Chern number.
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Figure 2
(a) A diagrammatic representation of the energy band structure in the case $ν \equiv k_{F}^{0} / k_{φ} = 1$ [see Fig. 1 for definitions of $k_{φ}$ and $k_{F}^{0}$ ]. The $y$ axis shows the wire index inside the unit cell, and the $x$ axis shows $k_{x}$ in units of $k_{φ}$ . The symbol $⊙$ $(\otimes)$ represents $k_{i}^{L}$ ( $k_{i}^{R}$ ). Colored arrows represent tunneling amplitudes between the wires. (b) The same diagram for a topological insulator with $ν = \frac{1}{3}$ . Colored arrows now represent the multielectron processes responsible for the creation of Laughlin-like states. These complex processes in terms of the electrons ( $ψ \sim e^{i ϕ}$ ) for $ν = 1 / 3$ are mapped to simple tunneling processes in terms of the fermions $\tilde{ψ} \sim e^{i η}$ . Thus, $ν = 1 / 3$ for $ψ$ is equivalent to $ν = 1$ for $\tilde{ψ}$ . In the presence of spin-orbit coupling, spin up (blue) and spin down (light red) experience opposite alternating effective magnetic fields.
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Figure 3
(a) A possible experimental realization of our construction. Green lines represent current carrying wires, which produce the alternating magnetic field needed for our construction. This way, the magnetic field is positive (i.e., out of the page, denoted by $⨀$ ) in $n$ regions that contain electrons, and negative ( $⨂$ ) in $p$ regions that contain holes. (b) Another possible realization of our construction where now the 2D plane is replaced by many $V$ grooves connected in parallel. Again, blue regions contain electrons and red regions contain holes. The arrows represent the constant external magnetic field. In this geometry, electrons and holes experience opposite magnetic fields.
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