Toric-code insulator enriched by translation symmetry

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Toric-code insulator enriched by translation symmetry

Pok Man Tam, Jörn W. F. Venderbos, and Charles L. Kane

Phys. Rev. B 105, 045106 – Published 4 January 2022

Abstract

We introduce a two-dimensional electronic insulator that possesses a toric-code topological order enriched by translation symmetry. This state can be realized from disordering a weak topological superconductor by double-vortex condensation. It is termed the toric-code insulator, whose anyonic excitations consist of a charge- $e$ chargon, a neutral fermion, and two types of visons. There are two types of visons because they have constrained motion as a consequence of the fractional Josephson effect of one-dimensional topological superconductor. Importantly, these two types of visons are related by a discrete translation symmetry and have a mutual semionic braiding statistics, leading to a symmetry enrichment akin to the type in Wen's plaquette model and Kitaev's honeycomb model. We construct this state using a three-fluid coupled-wire model, and analyze the anyon spectrum and braiding statistics in detail to unveil the nature of symmetry enrichment due to translation. We also discuss potential material realizations and present a band-theoretic understanding of the state, fitting it into a general framework for studying fractionalizaton in strongly interacting weak topological phases.

Received 20 July 2021
Revised 1 December 2021
Accepted 1 December 2021

DOI:https://doi.org/10.1103/PhysRevB.105.045106

Physics Subject Headings (PhySH)

Anyons Topological phases of matter Topological superconductors

Topological materials

Bosonization Luttinger liquid model Symmetries in condensed matter

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Pok Man Tam ¹, Jörn W. F. Venderbos ^2,3, and Charles L. Kane ¹

¹Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
²Department of Physics, Drexel University, Philadelphia, Pennsylvania 19104, USA
³Department of Materials Science & Engineering, Drexel University, Philadelphia, Pennsylvania 19104, USA

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Vol. 105, Iss. 4 — 15 January 2022

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Images

Figure 1
(a) Shows the wire model of a weak TSC, where each wire represents a 1D TSC (and black dots represent Majorana end modes). Two types of pointlike excitations in the bulk are depicted: the $h / 2 e$ vortex $(m)$ lives on links (dashed lines), and the Bogoliubov fermion $(f)$ lives on wires (blue lines). (b), (c) Illustrate the fractional Josephson effect: tunneling an $h / 2 e$ vortex across a wire leads to a $2 π$ phase slip, which in turn switches the ground-state fermion parity. Without creating additional excitations, a single vortex can only move across two wires at a time by exchanging a fermion between the wires, as shown in (a).
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Figure 2
(a) Two formulations for the Luttinger liquid of Cooper pairs. In one, $φ_{2}$ has $2 π$ compactification. Alternatively, $φ_{2}$ can have $4 π$ compactification and couple to a $Z_{2}$ gauge symmetry: $φ_{2} \mapsto φ_{2} + 2 π$ . (b) Three-fluid wire model of the toric-code insulator: two fluids for the charge and neutral sectors of a weak TSC, while the third fluid describes the $Z_{2}$ gauge sector emerging from double-vortex condensation. Anyons from each sector are labeled.
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Figure 3
Representative braiding processes in the toric-code insulator. The braiding phase can be computed by multiplying a string of operators that transport the anyon around a closed loop, depicted by a dashed line in each panel. (a), (b) Braiding between $e$ and $m$ leads to a $π$ phase. The same happens for the braiding between $f$ and $m$ , as $f$ is topologically equivalent to $e = ψ_{e} \times f$ . (c) Braiding between $m_{o}$ and $m_{e}$ leads to a $π$ phase. (d) The self-statistics of $m_{even/odd}$ is trivial. All these can be deduced from the expression of operator $T_{ℓ} (m)$ , which transports an $m$ particle across two wires.
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Figure 4
Toric-code insulator with an odd number of wires. Due to the periodic boundary condition, an even-link vison ( $m_{e}$ ) turns into an odd-link vison ( $m_{o}$ ) after going around $C_{y}$ once.
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Figure 5
Idealized square-lattice model for a weak TSC in two dimensions. (a) The square-lattice model of Eq. (4.3) is defined by staggered alternating hoppings in the $x$ direction, $t_{1}$ (intracell hopping) and $t_{2}$ (intercell hopping), giving rise to a two-site unit cell (dashed square). The horizontal rows, indicated in light blue, are interpreted as 1D wires (see text). (b) Brillouin zone of the two-site square lattice model. The location of the Dirac nodes which occur when $t_{x} = (t_{1} + t_{2}) / 2 = 0$ are shown by bold blue dots. The splitting of the Dirac nodes into Bogoliubov-Dirac nodes is indicated by open blue dots.
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