Symmetry-related transport on a fractional quantum Hall edge

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Symmetry-related transport on a fractional quantum Hall edge

Jinhong Park, Bernd Rosenow, and Yuval Gefen

Phys. Rev. Research 3, 023083 – Published 29 April 2021

Abstract

Low-energy transport in quantum Hall states is carried through edge modes and is dictated by bulk topological invariants and possibly microscopic Boltzmann kinetics at the edge. Here, we show how the presence or breaking of symmetries of the edge Hamiltonian underlie transport properties, specifically dc conductance and noise. We demonstrate this through the analysis of hole-conjugate states of the quantum Hall effect, specifically the $ν = 2 / 3$ case in a quantum point-contact geometry. We identify two symmetries, a continuous SU(3) symmetry and a discrete $Z_{3}$ symmetry, whose presence or absence (different symmetry scenarios) dictate qualitatively different types of behavior of conductance and shot noise. While recent measurements are consistent with one of these symmetry scenarios, others can be realized in future experiments.

Received 10 April 2020
Revised 31 December 2020
Accepted 24 March 2021

DOI:https://doi.org/10.1103/PhysRevResearch.3.023083

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Edge states Fractional quantum Hall effect Quantum transport

Bosonization

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Jinhong Park

Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel and Institute for Theoretical Physics, University of Cologne, Zülpicher Straße 77, 50937 Köln, Germany

Bernd Rosenow

Institut für Theoretische Physik, Universität Leipzig, D-04103 Leipzig, Germany and Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel

Yuval Gefen

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

Article Text

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Issue

Vol. 3, Iss. 2 — April - June 2021

Subject Areas

Condensed Matter Physics

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Images

Figure 1
Edge structures and excitations of the bulk filling factor $ν_{bulk} = 2 / 3$ . (a) Four bare edge modes are depicted with arrowheads indicating the direction of each mode [downstream (right movers) or upstream (left movers)]. A quasiparticle tunneling from the outermost edge mode to the inner ones turns into two quasiholes and one electron. Due to the presence of a vacuum region with $ν = 0$ , only electrons can tunnel to mode $ϕ_{4}$ , leaving the fractional charge on $ϕ_{4}$ invariant and giving rise to a $Z_{3}$ symmetry between sectors with fractional charge 0, $1 / 3$ , and $2 / 3$ . (b) Due to electron tunneling between the three inner edge modes ( $ϕ_{2}$ , $ϕ_{3}$ , and $ϕ_{4}$ ) and the bare interaction between them, the inner three edge modes are renormalized to a downstream charge mode (denoted as a black solid line) with the filling factor discontinuity $δ ν = 1 / 3$ and two upstream neutral modes (denoted as yellow dashed lines). The same tunneling process as in (a) now creates a chargeon and neutral charges $(- 1, 1)$ .
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Figure 2
A sketch of a two-step mechanism contributing to the conductance in a two-terminal setup with a quantum point contact (QPC). Potentials are tuned such that the outermost mode is fully transmitted through the QPC, while the inner ones are fully reflected. The outer charge mode supports quasiparticles (qp) and quasiholes (qh), while the inner one supports chargeons ( $c$ ) and antichargeons ( $\bar{c}$ ), all four carrying a charge of $\pm e / 3$ . Similarly, the two neutral modes support neutral excitations, neutralons ( $n$ ) or antineutralons ( $\bar{n}$ ). The size of the QPC region and the length of the arms between the contacts and the QPC are $L_{QPC}$ and $L_{arm}$ , respectively. (a) Equilibration processes involving charge tunneling and the creation of neutralons or antineutralons are $qp \to c + n$ (upper right corner) or $c \to qp + \bar{n}$ (lower left corner) represented in Eq. (10), with $ℓ_{ch-eq, 0} ≪ L_{arm}$ . (b) The excited (anti)neutralons arrive at the lower right (upper left) corner, and they decay giving rise to qp- $\bar{c}$ (qh-c) pairs. The excited qp, qh, c, or $\bar{c}$ flow to the drain D, contributing to the conductance. In the presence of $H_{n \bar{n}}$ [Eq. (13)], mixing between neutralons and antineutralons occurs over a distance $ℓ_{n \bar{n}, 0} ≪ L_{QPC}$ .
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Figure 3
SU(3) group representations. Quasiparticles described by charges $(m, n)$ [cf. Eq. (8)] form the fundamental representation of SU(3). In analogy to the quantum numbers $(I_{z}, Y)$ of isospin and hypercharge in flavor SU(3) [39], the elementary excitations (neutralons) are labeled $u$ , $d$ , and $s$ . The corresponding antineutralons form the conjugate representation; a conversion of neutralons into antineutralons is only possible if the $Z_{3}$ symmetry is broken. This can be achieved by allowing fractional quasiparticles to tunnel into and out of the innermost edge mode. More details pertaining to the $Z_{3}$ symmetry can be found in Eq. (14) and the discussion thereafter, and the SU(3) group lattice is described in Appendix pp1.
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Figure 4
A lattice for SU(3) group representations with allowed values of $m$ and $n$ for states of $e^{i m ϕ_{n_{1}} / \sqrt{2}} e^{i n ϕ_{n_{2}} / \sqrt{2}} | 0 〉$ . Here, $| 0 〉$ is the vacuum state. The points denoted as $\times$ are not allowed. The states are divided into three sublattices: the neutralon sector (denoted as red dots), the antineutralon sector (denoted as blue dots), and the vacuum sector (denoted as rectangles). Points within each sublattice are connected via the electron operators between the inner modes [written in the second line of Eq. (1)], while a sublattice is completely decoupled from the other sublattices due to Eq. (1). The presence of three different sectors manifests the $Z_{3}$ subgroup of the SU(3) group.
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Figure 5
The schematic dependence of the relevant length scales on the applied voltage $V$ (log-log plot); $ℓ_{0}$ and $ℓ_{ch-eq, 0}$ (blue dashed lines) are the voltage-independent elastic lengths. They correspond to elastic scattering within the inner modes and between the inner and outer modes, respectively. The voltage length $L_{V} \propto 1 / V$ (thin black line) acts as an infrared cutoff. $ℓ (V)$ (yellow curve) and $ℓ_{ch-eq} (V)$ (red curve) are the voltage-dependent inelastic lengths associated with inelastic scattering within the inner modes and between the inner and outer modes, respectively. Specifically, the scaling of $ℓ_{ch-eq}$ changes from $ℓ_{ch-eq} \propto 1 / V^{2}$ (low voltage) to $ℓ_{ch-eq} \propto V^{2 / 3}$ (high voltage) at the transition voltage $V = V^{*}$ . Here, the transition voltage $V^{*}$ is defined as the voltage that satisfies $L_{V} (V^{*}) = ℓ_{ch-eq, 0}$ . The inelastic length $ℓ_{in} (V)$ (at which the modes lose coherence) takes the smaller value between $ℓ$ and $ℓ_{ch-eq}$ . Our main focus in this paper is on the regime in which $L_{arm}$ is much larger than $ℓ_{ch-eq, 0}$ but smaller than $ℓ_{ch-eq}$ . The voltage $V$ is perfectly interchangeable with the temperature $T$ when $T ≫ V$ .
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Figure 6
A two-terminal setup with a quantum point contact (QPC). Source S is biased by $V_{0}$ . The two-terminal conductance is measured between S and D; the dc noise is measured at drain D. Black arrows denote the direction of tunneling charge currents at each corner.
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