Transport signatures of fractional quantum Hall binding transitions

Open Access

Transport signatures of fractional quantum Hall binding transitions

Christian Spånslätt, Ady Stern, and Alexander D. Mirlin

Phys. Rev. B 107, 245405 – Published 5 June 2023

Abstract

Certain fractional quantum Hall edges have been predicted to undergo quantum phase transitions which reduce the number of edge channels and at the same time bind electrons together. However, detailed studies of experimental signatures of such a “binding transition” remain lacking. Here, we propose quantum transport signatures with focus on the edge at filling $ν = 9 / 5$ . We demonstrate theoretically that in the regime of nonequilibrated edge transport, the bound and unbound edge phases have distinct conductance and noise characteristics. We also show that for a quantum point contact in the strong back-scattering (SBS) regime, the bound phase produces a minimum Fano factor $F_{SBS} = 3$ corresponding to three-electron tunneling, whereas single-electron tunneling is strongly suppressed at low energies. Together with recent experimental developments, our results will be useful for detecting binding transitions in the fractional quantum Hall regime.

Received 11 February 2023
Revised 22 May 2023
Accepted 23 May 2023

DOI:https://doi.org/10.1103/PhysRevB.107.245405

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. Funded by Bibsam.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Anyons Fractional quantum Hall effect Quantum transport

1-dimensional systems Strongly correlated systems

Bosonization Luttinger liquid model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Christian Spånslätt ^1,2,3,*, Ady Stern⁴, and Alexander D. Mirlin^2,3

¹Department of Microtechnology and Nanoscience (MC2), Chalmers University of Technology, S-412 96 Göteborg, Sweden
²Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany
³Institut für Theorie der Kondensierten Materie, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany
⁴Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 7610001, Israel

^*christian.spanslatt@chalmers.se

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Issue

Vol. 107, Iss. 24 — 15 June 2023

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Images

Figure 1
The FQH binding transition at filling $ν = 9 / 5$ . The free edge phase comprises two downstream propagating channels (right-pointing arrows), both with filling factor discontinuities $δ ν = 1$ and charge $q = e$ (here, the charge $q$ is the minimum charge that can tunnel across vacuum into the channel). There is also one upstream channel filling factor discontinuity $δ ν = - 1 / 5$ and charge $q = e$ (left-pointing arrow). This structure is equivalent [after a $S L (3, Z)$ basis change] to a Laughlin edge state with $δ ν = 1 / 5$ composed of charge $q = 3 e$ composite particles, plus two counterpropagating channels with $δ ν = \pm 1$ and $q = e$ . The binding transition is the phenomenon where interchannel interactions and tunneling cause the counterpropagating channels to localize (with characteristic low-temperature length scale $ξ_{0}$ ). In the bound phase, only the single composite channel remains.
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Figure 2
Schematic log-log “phase diagrams” of transport regimes in the parameter plane spanned by the temperature $T$ and the length scale $L$ at which the system is observed. The point $ξ_{0}$ on the $L$ axis in (a) and (b) is the zero-temperature localization length (63). A full line starting from this point is the temperature-dependent localization length $ξ (T)$ ; after crossing the line $T = T_{B}$ it becomes the equilibration length $ℓ_{eq} (T) \propto T^{2 - 2 Δ_{0}}$ , see Eq. (64), with $Δ_{0}$ being the scaling dimension of the null operator. Further, $ℓ_{φ} \propto 1 / T$ is the electron-electron scattering length (66). The vertical dashed line is the characteristic temperature scale $T_{B}$ for onset of localization, which is defined by the condition $ℓ_{φ} (T_{B}) = ℓ_{eq} (T_{B})$ and is given by Eq. (67). In (a) and (b), localization occurs in regime $I$ (yellow region), whereas regimes $II$ (blue region) and $III$ (red region) correspond to nonequilibrated and equilibrated transport, respectively. In (c), localization is absent. The blue and red, dashed lines depict crossovers $I \to III$ and $I \to II$ , respectively. Note that the latter crossover is only possible for $Δ_{0} < 1$ .
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Figure 3
Two-terminal setup for depicting edge transport at filling $ν = 9 / 5$ in the free [(a) and (b)] and bound [(c) and (d)] edge phases. The device length is $L$ . Two contacts at different potentials $V_{L} \neq V_{R}$ but the same temperature $T$ allows extraction of the two-terminal conductance $G$ (left figures). For contacts at the same potential $V_{L} = V_{R} = V$ but different temperatures $T_{L} \neq T_{R}$ (right figures), one can determine the two-terminal heat conductance $G^{Q}$ .
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Figure 4
(Top) Noise generation in regime of full charge equilibration but no thermal equilibration [condition (77)] in regime $II$ (cf. Table 1). With voltage bias $V_{0}$ , heat is generated by the voltage dropping in a region of size $\sim ℓ_{C}$ to the right contact: the hot spot. The dissipated power $P \sim V_{0}^{2}$ [see Eq. (B16)]. Dc noise by partitioning of the downstream (left to right direction) charge current is only generated in a region of size $\sim ℓ_{C}$ close to the left contact. This happens if heat is transported upstream (right to left) via the upstream channel, which is here only possible in the absence of thermal equilibration. (Bottom) By dissipation in the hot spot and reflection of plasmons at contacts (reflection coefficient $R$ ), the downstream and upstream modes acquire steady state temperatures $T_{+}$ respectively $T_{-}$ in the noise spot.
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Figure 5
Noise $S$ vs the bias current $I_{0}$ for different values of the scaling dimensions $δ_{\pm}$ [see Eq. (B14)] and the edge contact reflection coefficient $R$ . The calculation is done for the $ν = 9 / 5$ edge under the condition (77) for the edge equilibration length scales.
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Figure 6
Schematics of a quantum point contact (QPC) device at filling $ν = 3 / 5$ . [(a) and (b)] In the free phase, the weak back-scattering (WBS) regime favours tunneling of quasiparticles with charge $e / 5$ . For strong back-scattering (SBS), electrons with charge $e$ tunnel. (c) In the bound phase, the WBS regime is dominated by tunneling of fractional charges $3 e / 5$ . (d) In the SBS regime in the bound phase, single electron tunneling is strongly suppressed at low energies and three electron cotunneling, i.e., charge packets of $3 e$ is the most dominant process. All four types of charges can be detected in shot noise measurements.
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