Non-equilibrium bosonization of fractional quantum Hall edges

Abstract

Edge transport serves as a powerful probe of remarkable low-energy properties of fractional quantum Hall states, including the anyonic character of their excitations. Here, we develop a theory of fractional quantum Hall edges driven out of equilibrium, which is based on the Keldysh action for the bosonized chiral Luttinger liquid. With this non-equilibrium FQH bosonization framework, we first consider a single-mode Laughlin edge and analyze the full counting statistics of charge, the quasiparticle Green's functions, and tunneling transport properties through a quantum point contact, allowing for generic edge excitations. We then extend the formalism to multi-mode edges with inter-mode interactions, and explore, with focus on the $ν=4/3$ and $ν=2/3$ edges as paradigmatic examples, how interaction-induced fractionalization of anyons modifies the edge dynamics and the associated transport observables. While the full counting statistics probes the fractionalized charge of the excitations, the Green's functions and tunneling transport are governed by mutual braiding phases of fractionalized excitations and tunneling quasiparticles. We emphasize in particular the effect of interaction-induced fractionalization on the Fano factor $F$ and the differential Fano factor $F_d$, observables that can be measured experimentally. Our formalism, which provides a unified framework for non-equilibrium transport in FQH edges and Luttinger liquids, permits extracting anyonic braiding information from non-equilibrium edge-transport experiments, and paves the way to various extensions, including more involved experimental geometries and edge structures.

Figures (8)

• Figure 1: A single-mode (Laughlin) FQH edge characterized by a non-equilibrium distribution function $f(\epsilon)$, see Sec. \ref{['sec:formalism']} for description of the formalism. In this non-equilibrium state, we study the full counting statistics of charge (depicted by the yellow measurement pointer), Sec. \ref{['sec:noneq-Laughlin-FCS']}, and Green's functions (depicted by the eye symbol), Sec. \ref{['sec:Laughlin-equilibrium']}.
• Figure 2: Schematic representation of a setup for studying tunneling current and noise between non-equilibrium Laughlin edges, see Sec. \ref{['sec:tunneling']}. Two $\nu = 1/m$ Laughlin edges, characterized by distributions $f_u(\epsilon)$ and $f_d(\epsilon)$ of quasiparticles of charge $e_1^*=n_1 \nu e$ [see Eq. \ref{['n-epsilon-noneq-laughlin-ud']}], are connected by a central QPC where quasiparticles of charge $e_2^* = n_2 \nu e$ can tunnel.
• Figure 3: (a) Fano factor $F$ and (b) differential Fano factor $F_d$ for QPC tunneling transport between two Laughlin edges, see Fig. \ref{['fig:QPC']}, with $\nu=1/3$ (blue) and $\nu=1/5$ (orange), as functions of the mutual braiding phase $2\theta_{12}$ (normalized by $2\pi$). The edge $u$ is driven out of equilibrium by quasiparticle injection with charge $e_1^*=n_1\nu e$ under a bias $V_u>0$, while the edge $d$ is at zero-temperature equilibrium. The offset voltage $V_0$ is set to zero, $V_0 = 0$. The tunneling quasiparticle at the central QPC is taken to be elementary ($n_2=1$), so that the scaling dimension $\zeta=\nu n_2^2<1/2$. The solid lines represent Eq. \ref{['eq:Fano_single_bias']} in panel (a) and Eq. \ref{['eq:Fano_diff_balance']} in panel (b), with physical values (integer $n_1$) marked by stars. For the special point $2\theta_{12} = 2\pi$, the stars represent Eq. \ref{['eq:Fano_anyons']} in both panels.
• Figure 4: Setups with non-equilibrium complex FQH edges hosting two modes, either co-propagating [panel (a)] or counter-propagating [panel (b)]. In the central region II of length $L$ where the local GFs are studied (depicted by an eye symbol), there is an inter-mode interaction of strength $u$. In the regions I and III, the interaction is assumed to be zero. The interaction profile $u(x)$ is schematically shown in the bottom of panel (a) for smooth (solid line) and sharp (dotted line) interfaces I - II and II - III. The characteristic length scale on which the interactions spatially change is denoted $\Delta x$. The interfaces are described by the plasmon scattering matrices $S_{L/R}$. For the mode 1, a distribution $f_1(\epsilon)$ is created in region I. For the mode 2, a distribution $f_2(\epsilon)$ is created in the region I [panel (a)] or the region III [panel (b)]. At least one of these distributions propagating towards the central region II is non-equilibrium.
• Figure 5: Illustrations of fractionalization due to intermode interactions on the $\nu=4/3$ edge with a sharp interface. (a) Dynamics of the counting pulses \ref{['eq:totalcharge_xandt']} for FCS of charge passing across a point $x_0$ in the interacting region II during the time $\tau$. The pulse configurations are shown at three spatial locations. At $x=x_0 - 0$ (right), there is a single pulse in each mode, with quantized amplitudes $e$ and $e/3$, respectively For $-L/2 < x_0$ (middle), the pulses fractionalize into those corresponding to two eigenmodes labeled $+$ and $-$. For $x < -L/2$ (non-interacting region I, left), the eigenmode pulses separate into pulses in modes 1 and 1/3 that move with velocities $v_1$ and $v_2$. The amplitudes of these fractionalized pulses, $q_{1,\pm}^{\text{(p)}}$ and $q_{2,\pm}^{\text{(p)}}$, govern the counting phases $\delta_{1,\pm}$ and $\delta_{2,\pm}$ [see Eqs. \ref{['eq:phases_43_amplitudes']} and \ref{['eq:FCScharges_phases_correspondence']}] and the fractional charges of excitations as found by calculating the FCS in the interacting region, Eqs. \ref{['eq:43chargefluctuations_sharp']},\ref{['eq:pulsecharges43']}, and \ref{['average-charges-FCS-43']}. (b-c) Physical picture of charge fractionalization. (b) In the non-interacting region I, the mode 1 is driven out of equilibrium by dilute injection of excitations with charge $n_{1,1}e$, implying a random Poissonian train of such excitations. Upon entering the interacting region II, each excitation fractionalizes into two excitations that correspond to eigenmodes and are characterized by charges $n_{1,1}q_{1,+}^{\text{(p)}}$ and $n_{1,1}q_{1,-}^{\text{(p)}}$. (c) Similarly, if the 1/3 mode is driven out of equilibrium in region I by injection of excitations with charge $n_{1,2}e/3$, they will fractionalize into excitations with charges $n_{1,2}q_{2,+}^{\text{(p)}}$ and $n_{1,2}q_{2,-}^{\text{(p)}}$ upon entering the interacting region II. The FCS, Eqs. \ref{['eq:43chargefluctuations_sharp']}, \ref{['eq:pulsecharges43']}, and \ref{['average-charges-FCS-43']}, detects a superposition of Poissonian processes with these fractional charges.