We propose a method to extract the mutual exchange statistics of the anyonic excitations of a general Abelian fractional quantum Hall state, by comparing the tunneling characteristics of a quantum point contact in two different experimental conditions. In the first, the tunneling current between two edges at different chemical potentials is measured. In the second, one of these edges is strongly diluted by an earlier point contact. We describe the case of the dilute beam in terms of a time-domain interferometer between the anyons flowing along the edge and quasiparticle-quasihole excitations created at the tunneling quantum point contact. In both cases, temperature is kept large, such that the measured current is given to linear response. Remarkably, our proposal does not require the measurement of current correlations, and allows us to carefully separate effects of the fractional charge and statistics from effects of intra- and interedge interactions.
On the edge of certain fractional quantum Hall states, e.g., at 2/3 and 5/2 filling, a local fractional excitation, occurring by anyon tunneling at a quantum point contact, is further fractionalized into counterpropagating charge and neutral (Abelian or non-Abelian) anyonic excitations. We propose a scheme to detect the braiding statistics of the charge and neutral anyons separately. It is the injection of a dilute beam of a target (charge or neutral) anyon to a Fabry-Perot interferometer. The monodromy of the target anyon is obtained by comparing the amplitude and phase of the interference current with a reference signal of the same setup but without the injection. Our proposal relies on braiding between anyons on the edge, and applies even in the presence of bulk-edge couplings.
The discovery of the fractional quantum Hall effect stimulated the investigation of anyons, particles with fractional statistics which are neither bosons nor fermions. This thesis focuses on the study of quantum Hall states which may support non-Abelian anyons. We first address the validity of assumptions used in the numerical study of such states, and then proceed with analyzing different experiments which can detect non-Abelian fractional statistics. We quantitatively analyze the two-point contact interferometer experiment, which is hoped to display clear-cut, direct evidence of non-Abelian fractional statistics. We calculate the temperature and voltage dependence of the interference experiment outcome, and the signal attenuation due to finite temperature loss of coherence. We then analyze the edge theory of a family of non-Abelian quantum Hall states in the second Landau level, and examine the tunneling between these states and a quantum dot. This tunneling problem maps onto the multi-channel Kondo problem, and will allow distinguishing between different quantum Hall states. Finally, we use the same theoretical methods for analyzing Sagnac interference in the conductance of a carbon nanotube loop, a one-dimensional system.
An upstream neutral mode can influence quasiparticle tunneling even when electric charge propagates only downstream. We propose a three-path fractional quantum Hall interferometer that probes this directional structure through the flux dependence of average terminal currents. Interference between direct and two-step tunneling first contributes at cubic order in the tunneling amplitudes of three weak quantum point contacts. In the local theory, equal drain voltages separate the downstream and upstream responses: for a purely downstream tunneling excitation, the cubic fundamental Aharonov-Bohm harmonic vanishes at one drain, while the other remains bright. Averaging measurements with exchanged unequal drain voltages extends this separation to a configuration in which all contacts are biased. It requires independent calibration of the electrostatic phase shift and fixed or known tunneling magnitudes. For a selected Abelian excitation with two-point correlation exponent
0<Δℓ<1, the common low-temperature bias power and directional amplitude ratio determine its scaling dimension and principal exchange angle. We also show that a weak static density interaction spanning two tunneling points can activate the dark harmonic without adding an upstream mode. In the high-bias static limit, the first-order Laughlin activated amplitude scales as
E2ν−1, where
E is the source-drain bias energy and
ν is the filling factor. In the weak localized single-mode infrared regime, the bright harmonic retains its leading scaling and can serve as a reference in the same device. The calculation assumes one coherent tunneling species and common-temperature edge correlations with unresolved propagation times.
Two-particle interferometry is an important tool for extracting the exchange statistics of quantum particles. We theoretically investigate the prospects of such interferometry to probe the statistics of pointlike anyonic excitations injected in a Hong-Ou-Mandel (HOM) setup based on a quantum point contact device in the fractional quantum Hall regime. We compute the standard HOM ratio, i.e., the ratio of tunneling noises for two- and one-particle injections, and find that for pointlike anyons it only depends on the temperature and the anyon scaling dimension. Importantly, the latter is not necessarily related to the exchange phase. In fact, we establish that the HOM ratio does not reveal the exchange phase of the injected anyons: For injection-time delays that are small compared to the thermal timescale, we find that the exchange phase accumulated due to time-domain braiding between injected and thermally activated anyons is erased due to two mutually canceling subprocesses. In contrast, for time delays large compared to the thermal time, only a single subprocess contributes to the braiding, but the accumulated phase is canceled in the HOM ratio. These findings suggest caution when interpreting HOM interferometry experiments with anyons, and approaches beyond the standard HOM ratio are thus necessary to extract anyonic statistics with two-particle interferometry experiments.
We consider hierarchical quantum Hall edge states with
N modes and a spatially local quantum point contact (QPC). In general, the field of an injected anyon does not directly acquire the universal statistical phase
θ. Short-range inter-edge interactions split the universal anyon charge and phase into
N fractionalized charges associated with nonuniversal phases
πδm. In contrast, their sum
δ=∑m=1Nδm, which defines the local scaling dimension at the QPC, remains protected and is tied to the statistical angle through
πδ=θ. If the injected anyon is of the same species as the one dominating backscattering at the QPC, time-domain braiding with phase
θ is recovered either in the absence of inter-edge interactions with equal mode velocities, or by performing a spatially local anyon injection at the QPC. We then exploit a more robust \emph{local} anyonic time-exchange (ATE) link between anyons and quasiholes at the QPC, which is also a necessary ingredient for realizing such braiding. This allows us to propose minimal single-QPC protocols that do not rely on diluted anyon sources and that disentangle the role of
θ as a genuine statistical phase from that as a scaling dimension. From the ATE link we derive two novel nonequilibrium fluctuation--dissipation relations (FDRs) that isolate
θ. They relate the DC backscattering noise either to an integral over the DC current or to the phase shift of the AC current with respect to an applied AC voltage (i.e., the phase of the admittance), accessible down to low frequencies. For thermalized edges, we show that in the quantum regime this admittance phase directly yields
θ whenever
δ>1/2.
In fractional quantum Hall systems, quasiparticles of fractional charge can tunnel between the edges at a quantum point contact. Such tunneling (or backscattering) processes contribute to charge transport and provide information on both the charge and statistics of the quasiparticles involved. Here, we study quasiparticle tunneling in the Moore-Read state, in which quasiparticles of charges
e/4 (non-Abelian) and
e/2 (Abelian) may coexist and both contribute to edge transport. On a disk geometry, we calculate the matrix elements for
e/2 and
e/4 quasiholes to tunnel through the bulk of the Moore-Read state, in an attempt to understand their relative importance. We find that the tunneling amplitude for charge
e/2 quasihole is exponentially smaller than that for charge
e/4 quasihole, and the ratio between them can be (partially) attributed to their charge difference. We find that including long-range Coulomb interaction only has a weak effect on the ratio. We discuss briefly the relevance of these results to recent tunneling and interferometry experiments at filling factor $\ensuremath{\nu}=5/2$.