Topological states of matter are characterized by topological invariants, which are physical quantities whose values are quantized and do not depend on the details of the system (such as its shape, size and impurities). Of these quantities, the easiest to probe is the electrical Hall conductance, and fractional values (in units of e/h, where e is the electronic charge and h is the Planck constant) of this quantity attest to topologically ordered states, which carry quasiparticles with fractional charge and anyonic statistics. Another topological invariant is the thermal Hall conductance, which is harder to measure. For the quantized thermal Hall conductance, a fractional value in units of κ (κ = πk/(3h), where k is the Boltzmann constant) proves that the state of matter is non-Abelian. Such non-Abelian states lead to ground-state degeneracy and perform topological unitary transformations when braided, which can be useful for topological quantum computation. Here we report measurements of the thermal Hall conductance of several quantum Hall states in the first excited Landau level and find that the thermal Hall conductance of the 5/2 state is compatible with a half-integer value of 2.5κ, demonstrating its non-Abelian nature.

The quantum of thermal conductance of ballistic (collisionless) one-dimensional channels is a unique fundamental constant. Although the quantization of the electrical conductance of one-dimensional ballistic conductors has long been experimentally established, demonstrating the quantization of thermal conductance has been challenging as it necessitated an accurate measurement of very small temperature increase. It has been accomplished for weakly interacting systems of phonons, photons and electronic Fermi liquids; however, it should theoretically also hold in strongly interacting systems, such as those in which the fractional quantum Hall effect is observed. This effect describes the fractionalization of electrons into anyons and chargeless quasiparticles, which in some cases can be Majorana fermions. Because the bulk is incompressible in the fractional quantum Hall regime, it is not expected to contribute substantially to the thermal conductance, which is instead determined by chiral, one-dimensional edge modes. The thermal conductance thus reflects the topological properties of the fractional quantum Hall electronic system, to which measurements of the electrical conductance give no access. Here we report measurements of thermal conductance in particle-like (Laughlin-Jain series) states and the more complex (and less studied) hole-like states in a high-mobility two-dimensional electron gas in GaAs-AlGaAs heterostructures. Hole-like states, which have fractional Landau-level fillings of 1/2 to 1, support downstream charged modes as well as upstream neutral modes, and are expected to have a thermal conductance that is determined by the net chirality of all of their downstream and upstream edge modes. Our results establish the universality of the quantization of thermal conductance for fractionally charged and neutral modes. Measurements of anyonic heat flow provide access to information that is not easily accessible from measurements of conductance.

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We investigate rectification of a low-frequency ac bias in Y-junctions of one-channel Luttinger liquid wires with repulsive electron interaction. Rectification emerges due to three scatterers in the wires. We find that it is possible to achieve a higher rectification current in a Y-junction than in a single wire with an asymmetric scatterer at the same interaction strength and voltage bias. The rectification effect is the strongest in the absence of the time-reversal symmetry. In that case, the maximal rectification current can be comparable with the total current $\sim e^2V/h$ even for low voltages, weak scatterers and modest interaction strength. In a certain range of low voltages, the rectification current can grow as the voltage decreases. This leads to a bump in the $I$-$V$ curve.Comment: 14 pages, 4 figures; The latest versio

We study transport through an electronic Mach-Zehnder interferometer recently devised at the Weizmann Institute. We show that this device can be used to probe the statistics of quasiparticles in the fractional quantum Hall regime. We calculate the tunneling current through the interferometer as the function of the AharonovBohm flux, temperature, and voltage bias, and demonstrate that its flux-dependent component is strongly sensitive to the statistics of tunneling quasiparticles. More specifically, the flux-dependent and flux-independent contributions to the current are related by a power law, the exponent being a function of the quasiparticle statistics.

Numerical results suggest that the quantum Hall effect at ν = 5/2 is described by the Pfaffian or anti-Pfaffian state in the absence of disorder and Landau level mixing. Those states are incompatible with the observed transport properties of GaAs heterostructures, where disorder and Landau level mixing are strong. We show that the recent proposal of a PH-Pfaffian topological order by Son is consistent with all experiments. The absence of the particle-hole symmetry at ν = 5/2 is not an obstacle to the existence of the PH-Pfaffian order since the order is robust to symmetry breaking.One of the most interesting features of topological insulators and superconductors is their surface behavior. A great variety of gapless and topologically-ordered gapped surface states have been proposed [1]. Such states are anomalous, that is, they can only exist on the surface of a 3D bulk system and not in a stand-alone film. Finding experimental realizations of exotic surface states proved difficult and most of them have remained theoretical proposals. Thus, it came as a surprise when Son [2] argued that one such exotic state [3], made of Dirac composite fermions with the particle-hole symmetry (PHS), has long been observed experimentally in a two-dimensional system: the electron gas in the quantum Hall effect (QHE) with the filling factor ν = 1/2.At first sight, Son's idea violates the fermion doubling theorem [4]. However, the theorem does not apply to interacting systems such as the one Son considered. Besides, in contrast to the conditions of the doubling theorem, the action of PHS is nonlocal in QHE since it involves filling a Landau level. Interestingly, the picture of composite Dirac fermions sheds light on the geometrical resonance experiments [5] which the classic theory [6] of the 1/2 state could not explain. In this paper we show that a closely related idea [2] provides a natural explanation of the observed phenomenology on the enigmatic QHE plateau at ν = 5/2 in GaAs.Cooper pairing of Dirac composite fermions in the s channel results in a fractional QHE state dubbed PHPfaffian [2,7], where "PH" stands for "particle-hole". We argue that the PH-Pfaffian topological order is present on the QHE plateau at ν = 5/2. This might seem unlikely because the particle-hole symmetry is violated by the Landau level mixing (LLM) in the observed states of the second Landau level in GaAs. Besides, numerics [11][12][13][14][15][16] supports the Pfaffian [8] and anti-Pfaffian [9,10] states at ν = 5/2 even in the presence of PHS. At the same time, the existing numerical work, which always neglects strong disorder and typically neglects strong LLM, is not yet in the position to explain the physics of the 5/2 state, as is evidenced by a large discrepancy between numerical and experimental energy gaps [17]. Note that a recent attempt to incorporate LLM [16] into simulations led to the manifestly wrong conclusion that the 5/2 state does not exist at realistic LLM. Indeed, as discussed in Ref. [16], the existing perturbative methods are justifi...

Fractionally charged quasiparticles in the quantum Hall state with a filling factor 5=2 are expected to obey non-Abelian statistics. We demonstrate that their statistics can be probed by transport measurements in an electronic Mach-Zehnder interferometer. The tunneling current through the interferometer exhibits a characteristic dependence on the magnetic flux and a nonanalytic dependence on the tunneling amplitudes which can be controlled by gate voltages. DOI: 10.1103/PhysRevLett.97.186803 PACS numbers: 73.43.Jn, 73.43.Cd, 73.43.Fj One of the central features of the quantum Hall effect (QHE) is the fractional charge and statistics of quasiparticles. The quantum state of bosons or fermions does not change when one particle makes a full turn around another. On the other hand, Laughlin quasiparticles pick up nontrivial phases when they encircle each other. Non-Abelian statistics predicted in some QHE systems [1,2] is even more interesting: The state vector changes its direction in the Hilbert space after a particle makes a full circle.Shot noise experiments [3] allowed the observation of fractional charges in QHE liquids. Probing fractional statistics is more difficult. It was argued that the mutual statistics of nonidentical quasiparticles was detected in a recent experiment [4] by Camino et al.; however, the interpretation of the experimental results remains controversial [5,6]. There are several theoretical proposals for observing the statistics of identical Abelian quasiparticles [7][8][9][10] but none of them has been realized experimentally.Detecting non-Abelian anyons is of special interest due to their promise for fault-tolerant quantum computation [11]. One approach for their observation and probing their statistics is based on current noise in complex geometries [12,13]. A simpler proposal involves current through an Aharonov-Bohm interferometer with trapped quasiparticles [14,15]. This method should work if the number of trapped quasiparticles does not fluctuate on the measurement time scale [9]. Such a condition might be difficult to satisfy in non-Abelian systems, where the excitation gap is relatively low [16].We suggest another method, which is free from this limitation. It uses the electronic Mach-Zehnder interferometer recently designed at the Weizmann Institute [17]; see Fig. 1(a). We consider the 5=2 QHE state and show that the tunneling current I through the interferometer contains signatures of non-Abelian statistics. The current (8) is a periodic function of the magnetic flux through the interferometer with period 0 hc=e, but, in contrast with the Abelian case [10], it is not a simple sine wave. Note the nonanalytic dependence on the interedge tunneling amplitudes ÿ 1 and ÿ 2 at the quantum point contacts (constrictions) QPC1 and QPC2. In the limit of ÿ 2 ÿ 1 , the formula for the current assumes a sinusoidal form, I I 0 I cos 2 = 0 const , where the fluxdependent and flux-independent terms are related by the scaling law: The Letter is organized as follows. First, we briefly discuss the ...

We show how shot noise in an electronic Mach-Zehnder interferometer in the fractional quantum Hall regime probes the charge and statistics of quantum Hall quasiparticles. The dependence of the noise on the magnetic flux through the interferometer allows for a simple way to distinguish Abelian from non-Abelian quasiparticle statistics. In the Abelian case, the Fano factor ͑in units of the electron charge͒ is always lower than unity. In the non-Abelian case, the maximal Fano factor as a function of the magnetic flux exceeds 1.