Fractional Edge States Research Articles

Observation of fractional edge excitations in nanographene spin chains.

Fractionalization is a phenomenon in which strong interactions in a quantum system drive the emergence of excitations with quantum numbers that are absent in the building blocks. Outstanding examples are excitations with charge e/3 in the fractional quantum Hall effect1,2, solitons in one-dimensional conducting polymers3,4 and Majorana states in topological superconductors5. Fractionalization is also predicted to manifest itself in low-dimensional quantum magnets, such as one-dimensional antiferromagnetic S = 1 chains. The fundamental features of this system are gapped excitations in the bulk6 and, remarkably, S = 1/2 edge states at the chain termini7-9, leading to a four-fold degenerate ground state that reflects the underlying symmetry-protected topological order10,11. Here, we use on-surface synthesis12 to fabricate one-dimensional spin chains that contain the S = 1 polycyclic aromatic hydrocarbon triangulene as the building block. Using scanning tunnelling microscopy and spectroscopy at 4.5 K, we probe length-dependent magnetic excitations at the atomic scale in both open-ended and cyclic spin chains, and directly observe gapped spin excitations and fractional edge states therein. Exact diagonalization calculations provide conclusive evidence that the spin chains are described by the S = 1 bilinear-biquadratic Hamiltonian in the Haldane symmetry-protected topological phase. Our results open a bottom-up approach to study strongly correlated phases in purely organic materials, with the potential for the realization of measurement-based quantum computation13.

Parafermionic generalization of the topological Kondo effect

We propose and study a parafermionic generalization of the topological Kondo effect. The latter has been predicted to arise for a Coulomb-blockaded mesoscopic topological superconductor (Majorana box), where at least three normal leads are tunnel-coupled to different Majorana zero modes on the box. The Majorana states represent a quantum impurity spin that is partially screened due to cotunneling processes between leads, with a stable non-Fermi liquid ground state. Our theory studies a generalization where (i) Majorana states are replaced by topologically protected parafermionic zero modes, (ii) charging effects again define a spin-like quantum impurity on the resulting parafermion box, and (iii) normal leads are substituted by fractional edge states. In this multi-terminal problem, different fractional edge leads couple only via the parafermion box. We show that although the linear conductance tensor exhibits similar behavior as in the Majorana case, both at weak and strong coupling, our parafermionic generalization is actually not a Kondo problem but defines a rich new class of quantum impurity problems. At the strong-coupling fixed point, a current injected through a reference lead will be isotropically partitioned into outgoing currents in all other leads, together with a universal negative current scattered into the reference lead. The device can thus be operated as current extractor, where the current partitioning is noiseless at the fixed point. We describe a fractional quantum Hall setup proximitized by superconductors and ferromagnets, which could allow for an experimental realization in the near future.

Measurement and control of a Coulomb-blockaded parafermion box

Parafermionic zero modes are fractional topologically protected quasiparticles expected to arise in various platforms. We show that Coulomb charging effects define a parafermion box with unique access options via fractional edge states and/or quantum antidots. Basic protocols for the detection, manipulation, and control of parafermionic quantum states are formulated. With those tools, one may directly observe the dimension of the zero-mode Hilbert space, prove the degeneracy of this space, and perform on-demand digital operations satisfying a parafermionic algebra.

Kramers pairs of Majorana fermions and parafermions in fractional topological insulators

We propose a scheme based on topological insulators to generate Kramers pairs of Majorana fermions or parafermions in the complete absence of magnetic fields. Our setup consists of two topological insulators whose edge states are brought close to an s-wave superconductor. The resulting proximity effect leads to an interplay between a non-local crossed Andreev pairing, which is dominant in the strong electron-electron interaction regime, and usual superconducting pairing, which is dominant at large separation between the two topological insulator edges. As a result, there are zero-energy bound states localized at interfaces between spatial regions dominated by the two different types of pairing. Due to the preserved time-reversal symmetry, the bound states come in Kramers pairs. If the topological insulators carry fractional edge states, the zero-energy bound states are parafermions, otherwise, they are Majorana fermions.

Equilibration between edge states in the fractional quantum Hall effect regime at high imbalances

We experimentally study equilibration between edge states, copropagating at the edge of the fractional quantum Hall liquid, at high initial imbalances. We find an anomalous increase of the conductance between the fractional edge states at the filling factor $\ensuremath{\nu}=2∕5$ in comparison with the expected one for the model of independent edge states. We conclude that the model of independent fractional edge states is not suitable to describe the experimental situation at $\ensuremath{\nu}=2∕5$.

Separately contacted edge states in the fractional quantum Hall regime

Combining a quasi-Corbino geometry with the cross-gate technique, we developed a sample geometry that allows us to separately contact edge states in the integer and fractional quantum Hall regime. The energy barriers between edge states at integer filling factors give rise to pronounced steps in the I– V characteristics that directly reflect the gap structure of the reconstructed edge. The traces can readily be interpreted in terms of the Landauer–Büttiker formalism and the compressible/incompressible liquid picture. At a temperature of 30 mK and for the fractional filling factor combinations 1 : 2 3 and 1 : 1 3 , the slopes of the obtained I– V traces at currents up to 50 nA are all in very good agreement with the predictions of the Landauer–Büttiker formalism, assuming edge states of fractional charge 1 3 . From the nonlinearity of the I– V characteristics we estimate the energy barrier between fractional edge states of charge 1 3 to be of the order of 40 μeV .

Non-equilibrium transport along an edge of variable slope in the fractional quantum Hall regime

We have performed an experimental investigation of equilibration between edge states in the fractional quantum Hall regime. We have developed a technique to alter the electrostatic potential at the edge continuously and controllably from a smooth to a sharp potential gradient, using a set of gates. A non-equilibrium current distribution in the edge states is created which then travels along the edge to a detector. The scattering between the various fractional edge states is investigated, and a method developed to calculate scattering probabilities between pairs of edge states, revealing independent 1 5 , 1 3 and 2 5 edge states, for a bulk filling factor of 2/3. Equilibration is found to depend strongly on the potential slope.

Composite fermion theory, edge currents and the fractional quantum Hall effect

We present a mean field theory of composite fermion edge channel transport in the fractional and integer quantum Hall regimes. An expression relating the electrochemical potentials of composite fermions at the edges of a sample to those of the corresponding electrons is obtained, and a plausible form is assumed for the composite fermion Landau level energies near the edges. The theory yields the observed fractionally quantized Hall conductances and also explains other experimental results. We also discuss briefly some experiments that are relevant to the question as to whether fractional edge states in real devices should be described as Fermi or Luttinger liquids.

Understanding the dynamics of fractional edge states with composite fermions

Fractional edge states can be viewed as integer edge states of composite fermions. We exploit this to discuss the conductance of the fractional quantized Hall states and the velocity of edge magnetoplasmons.

Reflection of edge states in the fractional quantum Hall regime

We report on magnetoresistance experiments on two-dimensional electron gas (2DEG) split-gate devices fabricated on GaAs Al xGa 1−x As heterostructures in the vicinity of Landau level filling factor ν b = 1 2 . The filling factor in the devices is reduced by the application of a negative gate bias. As the filling factor in the constriction ν c decreases, plateaux are observed due to the successive reflection of fractional edge states. Plateau resistances are in good quantitative agreement with calculations based on fractional quantum Hall edge state transport in terms of the Landauer-Büttiker formalism, R= h e 2 ( 1 ν c − 1 ν b ). Plateaux ascribed to filling factors in the series p (2p+1) : 1 3 , 2 5 , 3 7 and 4 9 in the constriction are seen when transport in the unpatterned 2DEG is edge-like or bulk-like. The number of plateaux observed is dependent on the device geometry and is in quantitative agreement with the recent theory of Chklovskii.