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Register a new userOn the Interpretation of Thermal Conductance of the nu=5/2 Edge
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Steven H. Simon
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Recent experiments [Banerjee et al, arXiv:1710.00492] have measured thermal conductance of the nu=5/2 edge in a GaAs electron gas and found it to be quantized as K approx 5/2 (in appropriate dimensionless units). This result is unexpected, as prior numerical work predicts that the nu=5/2 state should be the Anti-Pfaffian phase of matter, which should have quantized K=3/2. The purpose of this paper is to propose a possible solution to this conflict: if the Majorana edge mode of the Anti-Pfaffian does not thermally equilibrate with the other edge modes, then K=5/2 is expected. I briefly discuss a possible reason for this nonequilibration, and what should be examined further to determine if this is the case.
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We address the interpretation proposed in [S. H. Simon, arXiv:1801.09687] of the thermal conductance data from [M. Banerjee et al., arXiv:1710.00492]. We show that the interpretation is inconsistent with experimental data and the sample structure. In particular, the paper misses the momentum mismatch between contra-propagating modes. Contrary to the claim of the paper, low energy tunneling involves a large momentum change. We consider only the small Majorana velocity mechanism [S. H. Simon, arXiv:1801.09687]. Other mechanisms, interpretations of the experiment, and their difficulties are beyond the scope of this Comment.
We study quasiparticle tunneling between the edges of a non-Abelian topological state. The simplest examples are a p+ip superconductor and the Moore-Read Pfaffian non-Abelian fractional quantum Hall state; the latter state may have been observed at Landau-level filling fraction nu=5/2. Formulating the problem is conceptually and technically non-trivial: edge quasiparticle correlation functions are elements of a vector space, and transform into each other as the quasiparticle coordinates are braided. We show in general how to resolve this difficulty and uniquely define the quasiparticle tunneling Hamiltonian. The tunneling operators in the simplest examples can then be rewritten in terms of a free boson. One key consequence of this bosonization is an emergent spin-1/2 degree of freedom. We show that vortex tunneling across a p+ip superconductor is equivalent to the single-channel Kondo problem, while quasiparticle tunneling across the Moore-Read state is analogous to the two-channel Kondo effect. Temperature and voltage dependences of the tunneling conductivity are given in the low- and high-temperature limits.
When a gas of electrons is confined to two dimensions, application of a strong magnetic field may lead to startling phenomena such as emergence of electron pairing. According to a theory this manifests itself as appearance of the fractional quantum Hall effect with a quantized conductivity at an unusual half-integer nu=5/2 Landau level filling. Here we show that similar electron pairing may occur in quantum dots where the gas of electrons is trapped by external electric potentials into small quantum Hall droplets. However, we also find theoretical and experimental evidence that, depending on the shape of the external potential, the paired electron state can break down, which leads to a fragmentation of charge and spin densities into incompressible domains. The fragmentation of the quantum Hall states could be an issue in the proposed experiments that aim to probe for non-abelian quasi-particle characteristics of the nu=5/2 quantum Hall state.
Recent schemes for experimentally probing non-abelian statistics in the quantum Hall effect are based on geometries where current-carrying quasiparticles flow along edges that encircle bulk quasiparticles, which are localized. Here we consider one such scheme, the Fabry-Perot interferometer, and analyze how its interference patterns are affected by a coupling that allows tunneling of neutral Majorana fermions between the bulk and edge. While at weak coupling this tunneling degrades the interference signal, we find that at strong coupling, the bulk quasiparticle becomes essentially absorbed by the edge and the intereference signal is fully restored.
Resistively detected nuclear magnetic resonance is used to measure the Knight shift of the As nuclei and determine the electron spin polarization of the fractional quantum Hall states of the second Landau level. We show that the 5/2 state is fully polarized within experimental error, thus confirming a fundamental assumption of the Moore-Read theory. We measure the electron heating under radio frequency excitation, and show that we are able to detect NMR at electron temperatures down to 30 mK.
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