Parafermions or Fibonacci anyons leading to universal quantum computing, require strongly interacting systems. A leading contender is the fractional quantum Hall effect, where helical channels can arise from counter- propagating chiral modes. These modes have been considered weakly interacting. However, experiments on transport in helical channels in the fractional quantum Hall effect at a 2/3 filling shows current passing through helical channels on the boundary between polarized and unpolarized quantum Hall liquids nine-fold smaller than expected. This current can increase three-fold when nuclei near the boundary are spin polarized. We develop a microscopic theory of strongly interacting helical states and show that emerging helical Luttinger liquid manifests itself as unequally populated charge, spin and neutral modes in polarized and unpolarized fractional quantum Hall liquids. We show that at strong coupling counter-propagating modes of opposite spin polarization emerge at the sample edges, providing a viable path for generating proximity topological superconductivity and parafermions. Current, calculated in strongly interacting picture is in agreement with the experimental data.
Search for Non-Abelions - particles whose exchange transforms the quantum state of a system noncommutatively- is driven both by the quest for deeper understanding of nature and prospects for universal topological quantum computation. Examples of non-Abelions are Majorana and parafermion zero modes. However, physical systems that can host these exotic excitations are rare and hard to realize in experiments. To demonstrate new examples of such experimentally feasible systems, we will describe the domain walls formed in spin transitions in the integer and fractional quantum Hall effects between spin- polarized and spin-unpolarized phases. We show that superconducting proximity coupling to domain walls between two topologically distinct fractional quantum Hall liquids leads to emergence of parafermions. While Majorana zero modes form a double degenerate state, parafermions emerge as a six-fold degenerate state. Emergence of parafermions in electron and hole systems is discussed.
I will introduce a new platform based on spin transitions in the
fractional quantum Hall effect regime where parafermions - higher
order non-abelian excitations - can be realized. Local (gate)
control of spin transition allows formation of isolated domain
walls, which consist of counter-propagating edge states of opposite
polarization with fractional charge excitations. When
superconductivity is induced into such a domain wall from
superconducting contacts via proximity effect, parafermions are
expected to be formed at the domain wall boundaries. In a
multi-gate device a re-configurable network of domain walls can be
formed allowing creation, braiding, manipulation and fusion of
parafermions. In respect to the quantum computing application
parafermons are more computationally intense than Majoranas and are
a building block for Fibonacci fermions, even high order
non-Abelian particles that can perform universal gate operations
within the topologically protected subspace.
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