Parafermion modes are non-Abelian anyons which were introduced as
ZN generalizations of
Z2 Majorana states. In particular,
Z3 parafermions can be used to produce Fibonacci anyons, laying a path towards universal topological quantum computation. Due to their fractional nature, much of the theoretical work on
Z3 parafermions has relied on bosonization methods or parafermionic quasiparticles. In this paper, we introduce a representation of
Z3 parafermions in terms of purely fermionic models. We establish the equivalency of a family of lattice fermionic models written in the basis of the $t\ensuremath{-}J$ model with a Kitaev-like chain supporting free
Z3 parafermionic modes at its ends. By using density matrix renormalization group calculations, we are able to characterize the topological phase transition and study the effect of local operators (doping and magnetic fields) on the spatial localization of the parafermionic modes and their stability. Moreover, we discuss the necessary ingredients towards realizing
Z3 parafermions in strongly interacting electronic systems.
The emergence of non-Abelian anyons from large collections of interacting elementary particles is a conceptually beautiful phenomenon with important ramifications for fault-tolerant quantum computing. Over the last few decades the field has evolved from a highly theoretical subject to an active experimental area, particularly following proposals for trapping non-Abelian anyons in 'engineered' structures built from well-understood components. In this short overview we briefly tour the impressive progress that has taken place in the quest for the simplest type of non-Abelian anyon—defects binding Majorana zero modes—and then turn to similar strategies for pursuing more exotic excitations. Specifically, we describe how interfacing simple quantum Hall systems with conventional superconductors yields 'parafermionic' generalizations of Majorana modes and even Fibonacci anyons—the latter enabling fully fault tolerant universal quantum computation. We structure our treatment in a manner that unifies these topics in a coherent way. The ideas synthesized here spotlight largely uncharted experimental territory in the field of quantum Hall physics that appears ripe for discovery.
Abstract Fibonacci anyon, an exotic quasi-particle excitation, plays a pivotal role in realization of a quantum computer. Starting from a SU (2)4 topological phase, in this paper we demonstrate a way to construct a Fibonacci topological phase which has only one non-trivial excitation described by the Fibonacci anyon. We show that arrays of anyonic chains created by excitations of the SU(2)4 phase leads to the Fibonacci phase. We further demonstrate that our theoretical propositions can be extended to other topological phases.
Can we find a physical intuitive language to represent the complex mathematical properties of anyon models? Is there a hidden organization in the space of anyon models? Can we systematically construct complex anyon models from other, more simple, anyon models? This work is an attempt to understand and explore the anatomy of the space of anyon models. I develop a construction to systematically generate and interconnect anyon models. The construction is formulated in a very physical, visual, and intuitive manner. The complex mathematical properties of anyon models, usually expressed in the complex language of braided modular tensor categories, are described here using familiar simple physical concepts like bosons in lattices, Fock states, and tunneling Hamiltonians. The construction formalism is conceptually simple and appealing. As in a Lego construction, a set of building block anyon models are assembled to construct towers of other, more complex, anyon models. A principle of assembly is established that dictates how to articulate the building blocks, setting out the global blueprint for the constructed anyon model. By assembling sets of the simple Abelian
Zn anyon models, the construction succeeds in generating towers of well known nontrivial anyon models. Moreover, novel anyon models (nontabulated, to my knowledge) arise. Furthermore, by applying the construction principle to other building blocks, such as Ising or Fibonacci, further series of anyon models are generated. In an intriguing connection, the boson-lattice construction unveils an alluring duality between anyon models and curved space geometries. Bosons induce curvature in the lattice geometry. I disclose a geography in the space of anyon models, where one is engendered from another by deforming the geometry of space. Moreover, the potential of the boson-lattice construction can be further explored via the quantum simulation of the corresponding boson-lattice systems in the versatile accessible playground of atomic systems.
Irrational CFTs in 1+1d with a discrete spectrum and no conserved currents other than the stress-tensor are expected to be generic, unsolvable by standard methods, and hard to construct explicitly. We introduce a lattice model that realizes a candidate for such a CFT as a conformal phase of matter without fine-tuning. The model is constructed by coupling N≥3 N ≥ 3 golden anyon chains together, preserving N N copies of the Fibonacci non-invertible symmetry. We use the MPS/DMRG approach to study this model numerically, which allows us to calculate the corresponding conformal data, obtaining hints of its irrationality. Along the way, we characterize the phase diagram for N=2 N = 2 coupled chains where we identify a weakly first-order phase transition as well as critical points that we are able to identify with known rational CFTs, except for one case. We also provide an extensive list of rational CFTs with 1<c<2.1 1 < c < 2.1 .
Parafermions are anyons with the potential for realizing nonlocal qubits that are resilient to local perturbations. Compared with Majorana zero modes, braiding of parafermions implements an extended set of topologically protected quantum gates. This, however, comes at the price that parafermionic zero modes cannot be realized in the absence of strong interactions, posing a challenge for their theoretical depiction. In this paper, we construct a simple lattice model for interacting spinful electrons with parafermionic zero-energy modes. The explicit microscopic nature of the considered model highlights realization avenues for these exotic excitations in recently fabricated quantum dot arrays. By density matrix renormalization group calculations, we identify a broad range of parameters, with well-localized zero modes, whose parafermionic nature is substantiated by their unique $8\ensuremath{\pi}$ periodic Josephson spectrum.