Exotic topologically protected zero modes with parafermionic statistics (also called fractionalized Majorana modes) have been proposed to emerge in devices fabricated from a fractional quantum Hall system and a superconductor. The fractionalized statistics of these modes takes them an important step beyond the simplest non-Abelian anyons, Majorana fermions. Building on recent advances towards the realization of fractional quantum Hall states of bosonic ultracold atoms, we propose a realization of parafermions in a system consisting of Bose-Einstein-condensate trenches within a bosonic fractional quantum Hall state. We show that parafermionic zero modes emerge at the end points of the trenches and give rise to a topologically protected degeneracy. We also discuss methods for preparing and detecting these modes.
Exotic topologically protected zero modes with parafermionic statistics (also called fractionalized Majorana modes) have been proposed to emerge in devices fabricated from a fractional quantum Hall system and a superconductor. The fractionalized statistics of these modes takes them an important step beyond the simplest non-Abelian anyons, Majorana fermions. Building on recent advances towards the realization of fractional quantum Hall states of bosonic ultracold atoms, we propose a realization of parafermions in a system consisting of Bose-Einstein-condensate trenches within a bosonic fractional quantum Hall state. We show that parafermionic zero modes emerge at the end points of the trenches and give rise to a topologically protected degeneracy. We also discuss methods for preparing and detecting these modes.
We study the topological classification of parafermionic chains in the presence of a modified time-reversal symmetry that satisfies T2=1. Such chains can be realized in one-dimensional structures embedded in fractionalized two-dimensional states of matter, e.g., at the edges of a fractional quantum spin Hall system, where counterpropagating modes may be gapped either by backscattering or by coupling to a superconductor. In the absence of any additional symmetries, a chain of Zm parafermions can belong to one of several distinct phases. We find that when the modified time-reversal symmetry is imposed, the classification becomes richer. If m is odd, each of the phases splits into two subclasses. We identify the symmetry-protected phase as a Haldane phase that carries a Kramers doublet at each end. When m is even, each phase splits into four subclasses. The origin of this split is in the emergent Majorana fermions associated with even values of m. We demonstrate the appearance of such emergent Majorana zero modes in a system where the constituent particles are either fermions or bosons.
Obtaining non-abelian anyons is one of the primary goals within the field of topological states of matter.The simplest of these anyons, Majorana bound states (MBSs) [1], have been proposed as candidates for implementing topological quantum computation relying on its non-abelian statistics [2][3][4][5].Parafermions bound states (PBSs) can be regarded as MBSs [6,7].PBS have richer non-abelian exchange statistics compared to MBS and would thus offer advantages for quantum computation.In contrast to MBS, PBS usually require strong interactions between electrons and has been proposed to exist, for instance, in fractional quantum Hall insulators with induced superconductivity [8-10].Tight-binding models have also been suggested to host PBSs [11][12][13], although in some cases they are nontopological [14].Recently, Kondo devices have also been proposed to host parafermionic states [15,16].Nonetheless, in all setups, many-body interactions are necessary for the existence of PBS.In this thesis, we are concerned with the properties of PBSs that can be used in experimental setups.While most of the work developed here concerns fermionic models [17,18], we also investigate a continuous model of fractional quantum Hall insulators with induced superconductivity [19].This work is, therefore, divided into two parts.In the first part, we examine the fermionic models using Density Matrix Renormalization Group (DMRG) [20][21][22] as well as some properties of Fock-parafermion [11].We start with a proposal on how to use Quantum dots (QDs) to probe the existence of Z 4 parafermions and differentiate it from 2 MBSs (2xZ 2 ).Then, we introduce two fermionic models that host Z 3 parafermions.We show the topological equivalence of the models and their properties.In the second part, we use a combination of analytical (semiclassical instanton approximation) and numerical (quantum Monte Carlo simulations) techniques to determine the effective parafermion Hamiltonian and its ground state splitting in a fractional quantum Hall insulators with induced superconductivity.
Parafermions are elusive fractional excitations potentially emerging in fractional quantum Hallsuperconductor junctions and represent one of the major milestones in fractional quantum matter.However, generic models of parafermions are not analytically solvable, and understanding their topological modes is a bigger challenge than conventional Majorana modes.Here, by using a combination of tensor network and kernel polynomial techniques, we demonstrate the emergence of zero modes and finite energy excitations in many-body parafermion chains.We show the appearance of zero-energy modes in the many-body spectral function at the edge of a topological parafermion chain, their relation with the topological degeneracy of the system, and we compare their physics with the Majorana bound states of topological superconductors.We demonstrate the robustness of parafermion topological modes with respect to a variety of perturbations, and we show how weakly coupled parafermion chains give rise to in-gap excitations.Our results exemplify the versatility of tensor network methods for studying dynamical excitations of interacting parafermion chains and highlight the robustness of topological modes in parafermion models.
Parafermion zero modes are generalizations of Majorana modes that underlie comparatively rich non-Abelian-anyon properties. We introduce exact mappings that connect parafermion chains, which can emerge in two-dimensional fractionalized media, to strictly one-dimensional fermionic systems. In particular, we show that parafermion zero modes in the former setting translate into symmetry-enriched Majorana modes that intertwine with a bulk order parameter---yielding braiding and fusion properties that are impossible in standard Majorana platforms. Fusion characteristics of symmetry-enriched Majorana modes are directly inherited from the associated parafermion setup and can be probed via two kinds of anomalous pumping cycles that we construct. Most notably, our mappings relate Z4 parafermions to conventional electrons with time-reversal symmetry. In this case, one of our pumping protocols entails fairly minimal experimental requirements: Cycling a weakly correlated wire between a trivial phase and time-reversal-invariant topological superconducting state produces an edge magnetization with quadrupled periodicity. Our work highlights new avenues for exploring beyond-Majorana physics in experimentally relevant one-dimensional electronic platforms, including proximitized ferromagnetic chains.
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.
Parafermions are non-Abelian anyons which generalize Majorana fermions and hold great promise for topological quantum computation. We study the braiding of Z2n parafermions which have been predicted to emerge as localized zero modes in fractional quantum Hall systems at filling factor $\ensuremath{\nu}=1/n$ (n odd). Using a combination of bosonization and refermionization, we calculate the energy splitting as a function of distance and chemical potential for a pair of parafermions separated by a gapped region. Braiding of parafermions in quantum Hall edge states can be implemented by repeated fusion and nucleation of parafermion pairs. We simulate the conventional braiding protocol of parafermions numerically, taking into account the finite separation and finite chemical potential. We show that a nonzero chemical potential poses challenges for the adiabaticity of the braiding process because it leads to accidental crossings in the spectrum. To remedy this, we propose an improved braiding protocol which avoids those degeneracies.