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arXiv:2111.10147 · Physical review. B./Physical review. B

Edge Z3 parafermions in fermionic lattices

Raphael L. R. C. Teixeira, Luis G. G. V. Dias da Silva

Published May 13, 2022• 5 citations
arXivHTMLPDF

Abstract

Parafermion modes are non-Abelian anyons which were introduced as ${\mathbb{Z}}_{N}$ generalizations of ${\mathbb{Z}}_{2}$ Majorana states. In particular, ${\mathbb{Z}}_{3}$ 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 ${\mathbb{Z}}_{3}$ parafermions has relied on bosonization methods or parafermionic quasiparticles. In this paper, we introduce a representation of ${\mathbb{Z}}_{3}$ 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 ${\mathbb{Z}}_{3}$ 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 ${\mathbb{Z}}_{3}$ parafermions in strongly interacting electronic systems.

DOI
10.1103/physrevb.105.195121

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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 Zn{\mathbb{Z}}_{n}Zn​ generalizations of Majorana quasiparticles, with fractional non-Abelian statistics. They can be used to encode topological qudits and perform Clifford operations by their braiding. Here we investigate the generation of quantum gates by allowing Z3{\mathbb{Z}}_{3}Z3​ parafermions to interact in order to achieve universality. In particular, we study the form of the nontopological gate that arises through direct short-range interaction of the parafermion edge modes in a Z3{\mathbb{Z}}_{3}Z3​ parafermion chain. We show that such an interaction gives rise to a dynamical phase gate on the encoded ground space, generating a non-Clifford gate which can be tuned to belong to even levels of the Clifford hierarchy. We illustrate how to access highly noncontextual states using this dynamical gate. Finally, we propose an experiment that simulates the braiding and dynamical evolutions of the Z3{\mathbb{Z}}_{3}Z3​ topological states with Rydberg atom technology.
In the Fock representation, we propose a framework to construct the generalized matrix product states (MPS) for topological phases with Zp{\mathbb{Z}}_{p}Zp​ parafermions. Unlike the Z2{\mathbb{Z}}_{2}Z2​ Majorana fermions, the Zp{\mathbb{Z}}_{p}Zp​ parafermions form intrinsically interacting systems. Here we explicitly construct two topologically distinct classes of irreducible Z3{\mathbb{Z}}_{3}Z3​ parafermionic MPS wave functions, characterized by one or two parafermionic zero modes at each end of an open chain. Their corresponding parent Hamiltonians are found as the fixed point models of the single Z3{\mathbb{Z}}_{3}Z3​ parafermion chain and two-coupled parafermion chains with ${\mathbb{Z}}_{3}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{3}$ symmetry. Our results thus pave the road to investigate all possible topological phases with Zp{\mathbb{Z}}_{p}Zp​ parafermions within the matrix product representation in one dimension.
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), have been proposed as candidates for implementing topological quantum computation relying on its non-abelian statistics. Parafermions bound states (PBSs) can be regarded as ZN generalizations of the Z2-symmetric MBSs. 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. Tight-binding models have also been suggested to host PBSs, although in some cases they are nontopological. Recently, Kondo devices have also been proposed to host parafermionic states. 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, we also investigate a continuous model of fractional quantum Hall insulators with induced superconductivity. This work is, therefore, divided into two parts. In the first part, we examine the fermionic models using Density Matrix Renormalization Group (DMRG) as well as some properties of Fock-parafermion. We start with a proposal on how to use Quantum dots (QDs) to probe the existence of Z4 parafermions and differentiate it from 2 MBSs (2xZ2). Then, we introduce two fermionic models that host Z3 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 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.
Abstract Moiré materials provide a remarkably tunable platform for topological and strongly correlated quantum phases of matter. Very recently, the first Abelian fractional Chern insulators (FCIs) at zero magnetic field have been experimentally demonstrated, and it has been theoretically predicted that non-Abelian states with Majorana fermion excitations may be realized in the nearly dispersionless minibands of these systems. Here, we provide telltale evidence based on many-body exact diagonalization for the even more exotic possibility of moiré-based non-Abelian FCIs exhibiting Fibonacci parafermion excitations. In particular, we obtain low-energy quantum numbers, spectral flow, many-body Chern numbers, and entanglement spectra consistent with the Z3{{\mathbb{Z}}}_{3}Z3​ Z 3 Read–Rezayi parafermion phase in an exemplary moiré system with tunable quantum geometry. Our results hint towards the robustness of moiré-based parafermions and encourage the pursuit in moiré systems of these non-Abelian quasiparticles that are superior candidates for topological quantum computing.