Nonlocal (topological) order and local (Landau-type) order are often viewed as opposing principles. Here, the authors study strongly interacting quantum wires that exhibit hybrid order, composed of local Ising magnetization and a topological order parameter. Based on an exact mapping of the corresponding strongly interacting electronic Hamiltonian onto \ensuremath{\mathbb{Z}}${}_{n}$ parafermions, they show that the system can be in a hybrid phase which is distinct from the topological and from the Ising phase. Moreover, the system has remarkable similarities to superconducting hybrid devices built from fractional quantum Hall edge states, which can host parafermions useful for quantum computation.
Nonlocal (topological) order and local (Landau-type) order are often viewed as opposing principles. Here, the authors study strongly interacting quantum wires that exhibit hybrid order, composed of local Ising magnetization and a topological order parameter. Based on an exact mapping of the corresponding strongly interacting electronic Hamiltonian onto \ensuremath{\mathbb{Z}}n parafermions, they show that the system can be in a hybrid phase which is distinct from the topological and from the Ising phase. Moreover, the system has remarkable similarities to superconducting hybrid devices built from fractional quantum Hall edge states, which can host parafermions useful for quantum computation.
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.
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.
We explore a scenario where local interactions form one-dimensional gapped interfaces between a pair of distinct chiral two-dimensional topological states-referred to as phases 1 and 2-such that each gapped region terminates at a domain wall separating the chiral gapless edge states of these phases. We show that this type of T junction supports pointlike fractionalized excitations obeying parafermion statistics, thus implying that the one-dimensional gapped interface forms an effective topological parafermionic wire possessing a nontrivial ground state degeneracy. The physical properties of the anyon condensate that gives rise to the gapped interface are investigated. Remarkably, this condensate causes the gapped interface to behave as a type of anyon "Andreev reflector" in the bulk, whereby anyons from one phase, upon hitting the interface, can be transformed into a combination of reflected anyons and outgoing anyons from the other phase. Thus, we conclude that while different topological orders can be connected via gapped interfaces, the interfaces are themselves topological.
We introduce a coupled wire model for a sequence of non-Abelian quantum Hall states that generalize the Z4 parafermion Read-Rezayi state. The Z4 orbifold quantum Hall states occur at filling factors $\ensuremath{\nu}=2/(2m\ensuremath{-}p)$ for odd integers m and p, and have a topological order with a neutral sector characterized by the orbifold conformal field theory with central charge c=1 at radius R=p/2. When p=1 the state is Abelian. The state with p=3 is the Z4 Read-Rezayi state, and the series of $p\ensuremath{\ge}3$ defines a sequence of non-Abelian states that resembles the Laughlin sequence. Our model is based on clustering of electrons in groups of four, and is formulated as a two fluid model in which each wire exhibits two phases: a weak clustered phase, where charge e electrons coexist with charge 4e bosons and a strong clustered phase where the electrons are strongly bound in groups of four. The transition between these two phases on a wire is mapped to the critical point of the four-state clock model, which in turn is described by the orbifold conformal field theory. For an array of wires coupled in the presence of a perpendicular magnetic field, strongly clustered wires form a charge 4e bosonic Laughlin state with a chiral charge mode at the edge, but no neutral mode and a gap for single electrons. Coupled wires near the critical state form quantum Hall states with a gapless neutral mode described by the orbifold theory. The coupled wire approach allows us to employ the Abelian bosonization technique to fully analyze the physics of single wire, and then to extract most topological properties of the resulting non-Abelian quantum Hall states. These include the list of quasiparticles, their fusion rules, the correspondence between bulk quasiparticles and edge topological sectors, and most of the phases associated with quasiparticles winding one another.
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.