Parafermions are emergent quasiparticles which generalize Majorana fermions and possess intriguing anyonic properties. The theoretical investigation of effective models hosting them is gaining considerable importance in view of present-day condensed-matter realizations where they have been predicted to appear. Here we study the simplest number-conserving model of particlelike Fock parafermions, namely a one-dimensional tight-binding model. By means of numerical simulations based on exact diagonalization and on the density-matrix renormalization group, we prove that this quadratic model is nonintegrable and displays bound states in the spectrum due to its peculiar anyonic properties. Moreover, we discuss its many-body physics, characterizing anyonic correlation functions and discussing the underlying Luttinger-liquid theory at low energies. In the case when Fock parafermions behave as fractionalized fermions, we are able to unveil interesting similarities with two counterpropagating edge modes of two neighboring Laughlin states at filling $1/3$.
Parafermions are emergent quasiparticles which generalize Majorana fermions and possess intriguing anyonic properties. The theoretical investigation of effective models hosting them is gaining considerable importance in view of present-day condensed-matter realizations where they have been predicted to appear. Here we study the simplest number-conserving model of particlelike Fock parafermions, namely a one-dimensional tight-binding model. By means of numerical simulations based on exact diagonalization and on the density-matrix renormalization group, we prove that this quadratic model is nonintegrable and displays bound states in the spectrum due to its peculiar anyonic properties. Moreover, we discuss its many-body physics, characterizing anyonic correlation functions and discussing the underlying Luttinger-liquid theory at low energies. In the case when Fock parafermions behave as fractionalized fermions, we are able to unveil interesting similarities with two counterpropagating edge modes of two neighboring Laughlin states at filling 1/3.
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.
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), 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.
We study the effects of a random magnetic field on a one-dimensional (1D) spin-1 chain with correlated nearest-neighbor XY interaction. We show that this spin model can be exactly mapped onto the 1D disordered tight-binding model of Z3 Fock parafermions (FPFs), exotic anyonic quasiparticles that generalize usual spinless fermions. Thus, we have a peculiar case of a disordered Hamiltonian that despite being bilinear in the creation and annihilation operators, exhibits a many-body localization (MBL) transition owing to the nontrivial statistics of FPFs. This is in sharp contrast to conventional bosonic and fermionic quadratic disordered Hamiltonians that show single-particle (Anderson) localization. We perform finite-size exact diagonalization calculations of level-spacing statistics, fractal dimensions, and entanglement entropy, and provide convincing evidence for the MBL transition at finite disorder strength.
We model p -state Fock parafermions on a lattice in one dimension (with occupation per orbital of 0 , 1 , ... , p − 1 ). For p a composite number, they may be mapped to q m -state parafermions where q m are the prime factors of p . For a Hamiltonian with a single-particle spectrum, the parafermions decompose into q m -state parafermions. When p is a power of two, the decomposition is into fermions. We use this to construct a parafermionic Hamiltonian for p = 4 with a single-particle spectrum using a fermionic tight-binding model which is bilinear in creation and annihilation operators. The single-particle levels may be determined by diagonalizing a square matrix whose order scales linearly with system size, and they are the same as those of the fermionic model. We show that the intermediate statistics of the thermodynamic distribution function for the occupation numbers (known as Gentile statistics) are consistent with the mapping to fermions, and we provide an example calculation of the internal energy and heat capacity for a simple linear chain.
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.