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.
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.
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 bound states (PBSs) are generalizations of Majorana bound states (MBSs) and have been predicted to exist as zero-energy eigenstates in proximitized fractional quantum Hall edge states. Similarly to MBSs, a finite distance between the PBS can split the ground-state degeneracy. However, parafermionic modes have a richer exchange statistics than MBSs, so several interaction terms are allowed by the underlying Z2n symmetry, rendering the effective Hamiltonian governing a pair of PBSs at a finite distance nontrivial. Here, we use a combination of analytical techniques (semiclassical instanton approximation) and numerical techniques (quantum Monte Carlo simulations) to determine the effective coupling Hamiltonian. For this purpose, we go beyond the dilute one-instanton gas approximation and show how finite-size effects can give rise to higher-order parafermion interactions. We find good quantitative agreement between the analytical results and Monte Carlo simulations. We estimate that these finite-size corrections should be observable in some of the recently proposed experiments to observe PBSs in strongly correlated systems.
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 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.
This thesis aims to investigate the novel physics around Majorana bound states (MBSs). MBSs are quasiparticle excitations predicted to exist in topological superconductors. Due to the inherent protection they obtain from their topology, they have long been regarded as a promising setup to build a topologically protected quantum computer. Apart from this very practical application, they continue to fascinate as platforms built with MBS exhibit unique effects like nonabelian braiding and non-Fermi liquid behaviour in the topological Kondo effect.The thesis starts with a general introduction to quantum computing, highlighting both the promises and challenges of that field. Afterwards follows an introduction to systems that are candidates to host MBSs. It focuses on the theoretical proposals and experimental realizations that are most relevant to the presented work. We continue with a summary of the most fundamental effects and the setups that make the physics of MBSs accessible. The author's contributions in this field are theoretical. However, each publication comes with the explicit goal of investigating setups that make the unique physics of MBSs directly accessible in experiments. The first three papers have been peer-reviewed and were published in scientific journals. The fourth one is currently under peer review. The first two papers investigate DC transport setups connecting to a qubit based on MBSs called the MBS tetron. We show that via an effect called parity blockade, it is possible to initialize the qubit, quantify its lifetime, and determine whether it holds MBSs or trivial (normal fermionic) states. Also employing parity blockade, the second paper establishes a transport setup that enables access to MBS fusion rules, an effect directly related to nonabelian braiding. The third paper translates the concept of a MBS tetron into the newly experimentally established approach of MBSs in quantum dot-based Kitaev chains. We denote this system the poor man's Majorana tetron. We find that the poor man's Majorana tetron in a transport setup with appropriate tuning exhibits the famed topological Kondo effect. Paper~IV focuses on nonabelian braiding. It investigates the hybridization-based braiding protocol and accounts for imperfections in MBS localization, breaking the ground state degeneracy. This detrimental effect is corrected by the introduction of a correction term, followed by a full analytical solution of the transformation induced by this generalized braiding protocol.
The concept of topology in condensed matter physics has led to the discovery of rich and exotic physics in recent years. Especially when strong correlations are included, phenomenons such as fractionalization and anyonic particle statistics can arise. In this thesis, we study several systems hosting topological phases of interacting fermions. In the first part, we consider one-dimensional systems of parafermions, which are generalizations of Majorana fermions, in the presence of a Z_N charge symmetry. We classify the symmetry-protected topological (SPT) phases that can occur in these systems using the projective representations of the symmetries and find a finite number of distinct phases depending on the prime factorization of N. The different phases exhibit characteristic degeneracies in their entanglement spectrum (ES). Apart from these SPT phases, we report the occurrence of parafermion condensate phases for certain values of N. When including an additional Z_N symmetry, we find a non-Abelian group structure under the addition of phases. In the second part of the thesis, we focus on two-dimensional lattice models of spinless fermions. First, we demonstrate the detection of a fractional Chern insulator (FCI) phase in the Haldane honeycomb model on an infinite cylinder by means of the density-matrix renormalization group (DMRG). We report the calculation of several quantities characterizing the topological order of the state, i.e., (i)~the Hall conductivity, (ii)~the spectral flow and level counting in the ES, (iii)~the topological entanglement entropy, and (iv)~the charge and topological spin of the quasiparticles. Since we have access to sufficiently large system sizes without band projection with DMRG, we are in addition able to investigate the transition from a metal to the FCI at small interactions which we find to be of first order. In a further study, we consider a time-reversal symmetric model on the honeycomb lattice where a Chern insulator (CI) induced by next-nearest neighbor interactions has been predicted by mean field theory. However, various subsequent studies challenged this picture and it was still unclear whether the CI would survive quantum fluctuations. We therefore map out the phase diagram of the model as a function of the interactions on an infinite cylinder with DMRG and find evidence for the absence of the CI phase. However, we report the detection of two novel charge-ordered phases and corroborate the existence of the remaining phases that had been predicted in mean field theory. Furthermore, we characterize the transitions between the various phases by studying the behavior of correlation length and entanglement entropy at the phase boundaries. Finally, we develop an improvement to the DMRG algorithm for fermionic lattice models on cylinders. By using a real space representation in the direction along the cylinder and a real space representation in the perpendicular direction, we are able to use the momentum around the cylinder as conserved quantity to reduce computational costs. We benchmark the method by studying the interacting Hofstadter model and report a considerable speedup in computation time and a severely reduced memory usage.