We analyse a class of 1D lattice models, known as models, which are characterised by an order- k clustering of spin-less fermions and by lattice supersymmetry. Our main result is the identification of a class of (bulk or edge) defects, that are in one-to-one correspondence with so-called spin fields in a corresponding parafermion CFT. In the gapped regime, injecting such defects leads to ground-state degeneracies that are protected by the supersymmetry. The defects, which are closely analogous to quasi-holes over the fermonic Read-Rezayi quantum Hall states, display characteristic fusion rules, which are of Ising type for k = 2 and of Fibonacci type for k = 3.We analyse a class of 1D lattice models, known as models, which are characterised by an order- k clustering of spin-less fermions and by lattice supersymmetry. Our main result is the identification of a class of (bulk or edge) defects, that are in one-to-one correspondence with so-called spin fields in a corresponding parafermion CFT. In the gapped regime, injecting such defects leads to ground-state degeneracies that are protected by the supersymmetry. The defects, which are closely analogous to quasi-holes over the fermonic Read-Rezayi quantum Hall states, display characteristic fusion rules, which are of Ising type for k = 2 and of Fibonacci type for k = 3.

arXiv:1504.02421Europhysics Letters (EPL)Aug 1, 20157 citations

Parafermionic modes in correlated electron systems

Raphael L. R. C. Teixeira

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.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.

Mar 21, 20230 citations

Ground states of Fermions on lattices

Taku Matsui

Communications in Mathematical PhysicsDec 1, 199613 citations

Bound states of a system of two fermions on a one-dimensional lattice

J. I. Abdullaev

Theoretical and Mathematical PhysicsApr 1, 200614 citations

Many-body localization of Z3 Fock parafermions

M. S. Bahovadinov, Wouter Buijsman, Aleksey K. Fedorov +2 more

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 ${\mathbb{Z}}_{3}$ 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 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 ${\mathbb{Z}}_{3}$ 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.

arXiv:2210.02947Physical review. B./Physical review. BDec 15, 20223 citations

{\mathbb{Z}}_{3}

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.