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arXiv:1706.01192

Numerical Observation of Parafermion Zero Modes and their Stability in 2D Topological States

Mohammad-Sadegh Vaezi, Abolhassan Vaezi

Published Jun 5, 2017• 6 citations
arXivHTMLPDF

Abstract

The possibility of realizing non-Abelian excitations (non-Abelions) in two-dimensional (2D) Abelian states of matter has generated a lot of interest recently. A well-known example of such non-Abelions are parafermion zeros modes (PFZMs) which can be realized at the endpoints of the so called genons in fractional quantum Hall (FQH) states or fractional Chern insulators (FCIs). In this letter, we discuss some known signatures of PFZMs and also introduce some novel ones. In particular, we show that the topological entanglement entropy (TEE) shifts by a quantized value after crossing PFZMs. Utilizing those signatures, we present the first large scale numerical study of PFZMs and their stability against perturbations in both FQH states and FCIs within the density-Matrix-Renormalization-Group (DMRG) framework. Our results can help build a closer connection with future experiments on FQH states with genons.

DOI
10.48550/arxiv.1706.01192

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The possibility of realizing non-Abelian excitations (non-Abelions) in two-dimensional (2D) Abelian states of matter has generated a lot of interest recently. A well-known example of such non-Abelions are parafermion zeros modes (PFZMs) which can be realized at the endpoints of the so called genons in fractional quantum Hall (FQH) states or fractional Chern insulators (FCIs). In this letter, we discuss some known signatures of PFZMs and also introduce some novel ones. In particular, we show that the topological entanglement entropy (TEE) shifts by a quantized value after crossing PFZMs. Utilizing those signatures, we present the first large scale numerical study of PFZMs and their stability against perturbations in both FQH states and FCIs within the density-Matrix-Renormalization-Group (DMRG) framework. Our results can help build a closer connection with future experiments on FQH states with genons.
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.
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.
Parafermionic zero modes are non-Abelian excitations which have been predicted to emerge at the boundary of topological phases of matter. Contrary to earlier proposals, here we show that such zero modes may also exist in multilegged star junctions of quantum Hall states. We demonstrate that the quantum states spanning the degenerate parafermionic Hilbert space may be detected and manipulated through protocols employing quantum antidots and fractional edge modes. Such star-shaped setups may be the building blocks of two-dimensional parafermionic networks.
Fractional quantum Hall-superconductor (FQH-SC) heterostructures are predicted to host defect-bound parafermion zero modes (PZMs). We propose two related configurations to probe their fusion structure. In a Josephson junction coupled to a single quantum point contact (QPC), quasiparticle tunneling switches the defect fusion channel, producing stochastic transitions between branches of the fractional Josephson spectrum. Embedding the junction in a two-QPC Fabry-Pérot interferometer provides a complementary probe. Weak zero mode tunneling produces fusion-channel-dependent interference while strong tunneling makes the interferometer probe a superposition of fusion channels and strongly suppresses the signal: in the topological limit it vanishes exactly, revealing the defects' non-Abelian nature even when the parent FQH state is Abelian.
Exotic topologically protected zero modes with parafermionic statistics (also called fractionalized Majorana modes) have been proposed to emerge in devices fabricated from a fractional quantum Hall system and a superconductor. The fractionalized statistics of these modes takes them an important step beyond the simplest non-Abelian anyons, Majorana fermions. Building on recent advances towards the realization of fractional quantum Hall states of bosonic ultracold atoms, we propose a realization of parafermions in a system consisting of Bose-Einstein-condensate trenches within a bosonic fractional quantum Hall state. We show that parafermionic zero modes emerge at the end points of the trenches and give rise to a topologically protected degeneracy. We also discuss methods for preparing and detecting these modes.
Chiral graviton modes are hallmark collective excitations of Fractional Quantum Hall (FQH) liquids. However, their existence on the lattice, where continuum symmetries that protect them from decay are lost, is still an open and urgent question, especially considering the recent advances in the realization of Fractional Chern Insulators (FCI) in transition metal dichalcogenides and rhombohedral pentalayer graphene. Here we present a comprehensive theoretical and numerical study of graviton-modes in fermionic FCI, and thoroughly demonstrate their existence. We first derive a lattice stress tensor operator in the context of the fermionic Harper-Hofstadter(HH) model which captures the graviton in the flat band limit. Importantly, we discover that such lattice stress-tensor operators are deeply connected to lattice quadrupolar density correlators, readily generalizable to generic Chern bands. We then explicitly show the adiabatic connection between FQH and FCI chiral graviton modes by interpolating from a low flux HH model to a Checkerboard lattice model that hosts a topological flat band. In particular, using state-of-the-art matrix product state and exact diagonalization simulations, we provide strong evidence that chiral graviton modes are long-lived excitations in FCIs despite the lack of continuous symmetries and the scattering with a two-magnetoroton continuum. By means of a careful finite-size analysis, we show that the lattice generates a finite but small intrinsic decay rate for the graviton mode. We discuss the relevance of our results for the exploration of graviton modes in FCI phases realized in solid state settings, as well as cold atom experiments.