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arXiv:2011.06462 · Physical Review Research

Dynamical topological excitations in parafermion chains

Vilja Kaskela, J. L. Lado

Published Jan 29, 2021• 18 citations

Abstract

Parafermions are elusive fractional excitations potentially emerging in fractional quantum Hallsuperconductor junctions and represent one of the major milestones in fractional quantum matter.However, generic models of parafermions are not analytically solvable, and understanding their topological modes is a bigger challenge than conventional Majorana modes.Here, by using a combination of tensor network and kernel polynomial techniques, we demonstrate the emergence of zero modes and finite energy excitations in many-body parafermion chains.We show the appearance of zero-energy modes in the many-body spectral function at the edge of a topological parafermion chain, their relation with the topological degeneracy of the system, and we compare their physics with the Majorana bound states of topological superconductors.We demonstrate the robustness of parafermion topological modes with respect to a variety of perturbations, and we show how weakly coupled parafermion chains give rise to in-gap excitations.Our results exemplify the versatility of tensor network methods for studying dynamical excitations of interacting parafermion chains and highlight the robustness of topological modes in parafermion models.

DOI: 10.1103/physrevresearch.3.013095

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Parafermions are elusive fractional excitations potentially emerging in fractional quantum Hallsuperconductor junctions and represent one of the major milestones in fractional quantum matter.However, generic models of parafermions are not analytically solvable, and understanding their topological modes is a bigger challenge than conventional Majorana modes.Here, by using a combination of tensor network and kernel polynomial techniques, we demonstrate the emergence of zero modes and finite energy excitations in many-body parafermion chains.We show the appearance of zero-energy modes in the many-body spectral function at the edge of a topological parafermion chain, their relation with the topological degeneracy of the system, and we compare their physics with the Majorana bound states of topological superconductors.We demonstrate the robustness of parafermion topological modes with respect to a variety of perturbations, and we show how weakly coupled parafermion chains give rise to in-gap excitations.Our results exemplify the versatility of tensor network methods for studying dynamical excitations of interacting parafermion chains and highlight the robustness of topological modes in parafermion models.

Parafermionic zero modes,

{\mathbb{Z}}_{n}

-symmetric generalizations of the well-known

{\mathbb{Z}}_{2}

Majorana zero modes, can emerge as edge states in topologically nontrivial strongly correlated systems displaying fractionalized excitations. In this paper, we investigate how signatures of parafermionic zero modes can be detected by its effects on the properties of a quantum dot tunnel-coupled to a system hosting such states. Concretely, we consider a strongly correlated one-dimensional fermionic model supporting

{\mathbb{Z}}_{4}

parafermionic zero modes coupled to an interacting quantum dot at one of its ends. By using a combination of density matrix renormalization group calculations and analytical approaches, we show that the dot's zero-energy spectral function and average occupation numbers can be used to distinguish between trivial,

{\mathbb{Z}}_{4}

and $2\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ phases of the system. The present work opens the prospect of using quantum dots as detection tools to probe nontrivial topological phases in strongly correlated systems.

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

{\mathbb{Z}}_{4}

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.

Parafermions are emergent excitations that generalize Majorana fermions and can also realize topological order. In this Letter, we present a nontrivial and quasi-exactly-solvable model for a chain of parafermions in a topological phase. We compute and characterize the ground-state wave functions, which are matrix-product states and have a particularly elegant interpretation in terms of Fock parafermions, reflecting the factorized nature of the ground states. Using these wave functions, we demonstrate analytically several signatures of topological order. Our study provides a starting point for the nonapproximate study of topological one-dimensional parafermionic chains with spatial inversion and time-reversal symmetry in the absence of strong edge modes.

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.

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.