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arXiv:1407.4002

Z$_N$ parafermion zero modes without Fractional Quantum Hall effect

Alexei M. Tsvelik

Published Jul 15, 2014• 3 citations
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Abstract

I discuss a one-dimensional model of interacting fermions which collective excitations are Z$_N$-parafermions. The phase diagram of this model contains ground states with Charge Density Wave and superconducting quasi long range order. Boundaries between these phases contain zero energy parafermion modes.

DOI
10.48550/arxiv.1407.4002

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ZN_NN​ parafermion zero modes without Fractional Quantum Hall effect
I discuss a one-dimensional model of interacting fermions which collective excitations are ZN_NN​-parafermions. The phase diagram of this model contains ground states with Charge Density Wave and superconducting quasi long range order. Boundaries between these phases contain zero energy parafermion modes.
We study the ground-state phase diagram of the strongly interacting Harper–Hofstadter–Mott model at quarter flux on a quasi-one-dimensional lattice consisting of a single magnetic flux quantum in y -direction. In addition to superfluid phases with various density patterns, the ground-state phase diagram features quasi-one-dimensional analogs of fractional quantum Hall phases at fillings ν = 1/2 and 3/2, where the latter is only found thanks to the hopping anisotropy and the quasi-one-dimensional geometry. At integer fillings—where in the full two-dimensional system the ground-state is expected to be gapless—we observe gapped non-degenerate ground-states: at ν = 1 it shows an odd 'fermionic' Hall conductance, while the Hall response at ν = 2 consists of the transverse transport of a single particle–hole pair, resulting in a net zero Hall conductance. The results are obtained by exact diagonalization and in the reciprocal mean-field approximation.
Parafermionic zero modes, Zn{\mathbb{Z}}_{n}Zn​-symmetric generalizations of the well-known Z2{\mathbb{Z}}_{2}Z2​ 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 Z4{\mathbb{Z}}_{4}Z4​ 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, Z4{\mathbb{Z}}_{4}Z4​ 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 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{\mathbb{Z}}_{2n}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.
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.
In this letter, we discuss the recently proposed fractional quantum Hall effect in the absence of Landau levels. It is shown that the parton construction can explain all properties of 1/3 state, including the effective charge of quasi-particles, their statistics and the many-body ground-state degeneracy. The low energy description of these states has been discussed. We also generalize our model to construct the hierarchical quantum Hall states at filling fractions other than 1/m1/m1/m.