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Topological phases with parafermions: theory and blueprints

Jason Alicea, Paul Fendley

Abstract

We concisely review the recent evolution in the study of parafermions -- exotic emergent excitations that generalize Majorana fermions and similarly underpin a host of novel phenomena. First we illustrate the intimate connection between Z_3-symmetric "spin" chains and one-dimensional parafermion lattice models, highlighting how the latter host a topological phase featuring protected edge zero modes. We then tour several blueprints for the laboratory realization of parafermion zero modes -- focusing on quantum Hall/superconductor hybrids, quantum Hall bilayers, and two-dimensional topological insulators -- and describe striking experimental fingerprints that they provide. Finally, we discuss how coupled parafermion arrays in quantum Hall architectures yield topological phases that potentially furnish hardware for a universal, intrinsically decoherence-free quantum computer.

Topological phases with parafermions: theory and blueprints

Abstract

We concisely review the recent evolution in the study of parafermions -- exotic emergent excitations that generalize Majorana fermions and similarly underpin a host of novel phenomena. First we illustrate the intimate connection between Z_3-symmetric "spin" chains and one-dimensional parafermion lattice models, highlighting how the latter host a topological phase featuring protected edge zero modes. We then tour several blueprints for the laboratory realization of parafermion zero modes -- focusing on quantum Hall/superconductor hybrids, quantum Hall bilayers, and two-dimensional topological insulators -- and describe striking experimental fingerprints that they provide. Finally, we discuss how coupled parafermion arrays in quantum Hall architectures yield topological phases that potentially furnish hardware for a universal, intrinsically decoherence-free quantum computer.

Paper Structure

This paper contains 14 sections, 18 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Parafermions in chains
  3. Lattice parafermions
  4. Edge zero modes
  5. Parafermions in conformal field theory
  6. Experimental blueprints for parafermion zero modes
  7. Quantum Hall/superconductor hybrids
  8. Setup and physical origin of zero modes
  9. Zero-mode detection
  10. Non-Abelian statistics
  11. Quantum Hall bilayers
  12. 2D topological insulator edges
  13. Fibonacci anyons from coupled parafermions
  14. Outlook

Figures (4)

  • Figure 1: Schematic representation of the parafermion chain [Eq. (\ref{['Hpf']})] deep in the (a) trivial and (b) topological phases, and (c) at the intervening critical point.
  • Figure 2: Select blueprints for parafermion zero modes $\alpha_{1,2}$. In (a) only the charged edge mode is shown for simplicity.
  • Figure 3: Detection schemes for parafermion zero modes, illustrated using the $\nu = 2/3$ setup from Fig. \ref{['Blueprints_fig']}(a). (a) Hybridizing a zero mode with the gapless edge states generates 'perfect Andreev conversion'. (b) Coupling $\mathbb{Z}_3$ parafermions across a Josephson junction yields a current (mediated by $2e/3$ tunneling) $6\pi$-periodic in the phase difference $\phi_L-\phi_R$.
  • Figure 4: (a) Correspondence between the $\mathbb{Z}_3$ Read-Rezayi state and the 'Fibonacci phase' formed in a $\nu = 2/3$ quantum Hall/superconductor heterostructure. (b) Triangular-lattice parafermion model (left) that realizes the Fibonacci phase in the weakly-coupled-chain limit (right). (c) Phase diagram of the setup in (b) obtained from the DMRG analysis of Ref. Stoudenmire:2015.