Partial fillings of the bosonic $E_8$ quantum Hall state

Abstract

We study bosonic topological phases constructed from electrons. In addition to a bulk excitation energy gap, these bosonic phases also have a fermion energy gap, below which all local excitations in the bulk and on the edge are even combinations of electrons. We focus on chiral phases, in which all low-energy edge excitations move in the same direction, that arise from the short-range entangled $E_8$ quantum Hall state, the bosonic analog of the filled lowest Landau level of electrons. The $E_8$ edge-state theory features an $E_8$ Kac-Moody symmetry that can be decomposed into ${\mathcal{G}}_A\times {\mathcal{G}}_B$ subalgebras, such as $SU(3)\times E_6$, $SO(M)\times SO(16-M)$, and $G_2\times F_4$. (Here, $\{SO(M)\}$, $\{SU(N)\}$, and $\{E_8,G_2,F_4\}$ denote orthogonal, unitary, and exceptional Lie algebras.) Using these symmetry decompositions, we construct exactly solvable coupled-wire model Hamiltonians for families of long-range entangled ${\mathcal{G}}_A$ or ${\mathcal{G}}_B$ bosonic fractional quantum Hall states that ``partially fill" the $E_8$ state and are pairwise related by a generalized particle-hole symmetry. These long-range entangled states feature either Abelian or non-Abelian topological order. Some support the emergence of non-local Dirac and Majorana fermions, Ising anyons, metaplectic anyons, Fibonacci anyons, as well as deconfined ${\mathbb{Z}}_2$ gauge fluxes and charges.

48 pages, 6 figures

References in corpus (12)

• Non-Abelian Anyons and Topological Quantum Computation
• Particle-hole symmetry and the Pfaffian state
• Particle-Hole Symmetry and the $\nu =5/2$ Quantum Hall State
• Condensate induced transitions between topologically ordered phases
• Level/rank Duality and Chern-Simons-Matter Theories
• Theory of Twist Liquids: Gauging an Anyonic Symmetry
• Thirty Years of Composite Fermions and Beyond
• Wavefunctionology: The Special Structure of Certain Fractional Quantum Hall Wavefunctions
• Assembling Fibonacci Anyons From a ${\mathbb{Z}}_3$ Parafermion Lattice Model
• Anyonic Symmetries and Topological Defects in Abelian Topological Phases: an application to the $ADE$ Classification
• From orbifolding conformal field theories to gauging topological phases
• Dihedral twist liquid models from emergent Majorana fermions

Cited by in corpus (1)

• Operator-state correspondence in simple current extended conformal field theories: Toward a general understanding of chiral conformal field theories and topological orders