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.

arXiv comment48 pages, 6 figures

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