TOPOLOGICAL ORDERS IN RIGID STATES

TLDR

Chiral spin states are described by topological Chern‑Simons theories in the continuum limit. The study introduces topological order in rigid states and demonstrates that such orders are characterized by a non‑Abelian gauge structure over the moduli space of model Hamiltonians. The authors characterize topological orders by analyzing chiral spin states in 2+1 dimensions and examining symmetry properties of their degenerate ground states. They find that non‑Abelian gauge structures determine quasiparticle statistics, that low‑lying excitations and ground‑state degeneracy are robust to disorder and symmetry breaking, and that some torus ground states carry non‑trivial 90° rotation quantum numbers.

Abstract

We study a new kind of ordering — topological order — in rigid states (the states with no local gapless excitations). We concentrate on characterization of the different topological orders. As an example we discuss in detail chiral spin states of 2 + 1 dimensional spin systems. Chiral spin states are described by the topological Chern-Simons theories in the continuum limit. We show that the topological orders can be characterized by a non-Abelian gauge structure over the moduli space which parametrizes a family of the model Hamiltonians supporting topologically ordered ground states. In 2 + 1 dimensions, the non-Abelian gauge structure determines possible fractional statistics of the quasi-particle excitations over the topologically ordered ground states. The dynamics of the low lying global excitations is shown to be independent of random spatial dependent perturbations. The ground state degeneracy and the non-Abelian gauge structures discussed in this paper are very robust, even against those perturbations that break translation symmetry. We also discuss the symmetry properties of the degenerate ground states of chiral spin states. We find that some degenerate ground states of chiral spin states on torus carry non-trivial quantum numbers of the 90° rotation.