Noncollinear magnetic order is typically characterized by a tetrad ground state manifold (GSM) of three perpendicular vectors or nematic directors. We study three types of tetrad orders in two spatial dimensions, whose GSMs are SO(3) = S(3)/Z(2), S(3)/Z(4), and S(3)/Q(8), respectively. Q(8) denotes the non-Abelian quaternion group with eight elements. We demonstrate that after quantum disordering these three types of tetrad orders, the systems enter fully gapped liquid phases described by Z(2), Z(4), and non-Abelian quaternion gauge field theories, respectively. The latter case realizes Kitaev's non-Abelian toric code in terms of a rather simple spin-1 SU(2) quantum magnet. This non-Abelian topological phase possesses a 22-fold ground state degeneracy on the torus arising from the 22 representations of the Drinfeld double of Q(8).
The topological defects of Spin(
n+1) nematics in two spatial dimensions, known as disclinations, are characterized by the ${\ensuremath{\pi}}_{1}(\mathbb{R}{P}^{n})={\text{Z}}_{2}$ homotopy group for $n\ensuremath{\ge}2$. We argue that incompressible quantum liquids of disclinations can exist as stable low-temperature phases and host composite quasiparticles which combine a fractional amount of fundamental
Z2 charge with a unit of topological charge. The fourfold topological ground state degeneracy on a torus admits a fermionic or semionic quasiparticle exchange statistics. The topological nontriviality of these states is visible in the existence of protected gapless edge modes. While the fermionic nematic and gapped
Z2 spin liquids have equivalent topological orders, they are still thermodynamically distinct due to having different edge modes, in analogy to the topologically nontrivial and trivial states of quantum spin-Hall systems. The analysis proceeds by recasting the
Z2 gauge theory of spin nematics as a continuum limit theory with a larger gauge structure. Nematic fractionalization parallels that of a quantum Hall liquid, but the large gauge symmetry restores the time-reversal symmetry and restricts the quasiparticle fusion rules and statistics. The conclusions from field theory analysis are complemented with the construction of plausible host microscopic models.
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.
We classify distinct types of quantum number fractionalization occurring in two-dimensional topologically ordered phases, focusing in particular on phases with
Z2 topological order, that is, on gapped
Z2 spin liquids. We find that the fractionalization class of each anyon is an equivalence class of projective representations of the symmetry group, corresponding to elements of the cohomology group
H2(G,Z2). This result leads us to a symmetry classification of gapped
Z2 spin liquids, such that two phases in different symmetry classes cannot be connected without breaking symmetry or crossing a phase transition. Symmetry classes are defined by specifying a fractionalization class for each type of anyon. The fusion rules of anyons play a crucial role in determining the symmetry classes. For translation and internal symmetries, braiding statistics plays no role, but can affect the classification when point group symmetries are present. For square lattice space group, time-reversal, and
SO(3) spin rotation symmetries, we find $2\phantom{\rule{0.16em}{0ex}}098\phantom{\rule{0.16em}{0ex}}176\ensuremath{\approx}{2}^{21}$ distinct symmetry classes. Our symmetry classification is not complete, as we exclude, by assumption, permutation of the different types of anyons by symmetry operations. We give an explicit construction of symmetry classes for square lattice space group symmetry in the toric code model. Via simple examples, we illustrate how information about fractionalization classes can, in principle, be obtained from the spectrum and quantum numbers of excited states. Moreover, the symmetry class can be partially determined from the quantum numbers of the four degenerate ground states on the torus. We also extend our results to arbitrary Abelian topological orders (limited, though, to translations and internal symmetries), and compare our classification with the related projective symmetry group classification of parton mean-field theories. Our results provide a framework for understanding and probing the sharp distinctions among symmetric
Z2 spin liquids and are a first step toward a full classification of symmetric topologically ordered phases.
Starting from an array of interacting fermionic quantum wires, we construct a family of non-Abelian topologically ordered states of matter in three spatial dimensions (3D). These states of matter inherit their non-Abelian topological properties from the
su(2)k conformal field theories that characterize the constituent interacting quantum wires in the decoupled limit. Thus, the resulting topological phases can be viewed as 3D generalizations of the (bosonic)
su(2)k Read-Rezayi sequence of fractional quantum Hall states. Focusing in detail on the
su(2)2 case, we first review how to determine the nature of the non-Abelian topological order (in particular, the topological degeneracy on the torus) in the two-dimensional (2D) case, before generalizing this approach to the 3D case. We also investigate the 2D boundary of the 3D phases, and show for the
su(2)2 case that there are anomalous gapless surface states protected by an analog of time-reversal symmetry, similar to the massless Dirac surface states of the noninteracting 3D topological insulator.