We investigate a family of spin-
S chain Hamiltonians recently introduced by one of us [Greiter, Mapping of Parent Hamiltonians, Springer Tracts in Modern Physics, Vol. 244 (Springer, Berlin, 2011)]. For
S=1/2, it corresponds to the Haldane-Shastry model. For general spin
S, we find indication that the low-energy theory of these spin chains is described by the SU(2)
k Wess-Zumino-Witten model with coupling
k=2S. In particular, we investigate the
S=1 model whose ground state is given by a Pfaffian for even number of sites
N. We reconcile aspects of the spectrum of the Hamiltonian for arbitrary
N with trial states obtained by Schwinger projection of two Haldane-Shastry chains.
I present a family of one-dimensional time reversal lattice Hamiltonians whose exact ground states are bosonic non-Abelian fractional quantum Hall liquids with well defined total chiral momentum. These Hamiltonians describe k-hard-core bosons and involve long-range tunneling and interaction processes with an amplitude decaying with the chord distance. For k=2 the ground state is a p-wave paired phase corresponding to the Pfaffian quantum Hall state. This family of non-Abelian liquids is shown to be in one-to-one correspondence with a family of non-Abelian spin-k/2 liquids exhibiting hidden non-local order. They are total singlets made out of k indistinguishable Resonating Valence Bond states. The corresponding spin parent Hamiltonians are obtained.
We use a family of critical spin chain models discovered recently by one of us [M. Greiter, Mapping of Parent Hamiltonians (Springer, Berlin, 2011)] to propose and elaborate that non-Abelian, SU(2) level-
k=2S anyon statistics manifests itself in one dimension through topological selection rules for fractional shifts in the spacings of linear momenta, which yield an internal Hilbert space of
2n (in the thermodynamic-limit) degenerate states. These shifts constitute the equivalent to the fractional shifts in the relative angular momenta of anyons in two dimensions. We derive the rules first for Ising anyons, and then generalize them to SU(2) level-
k anyons. We establish a one-to-one correspondence between the topological choices for the momentum spacings and the fusion rules of spin-
21 spinons in the SU(2) level-
k Wess-Zumino-Witten model, where the internal Hilbert space is spanned by the manifold of allowed fusion trees in the Bratteli diagrams. Finally, we show that the choices in the fusion trees may be interpreted as the choices of different domain walls between the
2S+1 possible, degenerate dimer configurations of the spin-
S chains at the multicritical point.
Recently, the one dimensional model of
N spins with
S=21 on a circle, interacting with an exchange that falls off with the inverse square of the separation: $H_{\rm ISE} =\sum_{i\neq j} \frac{1}{[\frac{N}π \sin(\frac{i-j}{N}π)]^2}(\svec_i\cdot \svec_j-\frac{1}{4})$, or ISE-model, has received ample attention. Its special features include: relatively simple eigenfunctions, non-interacting elementary excitations that obey semionic statistics (spinons), and a large ``quantum group'' symmetry algebra called the Yangian. This model is fully integrable, albeit in a slightly different sense than the more traditional nearest neighbor exchange (NNE) Heisenberg chain. This thesis comes in 4 chapters. Chapter 1 introduces the model and presents the construction of a subset of the eigenfunctions. The other eigenfunctions are shown to be generated by the action of the Yangian symmetry algebra of
HISE. Chapter 2 presents a method to construct the set of constants of the motion of the ISE-model. The ISE-model is tractable enough to obtain its zero-magnetic-field dynamical structure factors. Chapter 3 attempts to extend this to a non-zero magnetic field, where, due to the presence of spinons in the groundstate, more complicated excitations contribute: small numbers of magnons and spinons. discuss the relation to the more complicated NNE-model. Finally chapter 4 illustrates how a recently conjectured new form of Off Diagonal Long Range Order in antiferromagnetic spin chains can be reinterpreted as a spinon propagator in the ISE-model, and verified numerically. We briefly comment on its relevance to stabilizing superconductivity in the layered cuprates.
To begin with, we introduce several exact models for SU(3) spin chains: First is a translationally invariant parent Hamiltonian involving four-site interactions for the trimer chain, with a threefold degenerate ground state. We provide numerical evidence that the elementary excitations of this model transform under representation
3 of SU(3) if the original spins of the model transform under representation 3. Second is a family of parent Hamiltonians for valence bond solids of SU(3) chains with spin representations 6, 10, and 8 on each lattice site. We argue that of these three models, only the latter two exhibit spinon confinement and, hence, a Haldane gap in the excitation spectrum. We generalize some of our models to
SU(n). Finally, we use the emerging rules for the construction of valence bond solid states to argue that models of antiferromagnetic chains of
SU(n) spins, in general, possess a Haldane gap if the spins transform under a representation corresponding to a Young tableau consisting of a number of boxes $\ensuremath{\lambda}$ which is divisible by
n. If $\ensuremath{\lambda}$ and
n have no common divisor, the spin chain will support deconfined spinons and not exhibit a Haldane gap. If $\ensuremath{\lambda}$ and
n have a common divisor different from
n, it will depend on the specifics of the model including the range of the interaction.