We introduce a spin-1/2 model in three dimensions which is a generalization of the well-known Kitaev model on a honeycomb lattice. Following Kitaev, we solve the model exactly by mapping it to a theory of non-interacting fermions in the background of a static Z_2 gauge field. The phase diagram consists of a gapped phase and a gapless one, similar to the two-dimensional case. Interestingly, unlike in the two-dimensional model, in the gapless phase the gap vanishes on a contour in the k space. Furthermore, we show that the flux excitations of the gauge field, due to some local constraints, form loop like structures; such loops exist on a lattice formed by the plaquettes in the original lattice and is topologically equivalent to the pyrochlore lattice. Finally, we derive a low-energy effective Hamiltonian that can be used to study the properties of the excitations in the gapped phase.Comment: 9 pages, 7 figures; published version; a new section and more references adde
We study the locations of the gapless points which occur for quantum spin chains of finite length (with a twisted boundary condition) at particular values of the nearest neighbor dimerization, as a function of the spin S and the number of sites. For strong dimerization and large values of S, a tunneling calculation reproduces the same results as those obtained from more involved field theoretic methods using the non-linear σ-model approach. A different analytical calculation of the matrix element between the two Néel states gives a set of gapless points; for strong dimerization, these differ significantly from the tunneling values. Finally, the exact diagonalization method for a finite number of sites yields a set of gapless points which are in good agreement with the Néel state calculations for all values of the dimerization, but the agreement with the tunneling values is not very good even for large S. This raises questions about possible corrections to the tunneling results.
We introduce the concept of super universality in quantum Hall liquids and spin liquids. This concept has emerged from previous studies of the quantum Hall effect and states that all the fundamental features of the quantum Hall effect are generically displayed as general topological features of the θ parameter in nonlinear σ models in two dimensions.To establish super universality in spin liquids we revisit the mapping by Haldane who argued that the anti ferromagnetic Heisenberg spin s chain in 1 + 1 space-time dimensions is effectively described by the O(3) nonlinear σ model with a θ term. By combining the path integral representation for the dimerized spin s = 1/2 chain with renormalization group decimation techniques we generalize the Haldane approach to include a more complicated theory, the fermionic rotor chain, involving four different renormalization group parameters. We show how the renormalization group calculation technique can be used to lay the bridge between the fermionic rotor chain and the O(3) nonlinear σ model with the θ term.As an integral and fundamental aspect of the mapping we establish the topological significance of the dangling spin at the edge of the chain. The edge spin in spin liquids is in all respects identical to the massless chiral edge excitations in quantum Hall liquids. We consider various different geometries of the spin chain such as open and closed chains, chains with an even and odd number of sides. We show that for each of the different geometries the θ term has a distinctly different physical meaning. We compare each case with a topologically equivalent quantum Hall liquid.
We study an exactly solvable toric code type of Hamiltonian in three dimensions, defined on the diamond lattice with spin-1/2 degrees of freedom at each site. The Hamiltonian is a sum of mutually commuting plaquette operators B p , all of which have eigenvalue +1 in the ground state. The excitations are "fluxes," which are plaquettes with B p = −1. Due to certain local kinematic constraints, fluxes form loops. The elementary flux-loop excitations are fermions, in contrast to other solvable spin-1/2 models in three dimensions, where the excitations are bosons. Furthermore, the flux loops braid nontrivially, giving rise to Abelian anyonlike statistics.
PACS 73.43.Cd -Quantum Hall effects: Theory and Modelling PACS 75.10.Jm -Quantized spin models PACS 11.10.Kk -Field theory in dimensions other than four Abstract. -The low energy dynamics of the anti-ferromagnetic Heisenberg spin S chain in the semiclassical limit S → ∞ is known to map onto the O(3) nonlinear σ model with a θ term in 1+1 dimension. Guided by the underlying dual symmetry of the spin chain, as well as the recently established topological significance of "dangling edge spins," we report an exact mapping onto the O(3) model that avoids the conventional large S approximation altogether. Our new methodology demonstrates all the super universal features of the θ angle concept that previously arose in the theory of the quantum Hall effect. It explains why Haldane's original ideas remarkably yield the correct answer in spite of the fundamental complications that generally exist in the idea of semiclassical expansions.
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