The S=1/2 Heisenberg Hamiltonian H=1/2${\mathcal{J}}_{\mathrm{n}=1\mathrm{}}^{\mathrm{N}\mathrm{-}1}$ ${\mathcal{J}}_{\mathit{m}=1\mathrm{}}^{\mathit{N}}$ ${\mathit{J}}_{\mathit{n}}$ ${\mathrm{\sigma }}_{\mathit{m}}$⋅ ${\mathrm{\sigma }}_{\mathit{m}+\mathit{n}}$, with $J_n$=$J_0$/${\sin }^2$(nπ/N), is shown to have a simple singlet ground state in the form of a Jastrow function. The spectrum and correlations are explicitly known and the magnetic susceptibility is shown to be Pauli type at T=0. The model has a striking similarity to the nearest-neighbor isotropic Heisenberg model and may be viewed as a discretized version of the Sutherland-Calogero-Moser system.

We identify the boundary energy of a many-body system of fermions on a lattice under twisted boundary conditions as the inverse of the effective charge-carrying mass, or the stiffness, renormalizing nontrivially under interactions due to the absence of Galilean invariance. We point out that this quantity is a sensitive and direct probe of the metal-insulator transitions possible in these systems, i.e., the Mott-Hubbard transition or Density-wave formation. We calculate exactly the stiffness, or the effective mass, in the 1D Heisenberg-Ising ring and the 1D Hubbard model by using the ansatz of Bethe. For the Hubbard ring we also calculate a spin stiffness by extending the nested ansatz of Bethe-Yang to this case.

The excitation spectrum of an antiferromagnetic chain exhibiting the effects of both dimerization and frustration is studied. The method used is based on an exact solution for the doubly degenerate ground state, and views the excitations as propagating defect boundaries between the two exact ground states. These excitations are analogous to "solitons," and can bind into a second type of excitation, analogous to "breathers."

We present a theory of Raman scattering in the Hubbard model. The scattering of light has two contributions. One gives rise to scattering by spin degrees of freedom in the insulating case where the general form of the scattering Hamiltonian is derived. The fluctuations of the ‘‘chiral’’ spin operator ${\mathrm{\Sigma }}_{{\mathit{S}}_{\mathit{i}}}$⋅(${\mathrm{S}}_{\mathit{i}}$×${\mathrm{S}}_{\mathit{k}}$) are shown to contribute in the ${\mathit{B}}_{2\mathit{g}}$ scattering geometry. The other contributes in the doped case and is shown to probe the fluctuations of the ‘‘stress tensor.’’ This quantity is not conserved, and hence its fluctuations at small q inherent in optical experiments need not be small, in striking contrast to density fluctuations in usual metals.

Pyrochlore magnets are candidates for what Harris et al. [Phys. Rev. Lett. 79, 2554 (1997)] call “spin-ice” behavior. We present theoretical simulations of relevance for the pyrochlore family $R_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ ( $R=\mathrm{rare}\mathrm{earth}$) supported by magnetothermal measurements on selected systems. By considering long-ranged dipole-dipole as well as short-ranged superexchange interactions, we get three distinct behaviors: (i) an ordered doubly degenerate state, (ii) a highly disordered state with a broad transition to paramagnetism, and (iii) a partially ordered state with a sharp transition to paramagnetism. Closely corresponding behavior is seen in the real compounds.