On the basis of Bogoliubov's 1/k2 theorem the relation between the long-range order in the ground state and the collective mode, namely, the Goldstone particle is investigated. It is pointed out that Bogoliubov's inequality rules out the long-range orders in the ground states of the isotropic Heisenberg model, the hafl-filled Hubbard model and the interacting Bose system for one dimension while it admits the long-range orders for two dimensions. The present proof is based on the fact that the lowest-excited state that can be regarded as the Goldstone particle has the energy E(k)∝|k| for small k. This energy spectrum is exactly given in the one-dimensional models and is shown to be proven in the ordered state on a reasonable assumption except for the ferromagnetic case.
We apply the method of infrared bounds with the technique of Kennedy, Lieb and Shastry to the isotropic spin-1 Hamiltonian with bilinear (- J ) and biquadratic (- J ') exchange interactions to examine the existence of antiferro-dipole long-range order (DLRO) and ferro-quadrupole long-range order (QLRO). We prove that DLRO exists at zero temperature in two and three dimensions for 0⩽ J '<-0.188 J and 0⩽ J '<-1.954 J , respectively. In three dimensions we also prove the existence of QLRO in the ground state for 0<2 J ⩽ J '<2.664 J .
Randomly diluted quantum boson and spin models in two dimensions combine the physics of classical percolation with the well-known dimensionality dependence of ordering in quantum lattice models. This combination is rather subtle for models that order in two dimensions but have no true order in one dimension, as the percolation cluster near threshold is a fractal of dimension between 1 and 2: two experimentally relevant examples are the
O(2) quantum rotor and the Heisenberg antiferromagnet. We study two analytic descriptions of the
O(2) quantum rotor near the percolation threshold. First a spin-wave expansion is shown to predict long-ranged order, but there are statistically rare points on the cluster that violate the standard assumptions of spin-wave theory. A real-space renormalization group (RSRG) approach is then used to understand how these rare points modify ordering of the
O(2) rotor. A new class of fixed points of the RSRG equations for disordered one-dimensional bosons is identified and shown to support the existence of long-range order on the percolation backbone in two dimensions. These results are relevant to experiments on bosons in optical lattices and superconducting arrays, and also (qualitatively) for the diluted Heisenberg antiferromagnet ${\mathrm{La}}_{2}{(\mathrm{Zn},\mathrm{Mg})}_{x}{\mathrm{Cu}}_{1\ensuremath{-}x}{\mathrm{O}}_{4}$.