Theory of valence-bond lattice on spin-lattices

Quantum spin-lattice systems in low dimensions exhibit a variety of interesting zero-temperature phases, some of which show non-classical (i.e., non-magnetic) long-range orders, such as dimer or trimer valence-bond order. These symmetry-breaking systems with localized valence bonds are referred to as valence-bond lattices (VBL) in this article. A review of our systematic microscopic formalism based on a proper set of composite operators for the ground and excited states of the VBL systems is given. The one-dimensional (1D) spin-12 frustrated model is investigated in detail. Several possible VBL systems on the 1D spin-1 chains, the 2D square and {\it kagom\'e} lattices are also discussed. That our microscopic theory guarantees the rotational symmetry of the VBL systems is emphasized.