Some properties of frustrated spin systems: extensions and applications of Lieb-Schupp approach

Lieb and Schupp have obtained a number of ground-state properties for frustrated Heisenberg models. The basic tool used was certain version of spin-reflection positivity method. One group of these results is related to singlet nature of ground state. It needs an assumption of reflection symmetry present in the system. In this paper, it is shown that analogous results hold also for other symmetries (inversion etc.). The second Lieb-Schupp result is matrix inequality, which imply inequalities between ground-state energies of certain systems. In the paper, the Lieb-Schupp inequality is applied to relate ground-state energies of various systems: spin chains, ladders and multidimensional lattices.