Valence bond ground states in quantum antiferromagnets and quadratic algebras

The wavefunctions corresponding to the zero-energy eigenvalue of a one-dimensional quantum chain Hamiltonian can be written in a simple way using quadratic algebras. Hamiltonians describing stochastic processes have stationary states given by such wavefunctions and various quadratic algebras have been found and applied to several diffusion processes. We show that similar methods can also be applied for equilibrium processes. As an example, for a class of q-deformed O(N) symmetric antiferromagnetic quantum chains, we give the zero-energy wavefunctions for periodic boundary conditions corresponding to momenta zero and π. We also consider free and various non-diagonal boundary conditions and give the corresponding wavefunctions. All correlation lengths are derived.