The spin-1/2 Heisenberg Chains in Constant Magnetic Field and Coherent states1

In the ferromagnetic Heisenberg chains of XXX and XXZ types with the hidden symmetries of Lie bi-algebra su(2) and quantum bi-algebra su_q(2), we show that at thermodynamic limit the algebra contractions give the boson algebra h(4) and the q-deformed boion algebra h_q(2) as the hidden symmetries respectively. The chains in constant magnetic field are studied and the ground states and lowest excited states are given explicitly with energy spectra. The phonon (or angular momentum) excitations are shown to be bosonic for the isotropic case and q-bosonic for the anisotropic case, and the ground states and lowest excited states of the systems of the chains in field are given explicitly. We give the phonon coherent states in the isotropic Heisenberg chain and the q-coherent states of the anisotropic chain at the thermodynamic limit. The q-coherent states are shown to be a squeezed states of phonon excitations.