Quantum spin chains by effective Hamiltonian

The quantum thermodynamic properties of the ferromagnetic spin chain with easy-plane single-ion anisotropy are evaluated by a recently proposed theoretical method, called the pure-quantum self-consistent harmonic approximation (PQSCHA), that develops in the framework of the path-integral formulation of quantum statistical mechanics. Starting from the original Hamiltonian operator, one is lead to the practical use of an effective Hamiltonian, by means of which classical-like formulas for the quantum system can be written. For the spin operators we use the Villain transformation to canonical conjugate variables, in order to treat the quantum renormalizations within this framework, and in the resulting effective Hamiltonian we restore classical spin variables, according to the classical counterpart of the Villain transformation. The effective Hamiltonian bears the form of the classical counterpart of the original one, but with suitably renormalized values of applied field, anisotropy, and exchange constant. The thermodynamic quantities and (static) spin correlation functions can then be evaluated by the classical transfer matrix method, with little computational effort. We consider the case of the real ferromagnet CsNiF3, and show that the experimental data for this system, as well as quantum Monte Carlo data, are very well reproduced.