Ground state of a spin-phonon system. II. Adiabatic limit

The phase transition for a spin in a magnetic field B coupled to acoustic phonons by a coupling constant α is studied. The case B much greater than 1 with an upper cutoff of unity for the phonons is studied systematically by using an adiabatic canonical transformation. In leading order the transition line is at γ = 2α/B = 1. In the normal phase (γ < 1) the ground-state energy is -B/2 plus a function of γ that is given explicitly as the solution of a pair theory. In the broken symmetry phase (γ > 1) the energy is the classical energy plus the same function of σ = 1/γ2. It is found that the first derivatives of the energy with respect to α and with respect to B have finite jumps across the transition line. Quantum fluctuations in both phases are treated. Higher-order terms are a series of powers of 1/B times functions of γ. The case of a small transverse field B is also studied. The sharp transition disappears and is replaced by rapid variation in a region of order (B→/B)/sup 2/3/ about γ = 1