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

Abstract

The phase transition for a spin in a magnetic fieldB coupled to acoustic phonons by a coupling constant α is studied. The caseB≫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 toB 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 fieldB is also studied. The sharp transition disappears and is replaced by rapid variation in a region of order (B₁/B)^2/3 about γ=1.