Ground state of a spin-phonon system. I. Variational estimates

Abstract

A study is made of the ground-state energy of a spin-one-half particle in a fieldB and interacting with a phonon bath. The infrared-sensitive case of acoustic phonons with point coupling in three dimensions is characterized by two parameters, a coupling constant α andB. Units are used where the high-momentum phonon cutoff is unity. There is a curve α(B) separating a symmetry-breaking region with a long-range phonon field from a normal region. Two simple, well-known, approximations are compared. The source theory yields discontinuities in the first derivatives of the energy with respect toB and α whenB>e ⁻¹ and an infinite-order transition whenB<e ⁻¹, but is trivial in the large-α region. The classical theory yields discontinuities in the second derivatives but is trivial in the small-α region. An improved variationally fixed ground-state wave function is analyzed. It gives a new α(B) curve with an infinite-order transition with continuous energy derivatives whenB<e/(e ²−1/4) and with discontinuous derivatives whenB is larger than this value. It is nontrivial in the entire α(B) plane. The crossover to classical behavior occurs near α=1/2 forB≪1. But the wave function does not describe quantum fluctuations in the large-α phase. A second way of combining source and classical effects is described. It yields a second-order transition (near α=1/2 forB≪1) everywhere. These theories are special cases of a symmetry-breaking transformation together with a one-mode treatment of quantum fluctuations. The transition is viewed in terms of a single mode with a variable length, coupled dynamically to the spin.