Journal of Statistical Physics

, Volume 54, Issue 1–2, pp 405–427 | Cite as

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

  • Eugene P. Gross
Article

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/(e2−1/4) 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.

Key words

Spin phonon transition Spin phonon ground state 

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Copyright information

© Plenum Publishing Corporation 1989

Authors and Affiliations

  • Eugene P. Gross
    • 1
  1. 1.Department of PhysicsBrandeis UniversityWaltham

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