Ground state of a spin-phonon system. I. Variational estimates
- 25 Downloads
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 wordsSpin phonon transition Spin phonon ground state
Unable to display preview. Download preview PDF.
- 1.A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. T. Fisher, A. Garg, and W. Zwerger,Rev. Mod. Phys. 59:1 (1987).Google Scholar
- 2.H. Spohn and R. Dumcke,J. Stat. Phys. 41:389 (1983).Google Scholar
- 3.V. J. Emery and A. Luther,Phys. Rev. B 9:215 (1974).Google Scholar
- 4.S. Chekravarty,Phys. Rev. Lett. 49:681 (1982).Google Scholar
- 5.R. Silbey and R. A. Harris,J. Chem. Phys. 80:2615 (1984).Google Scholar
- 6.A. Tanaka and A. Sakurai,Prog. Theor. Phys. 76:999 (1986).Google Scholar
- 7.P. Prelovsek,J. Phys. C 12:1855 (1979).Google Scholar
- 8.R. Beck, N. Götze, and P. Prelovsek,Phys. Rev. A 20:1140 (1979).Google Scholar
- 9.H. B. Shore and L. M. Sander,Phys. Rev. B 7:4537 (1973).Google Scholar
- 10.M. Wagner,Z. Physik B 37:225 (1979).Google Scholar