Journal of Statistical Physics

, Volume 41, Issue 3–4, pp 389–423 | Cite as

Quantum tunneling with dissipation and the Ising model over ℝ

  • Herbert Spohn1
  • Rolf Dümcke1


We consider a quantum particle in a double-well potential, for simplicity in the two-level approximation, coupled to a phonon field. We show that static and dynamical ground state correlations of the particle and of the field are expressible through expectations in an Ising model over ℝ (rather than ℤ). Its free measure is a spin flip process with flip rate ɛ, the difference in energy between the ground state and the first excited state. The Ising model has a ferromagnetic pair interaction whose form depends on the couplings to the phonon field and on the dispersion relation of the phonon field. In physical applications the interaction is long ranged and decays ast −2 for large distances. In this case we prove that for sufficiently strong coupling the particle becomes localized in one of the wells. The effective tunnel rate is zero. The transition to localization is associated with the generation of an infinite number of low momentum phonons. We apply the Ising technology to our problem and discuss the phase diagram in some detail.

Key words

Two-level system coupled to an ideal heat bath one-dimensional Ising model with long-range forces absence and existence of a phase transition (= degeneracy of the ground state) 


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Copyright information

© Plenum Publishing Corporation 1985

Authors and Affiliations

  • Herbert Spohn1
    • 1
  • Rolf Dümcke1
    • 1
  1. 1.Theoretische PhysikLudwig-Maximilians-UniversitÄt MünchenMünchen 2Federal Republic of Germany

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