Applying the variational principle to a spin-boson Hamiltonian

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Abstract
Ground states of a spin‐boson Hamiltonian, describing o n e two‐level system (a spin) coupled to i n f i n i t e l y many harmonic oscillators (bosons), are studied. This spin‐boson Hamiltonian is a prototype for the description of a ‘‘s m a l l’’ s y s t e m (e.g., a m o l e c u l e) c o u p l e d t o i t s e n v i r o n m e n t. The respective ground‐state vector(s) are approximated by a linear combination of two c o h e r e n t‐state vectors corresponding to the two levels of the spin. Interest concentrates mainly on phase‐transition phenomena (generation of superselection rules) in case the parameters of the Hamiltonian (level splitting of the spin, frequencies of the field modes, and coupling constants) exhibit an infrared singularity. The resulting phase diagrams are shown to satisfy reasonably well the rigorous bounds derived by Spohn, and in particular distinguish between the superohmic, ohmic, and subohmic regime in the sense of Leggett. Nevertheless, the approximation method used is simple enough so that everything can be explicitly calculated. Former results by Pfeifer as well as results by Emery and Luther, Zwerger and Harris and Silbey are extended and discussed.

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