Bosonization of a two-level system with dissipation

We summarize in this work the relations of the spin-boson Hamiltonian, which was recently studied in connection with the phenomenon of quantum coherence in the presence of dissipation, to three different fermionic Hamiltonians. These relations were obtained through well-known equivalences between Fermi and Bose operators in one dimension. The fermionic Hamiltonians correspond to (a) a two-level system coupled linearly with a fermionic bath, (b) the resonant-level model, and (c) the anisotropic Kondo model. For the first Hamiltonian we reobtain and discuss the relationship between the dimensionless dissipation coefficient and the phase shift of the fermions. The resonant-level model allows us to study the properties of the two-level system for values of around (1/2). At = (1/2) the model reduces to the Toulouse limit, where the Hamiltonian is exactly soluble. A pure exponential decay is obtained for the relaxation of P (t) =〈ₙ (t) 〉 for t>0, given that for t<0 the system is known to be localized in one of the two states. The comparison with the Kondo model gives the long-time-limit behavior of the system for (1/2) 1 and provides a connection between universal numbers obtained previously for the spin-boson Hamiltonian and the Kondo model.