Spin-Boson Model — Ohmic Bath & Kondo Mapping

The spin-boson model couples a two-level system (qubit) to an Ohmic bath of harmonic oscillators with spectral density J(ω) = 2α·ω·e^{-ω/ω_c}. The dimensionless coupling α controls a quantum phase transition: for α<1 the spin tunnels (delocalized phase); for α>1 the spin is trapped (localized phase). This maps exactly onto the Kondo problem: the bath modes play the role of conduction electrons, α↔J_K. The renormalized tunneling Δ_r ~ Δ·(Δ/ω_c)^{α/(1-α)} vanishes at α=1.