Quantum dissipation, J(ω)=2παω e^(−ω/ωc), Kosterlitz-Thouless quantum phase transition at α=1
J(ω) = 2παω exp(−ω/ωc) — ohmic spectral function; α = dimensionless coupling
Phase diagram: α < 1/2 → delocalized (tunneling preserved); α > 1 → localized (KT transition)
The spin-boson model is the paradigmatic open quantum system: a two-level system coupled to a harmonic oscillator bath. Ohmic dissipation (s=1) shows a quantum phase transition at α=1 (Kosterlitz-Thouless type). Below α=1/2: coherent oscillations. Between 1/2 and 1: incoherent tunneling. Above 1: localized, tunneling quenched. The effective tunneling Δ_eff = Δ(Δ/ωc)^(α/(1-α)) renormalizes to zero at the QPT.