Spin-Boson Model — Quantum Dissipation

Two-Level System in a Bosonic Bath: Decoherence & Localization Transition
Coupling α0.10
Bias ε0.0
Tunneling Δ1.0
Spectral density
Phase: Delocalized Renorm. Δ_r: — Coherence time τ: — α_c = 0.5
H = (ε/2)σ_z + (Δ/2)σ_x + Σ_k ω_k a†_k a_k + (σ_z/2) Σ_k λ_k(a†_k + a_k). The Ohmic spin-boson model has a quantum phase transition at α_c ≈ 0.5 (for Δ/ω_c → 0): below α_c the spin tunnels (delocalized phase); above α_c it localizes (⟨σ_z⟩ ≠ 0). For Ohmic bath: Δ_r = Δ(Δ/ω_c)^{α/(1-α)} — renormalized tunneling. Sub-Ohmic: localization for any α>0 at zero temperature. Super-Ohmic: always delocalized.