spinboson

The spin-boson model is a fundamental theoretical framework in quantum mechanics used to study dissipation and decoherence of a quantum two-level system (a spin-1/2 or qubit) coupled to an environment represented by a bath of harmonic oscillators. It is widely employed in condensed matter physics, quantum information, and chemical physics to understand how an open quantum system interacts with its surroundings.

The Hamiltonian typically takes the form H = -Δ/2 σx - ε/2 σz + ∑k ωk bk† bk + σz ∑k λk (bk + bk†), where Δ is the tunneling amplitude between the two levels, ε is a bias, σi are Pauli matrices, and bk†, bk are bosonic creation and annihilation operators for the bath modes with frequencies ωk. The environment is characterized by the spectral density J(ω) = ∑k λk^2 δ(ω - ωk), which in a continuum description is written as J(ω) ≈ 2 α ω^s ωc^{1-s} for ω ≪ ωc, with ωc a high-frequency cutoff. The exponent s defines the bath type: Ohmic (s = 1), sub-Ohmic (s < 1), or super-Ohmic (s > 1); α is a dimensionless coupling strength.

The spin-boson model captures how dissipation drives relaxation, dephasing, and, in certain regimes, quantum phase transitions between localized and delocalized spin states. In Ohmic environments, a transition at α ≈ 1 can occur for zero bias; sub-Ohmic baths can host a quantum critical point at finite α, while super-Ohmic baths typically exhibit weaker dissipation with no localization transition. The model is analyzed via path integrals and the Feynman-Vernon influence functional, with approximations such as the noninteracting blip approximation and numerically exact methods like NRG and QMC.

Historically introduced in the late 1980s by Leggett and colleagues, the spin-boson model remains a standard tool for exploring quantum dissipation, decoherence, and their implications for quantum devices.