--- title: 'Ohmic Spin-Boson Model: Dynamics & Phase Transitions' url: https://www.emergentmind.com/topics/ohmic-spin-boson-model type: topic --- # Ohmic Spin-Boson Model: Dynamics & Phase Transitions The Ohmic spin-boson model is the standard dissipative two-level-system model in which a spin-\(\tfrac12\) impurity, qubit, or tunneling degree of freedom is linearly coupled to a bosonic bath whose low-frequency spectral density is linear in frequency, \(J(\omega)\propto \omega\). In common conventions this is written either as \(J(\omega)=2\pi\alpha\,\omega\,e^{-\omega/\omega_c}\) or, in a Schwinger–Keldysh formulation, \(f(\omega)=\alpha|\omega|\) with bath correlator \(\langle XX\rangle_\omega=f(\omega)\coth(\omega/2T)\) [2305.00110] [2501.12457]. The model is a canonical quantum-impurity problem because the competition between tunneling and dissipation produces renormalized dynamics, coherent-to-incoherent crossover, and a zero-temperature localization transition, while also admitting formulations in terms of anisotropic Kondo physics, boundary field theory, Majorana fermions, non-interacting blip approximation (NIBA), hierarchical equations of motion (HEOM), stochastic Schrödinger equations (SSE), and circuit-QED realizations [1106.2654] [1711.07463]. ## 1. Hamiltonian structure and meaning of the Ohmic bath A standard Hamiltonian is \[ H=\frac{\Delta}{2}\sigma_x+\sum_k \omega_k b_k^\dagger b_k+\frac{\sigma_z}{2}\sum_k \lambda_k(b_k^\dagger+b_k), \] with spectral density \[ J(\omega)=\pi\sum_k \lambda_k^2\delta(\omega-\omega_k)=2\pi\alpha\,\omega\,e^{-\omega/\omega_c}, \] where \(\Delta\) is the tunneling amplitude, \(\alpha\) the dimensionless coupling, and \(\omega_c\) a high-frequency cutoff [2305.00110]. Equivalent notational choices are widely used. For biased systems one often writes \[ H=\frac{\epsilon}{2}\sigma_z-\frac{\Delta}{2}\sigma_x+\sum_q \omega_q a_q^\dagger a_q+\frac{1}{2}\sigma_z\sum_q g_q(a_q+a_q^\dagger), \] with the same Ohmic low-frequency structure [1802.09846]. A Schwinger–Keldysh/Majorana formulation instead starts from \[ \mathcal H=-B S_z - b S_x + \lambda S_x X + \mathcal H_X, \qquad f(\omega)=\alpha |\omega|^s, \] so that the Ohmic specialization is \(s=1\) and \(f(\omega)=\alpha|\omega|\) [2501.12457]. The defining physical input is the bath spectral law. In the broader spin-boson taxonomy, \(s=1\) is Ohmic, \(01\) super-Ohmic [1106.2654]. The Ohmic case is therefore the borderline between the stronger infrared weight of sub-Ohmic baths and the weaker infrared dissipation of super-Ohmic baths. This borderline character is one reason the model is central in discussions of both equilibrium quantum criticality and real-time dissipative dynamics [2402.18561]. ## 2. Phase structure, renormalization-group flows, and the order of the transition In the canonical picture, the Ohmic model has a delocalized phase dominated by tunneling and a localized phase in which dissipation suppresses tunneling. A Majorana-based perturbative RG treatment yields \[ \frac{d\alpha}{d\ell}=-\alpha h^2, \qquad \frac{dh}{d\ell}=(1-\alpha)h, \] where \(h\) is the dimensionless transverse field. For small nonzero \(\Delta\), the critical dissipation is \[ \alpha_c=1, \] and the transition is described as Kosterlitz–Thouless (KT), with a separatrix and a line of fixed points on the localized side [1106.2654]. The same Ohmic regime is exactly mappable to the anisotropic Kondo model, and in the boundary-field-theory language the KT point occurs at \(g=1\) [1506.01358]. The coherent–incoherent crossover is distinct from the localization transition. Near the Ohmic limit, transient real-time studies identify a frequency-driven incoherence criterion \(\Omega/\Delta\to 0\) and note that, for the Ohmic model, the incoherence transition is known analytically at the Toulouse point \[ \alpha=0.5, \] whereas localization in the scaling limit remains associated with the stronger-coupling regime near \(\alpha_c\simeq 1\) [2402.18561] [2302.01908]. This separation between incoherence and localization is essential: overdamping and tunneling suppression are not interchangeable diagnostics. A more recent Schwinger–Keldysh/Majorana treatment proposes a different interpretation. Specializing its one-loop flow to \(s=1\) gives \[ \frac{d\lambda}{d\ln\Lambda}\approx \lambda\left(\frac12-\frac{\lambda^2\alpha}{2\pi}\right), \] with fixed point \[ b^*=0,\qquad \lambda^*=\sqrt{\pi/\alpha}, \] and interprets the Ohmic localization transition as a continuous second-order quantum phase transition rather than a BKT transition [2501.12457]. In that framework the critical magnetization exponent obeys \[ \frac{1}{\delta}=\frac{1-s}{1+s}, \] so that in the Ohmic limit \(1/\delta=0\), which is taken to reflect the logarithmic character of the transition [2501.12457]. | Aspect | Canonical interpretation | Keldysh/Majorana critical-dynamics interpretation | |---|---|---| | Zero-temperature Ohmic transition | KT at \(\alpha_c=1\) for small nonzero \(\Delta\) [1106.2654] | Continuous second order with \(\lambda^*=\sqrt{\pi/\alpha}\) [2501.12457] | | Critical structure | Separatrix and line of fixed points [1106.2654] | Wilson–Fisher-like fixed point [2501.12457] | | Ohmic order-parameter scaling | No ordinary power-law singularity emphasized [1106.2654] | \(1/\delta=0\) in the Ohmic limit [2501.12457] | This disagreement is an active interpretive issue rather than a settled replacement of the standard KT picture. A plausible implication is that the inferred order of the Ohmic transition depends sensitively on formulation, infrared treatment, and the status assigned to quantum-to-classical mapping. ## 3. Real-time dynamics, renormalized tunneling, and crossover structure For the zero-temperature Ohmic bath in the regime \[ 0<\alpha<\frac12, \] the spin exhibits underdamped oscillations with strongly renormalized frequency \[ \Delta_r=\Delta\left(\frac{\Delta}{\omega_c}\right)^{\frac{\alpha}{1-\alpha}}, \] a standard manifestation of non-Markovian bath dressing [2305.00110]. When additional Lindblad channels are added on top of the Ohmic bath, the interplay is highly anisotropic: for \(L_{\rm dph}=\sqrt{\Gamma_\phi}\sigma_z\), \[ \Omega \approx \sqrt{\Omega_0^2-\Gamma_\phi^2}, \qquad \gamma \approx \gamma_0+\Gamma_\phi, \] whereas for \(L_x=\sqrt{\Gamma_x}\sigma_x\), \[ \Omega \approx \Omega_0, \qquad \gamma \approx \gamma_0+\frac{3}{2}\Gamma_x. \] Thus \(z\)-dephasing reduces the oscillation frequency and can drive overdamping, while \(x\)-depolarization primarily broadens the decay rate [2305.00110]. At weak coupling and arbitrary bias, real-time RG and renormalized perturbation theory show that the reduced density matrix contains one purely decaying mode and two oscillatory modes, \[ z_0=-i\Gamma,\qquad z_\pm=\pm\Omega-\frac{i}{2}\Gamma, \] with self-consistent renormalized tunneling \[ \tilde{\Delta}=\Delta\left(\frac{\Omega}{D}\right)^\alpha, \qquad \Omega=\sqrt{\epsilon^2+\tilde{\Delta}^2}, \] and decay rate \[ \Gamma=\pi\alpha\,\frac{\tilde{\Delta}^2}{\Omega}. \] The same analysis finds logarithmic corrections, bias-dependent long-time power laws, and \(1/t\) tails that are absent in the unbiased case [1802.09846]. Zero-temperature HEOM calculations connect these dynamical features to spectral observables. In the Ohmic case the linear absorption spectrum is dominated by a single peak that moves from \(\omega\approx 2\Delta\) toward \(\omega\approx 0\) as \(\alpha\) increases, while the short-time oscillation period grows and the motion becomes overdamped. In that study the coherent–incoherent transition occurs at approximately \[ \alpha_{\rm CI}\approx 0.5 \] in their numerics, whereas the delocalized–localized transition is expected near \(\alpha_c\simeq 1\) in the scaling limit [2302.01908]. This reinforces the separation between loss of oscillatory coherence and true localization. ## 4. Correlation functions, susceptibilities, and heat transport A Green’s-function treatment based on Majorana fermions and a polaron transformation yields closed expressions for the symmetrized spin correlation function (SSCF) and susceptibility. The central SSCF formula is \[ S_z(\omega)=\frac{2\Gamma}{(\omega-2\Lambda)^2+\Gamma^2}, \] where \(\Gamma(\omega)\) and \(\Lambda(\omega)\) are determined by bath correlation functions after the transformed interaction is treated perturbatively [1608.04854]. In the unbiased Ohmic case, the kernel becomes identical to the NIBA result at the SSCF level, while in biased systems the same framework is stated to remain reliable over a wider temperature range than NIBA, especially in the quasi-elastic low-frequency sector [1608.04854]. The low-frequency relation between bath and spin response is also captured by Shiba-type structure. In the delocalized phase one has \[ C(\nu)=\frac14 J(|\nu|)\,[\chi_z'(0)]^2, \] so for the Ohmic bath the spin fluctuation spectrum inherits the linear bath spectrum [1106.2654]. In the quantum critical regime the longitudinal susceptibility scales as \[ \chi_z(T)\sim \frac{1}{T^s}, \] which for \(s=1\) becomes \(\chi_z(T)\sim 1/T\) [1106.2654]. For non-equilibrium heat transport, the Ohmic spin-boson model with two reservoirs is treated in the incoherent tunneling regime \(\omega_c\gg \Delta\). In linear response the exact thermal conductance benchmark is \[ \kappa = k_B\hbar \frac{\alpha_L\alpha_R}{\alpha_L+\alpha_R} \int_0^{\omega_c} d\omega\, S_\alpha(\omega)\,\omega^2 \left[\frac{\beta\hbar\omega/2}{\sinh(\beta\hbar\omega/2)}\right]^2, \] while NIBA gives a closed conductance formula valid for \[ T>T_K, \] with \[ T_K = \frac{\hbar\Delta}{k_B} \left(\frac{\Delta}{\omega_c}\right)^{\alpha/(1-\alpha)} \left[\Gamma(1-2\alpha)\cos(\pi\alpha)\right]^{1/[2(1-\alpha)]} \] for \(0<\alpha<1\) [1405.6358]. In the weak-coupling limit the result reduces to the Born–Markov form, whereas at strong coupling/high temperature the conductance follows \[ \kappa \simeq \mathcal A \frac{k_B\Delta^2}{\omega_c} \left(\frac{\hbar\omega_c}{k_B T}\right)^{1-2\alpha}, \] equivalently \(\kappa\sim T^{2\alpha-1}\) [1405.6358]. ## 5. Integrability, boundary field theory, and orthogonality The Ohmic model is also an integrable boundary quantum field theory. In one formulation the Hamiltonian is \[ {\boldsymbol H}= { H}_{\rm free }\,\sigma_0 - \Big(\frac{1}{2}\Pi_B+h\Big)\sigma_3-J\sigma_1, \] with the bath represented by a massless Gaussian field on the half-line and the dissipation parameter denoted by \(g\) [1506.01358]. In the delocalized regime \[ 0