Ohmic Spin-Boson Model: Dynamics & Phase Transitions

• The Ohmic spin-boson model is a canonical dissipative two-level system featuring a linear bosonic bath spectral density that drives quantum critical behavior.
• It captures key phenomena including a coherent–incoherent crossover and localization transition, with interpretations via KT or continuous second-order frameworks.
• Real-time dynamics reveal renormalized tunneling and non-Markovian effects, with experimental realizations in circuit-QED and superconducting platforms.

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 $\langle XX\rangle_\omega=f(\omega)\coth(\omega/2T)$ (Kamar et al., 2023, Vasin et al., 21 Jan 2025). 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 (Florens et al., 2011, Leppäkangas et al., 2017).

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 (Kamar et al., 2023). Equivalent notational choices are widely used. For biased systems one often writes

$J(\omega)\propto \omega$0

with the same Ohmic low-frequency structure (Lindner et al., 2018). A Schwinger–Keldysh/Majorana formulation instead starts from

$J(\omega)\propto \omega$1

so that the Ohmic specialization is $J(\omega)\propto \omega$2 and $J(\omega)\propto \omega$3 (Vasin et al., 21 Jan 2025).

The defining physical input is the bath spectral law. In the broader spin-boson taxonomy, $J(\omega)\propto \omega$4 is Ohmic, $J(\omega)\propto \omega$5 sub-Ohmic, and $J(\omega)\propto \omega$6 super-Ohmic (Florens et al., 2011). 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 (Goulko et al., 2024).

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

$J(\omega)\propto \omega$7

where $J(\omega)\propto \omega$8 is the dimensionless transverse field. For small nonzero $J(\omega)\propto \omega$9, the critical dissipation is

$J(\omega)=2\pi\alpha\,\omega\,e^{-\omega/\omega_c}$0

and the transition is described as Kosterlitz–Thouless (KT), with a separatrix and a line of fixed points on the localized side (Florens et al., 2011). The same Ohmic regime is exactly mappable to the anisotropic Kondo model, and in the boundary-field-theory language the KT point occurs at $J(\omega)=2\pi\alpha\,\omega\,e^{-\omega/\omega_c}$1 (Lukyanov, 2015).

The coherent–incoherent crossover is distinct from the localization transition. Near the Ohmic limit, transient real-time studies identify a frequency-driven incoherence criterion $J(\omega)=2\pi\alpha\,\omega\,e^{-\omega/\omega_c}$2 and note that, for the Ohmic model, the incoherence transition is known analytically at the Toulouse point

$J(\omega)=2\pi\alpha\,\omega\,e^{-\omega/\omega_c}$3

whereas localization in the scaling limit remains associated with the stronger-coupling regime near $J(\omega)=2\pi\alpha\,\omega\,e^{-\omega/\omega_c}$4 (Goulko et al., 2024, Wang et al., 2023). 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 $J(\omega)=2\pi\alpha\,\omega\,e^{-\omega/\omega_c}$5 gives

$J(\omega)=2\pi\alpha\,\omega\,e^{-\omega/\omega_c}$6

with fixed point

$J(\omega)=2\pi\alpha\,\omega\,e^{-\omega/\omega_c}$7

and interprets the Ohmic localization transition as a continuous second-order quantum phase transition rather than a BKT transition (Vasin et al., 21 Jan 2025). In that framework the critical magnetization exponent obeys

$J(\omega)=2\pi\alpha\,\omega\,e^{-\omega/\omega_c}$8

so that in the Ohmic limit $J(\omega)=2\pi\alpha\,\omega\,e^{-\omega/\omega_c}$9, which is taken to reflect the logarithmic character of the transition (Vasin et al., 21 Jan 2025).

Aspect	Canonical interpretation	Keldysh/Majorana critical-dynamics interpretation
Zero-temperature Ohmic transition	KT at $f(\omega)=\alpha|\omega|$0 for small nonzero $f(\omega)=\alpha|\omega|$1 (Florens et al., 2011)	Continuous second order with $f(\omega)=\alpha|\omega|$2 (Vasin et al., 21 Jan 2025)
Critical structure	Separatrix and line of fixed points (Florens et al., 2011)	Wilson–Fisher-like fixed point (Vasin et al., 21 Jan 2025)
Ohmic order-parameter scaling	No ordinary power-law singularity emphasized (Florens et al., 2011)	$f(\omega)=\alpha|\omega|$3 in the Ohmic limit (Vasin et al., 21 Jan 2025)

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

$f(\omega)=\alpha|\omega|$4

the spin exhibits underdamped oscillations with strongly renormalized frequency

$f(\omega)=\alpha|\omega|$5

a standard manifestation of non-Markovian bath dressing (Kamar et al., 2023). When additional Lindblad channels are added on top of the Ohmic bath, the interplay is highly anisotropic: for $f(\omega)=\alpha|\omega|$6,

$f(\omega)=\alpha|\omega|$7

whereas for $f(\omega)=\alpha|\omega|$8,

$f(\omega)=\alpha|\omega|$9

Thus $\langle XX\rangle_\omega=f(\omega)\coth(\omega/2T)$0-dephasing reduces the oscillation frequency and can drive overdamping, while $\langle XX\rangle_\omega=f(\omega)\coth(\omega/2T)$1-depolarization primarily broadens the decay rate (Kamar et al., 2023).

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,

$\langle XX\rangle_\omega=f(\omega)\coth(\omega/2T)$2

with self-consistent renormalized tunneling

$\langle XX\rangle_\omega=f(\omega)\coth(\omega/2T)$3

and decay rate

$\langle XX\rangle_\omega=f(\omega)\coth(\omega/2T)$4

The same analysis finds logarithmic corrections, bias-dependent long-time power laws, and $\langle XX\rangle_\omega=f(\omega)\coth(\omega/2T)$5 tails that are absent in the unbiased case (Lindner et al., 2018).

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 $\langle XX\rangle_\omega=f(\omega)\coth(\omega/2T)$6 toward $\langle XX\rangle_\omega=f(\omega)\coth(\omega/2T)$7 as $\langle XX\rangle_\omega=f(\omega)\coth(\omega/2T)$8 increases, while the short-time oscillation period grows and the motion becomes overdamped. In that study the coherent–incoherent transition occurs at approximately

$\langle XX\rangle_\omega=f(\omega)\coth(\omega/2T)$9

in their numerics, whereas the delocalized–localized transition is expected near $$H=\frac{\Delta}{2}\sigma_x+\sum_k \omega_k b_k^\dagger b_k+\frac{\sigma_z}{2}\sum_k \lambda_k(b_k^\dagger+b_k),$$0 in the scaling limit (Wang et al., 2023). 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

$$H=\frac{\Delta}{2}\sigma_x+\sum_k \omega_k b_k^\dagger b_k+\frac{\sigma_z}{2}\sum_k \lambda_k(b_k^\dagger+b_k),$$1

where $$H=\frac{\Delta}{2}\sigma_x+\sum_k \omega_k b_k^\dagger b_k+\frac{\sigma_z}{2}\sum_k \lambda_k(b_k^\dagger+b_k),$$2 and $$H=\frac{\Delta}{2}\sigma_x+\sum_k \omega_k b_k^\dagger b_k+\frac{\sigma_z}{2}\sum_k \lambda_k(b_k^\dagger+b_k),$$3 are determined by bath correlation functions after the transformed interaction is treated perturbatively (Liu et al., 2016). 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 (Liu et al., 2016).

The low-frequency relation between bath and spin response is also captured by Shiba-type structure. In the delocalized phase one has

$$H=\frac{\Delta}{2}\sigma_x+\sum_k \omega_k b_k^\dagger b_k+\frac{\sigma_z}{2}\sum_k \lambda_k(b_k^\dagger+b_k),$$4

so for the Ohmic bath the spin fluctuation spectrum inherits the linear bath spectrum (Florens et al., 2011). In the quantum critical regime the longitudinal susceptibility scales as

$$H=\frac{\Delta}{2}\sigma_x+\sum_k \omega_k b_k^\dagger b_k+\frac{\sigma_z}{2}\sum_k \lambda_k(b_k^\dagger+b_k),$$5

which for $$H=\frac{\Delta}{2}\sigma_x+\sum_k \omega_k b_k^\dagger b_k+\frac{\sigma_z}{2}\sum_k \lambda_k(b_k^\dagger+b_k),$$6 becomes $$H=\frac{\Delta}{2}\sigma_x+\sum_k \omega_k b_k^\dagger b_k+\frac{\sigma_z}{2}\sum_k \lambda_k(b_k^\dagger+b_k),$$7 (Florens et al., 2011).

For non-equilibrium heat transport, the Ohmic spin-boson model with two reservoirs is treated in the incoherent tunneling regime $$H=\frac{\Delta}{2}\sigma_x+\sum_k \omega_k b_k^\dagger b_k+\frac{\sigma_z}{2}\sum_k \lambda_k(b_k^\dagger+b_k),$$8. In linear response the exact thermal conductance benchmark is

$$H=\frac{\Delta}{2}\sigma_x+\sum_k \omega_k b_k^\dagger b_k+\frac{\sigma_z}{2}\sum_k \lambda_k(b_k^\dagger+b_k),$$9

while NIBA gives a closed conductance formula valid for

$$J(\omega)=\pi\sum_k \lambda_k^2\delta(\omega-\omega_k)=2\pi\alpha\,\omega\,e^{-\omega/\omega_c},$$0

with

$$J(\omega)=\pi\sum_k \lambda_k^2\delta(\omega-\omega_k)=2\pi\alpha\,\omega\,e^{-\omega/\omega_c},$$1

for $$J(\omega)=\pi\sum_k \lambda_k^2\delta(\omega-\omega_k)=2\pi\alpha\,\omega\,e^{-\omega/\omega_c},$$2 (Segal, 2014). In the weak-coupling limit the result reduces to the Born–Markov form, whereas at strong coupling/high temperature the conductance follows

$$J(\omega)=\pi\sum_k \lambda_k^2\delta(\omega-\omega_k)=2\pi\alpha\,\omega\,e^{-\omega/\omega_c},$$3

equivalently $$J(\omega)=\pi\sum_k \lambda_k^2\delta(\omega-\omega_k)=2\pi\alpha\,\omega\,e^{-\omega/\omega_c},$$4 (Segal, 2014).

5. Integrability, boundary field theory, and orthogonality

The Ohmic model is also an integrable boundary quantum field theory. In one formulation the Hamiltonian is

$$J(\omega)=\pi\sum_k \lambda_k^2\delta(\omega-\omega_k)=2\pi\alpha\,\omega\,e^{-\omega/\omega_c},$$5

with the bath represented by a massless Gaussian field on the half-line and the dissipation parameter denoted by $$J(\omega)=\pi\sum_k \lambda_k^2\delta(\omega-\omega_k)=2\pi\alpha\,\omega\,e^{-\omega/\omega_c},$$6 (Lukyanov, 2015). In the delocalized regime

$$J(\omega)=\pi\sum_k \lambda_k^2\delta(\omega-\omega_k)=2\pi\alpha\,\omega\,e^{-\omega/\omega_c},$$7

the RG equation

$$J(\omega)=\pi\sum_k \lambda_k^2\delta(\omega-\omega_k)=2\pi\alpha\,\omega\,e^{-\omega/\omega_c},$$8

trades the bare tunneling for the invariant scale

$$J(\omega)=\pi\sum_k \lambda_k^2\delta(\omega-\omega_k)=2\pi\alpha\,\omega\,e^{-\omega/\omega_c},$$9

which is the impurity scale controlling the boundary flow (Lukyanov, 2015).

Within this integrable structure, ground-state overlaps display Anderson orthogonality. If $\Delta$0 and $\Delta$1 are vacua at different couplings, then

$\Delta$2

with orthogonality exponent

$\Delta$3

At the Toulouse limit $\Delta$4, the same exponent coincides with the resonant-level-model phase-shift expression (Lukyanov, 2015). This places the Ohmic spin-boson model in the same exact-structure class as the anisotropic Kondo and resonant-level models, with fidelity amplitudes constrained by Yang–Baxter relations and quantum Jost-operator algebra.

6. Bath-side criticality, dense-spectrum numerics, and experimental realizations

Dense-spectrum variational simulations emphasize that accurate Ohmic criticality requires approaching the continuum limit with $\Delta$5. Using a coherent-state expansion with $\Delta$6 and $\Delta$7, the unbiased Ohmic model exhibits a sharp symmetry change diagnosed by

$\Delta$8

with the symmetry parameter $\Delta$9 jumping from 1 in the delocalized phase to 0 in the localized phase. For $\alpha$0 and $\alpha$1, the critical coupling is found near

$\alpha$2

and extrapolation to $\alpha$3 gives

$\alpha$4

consistent with the expected Ohmic result $\alpha$5 (Qian et al., 2021). The same work identifies bath criticality directly through displacements, uncertainty products, and induced bath-mode correlations, with an emergent scale $\alpha$6 and associated correlation length

$\alpha$7

and reports effective exponents $\alpha$8 in the delocalized phase that become effectively zero at criticality (Qian et al., 2021).

On the experimental side, a superconducting flux qubit in an open transmission line directly realizes the Ohmic model. In that system the bare line obeys

$\alpha$9

over the measured $\omega_c$0 GHz range, with

$\omega_c$1

showing broadband Ohmic behavior (Haeberlein et al., 2015). Introducing partial reflectors converts the nearly featureless Ohmic bath into a structured reservoir with a peaked $\omega_c$2, allowing direct spectroscopy of the spectral function and controlled departure from pure Ohmicity (Haeberlein et al., 2015).

A complementary circuit-QED proposal engineers an effective Ohmic bath for a transmon via many broadened microwave resonators, with

$\omega_c$3

Using two-tone driving, the rotating-frame Hamiltonian acquires

$\omega_c$4

and the effective bath can be made approximately linear over a narrow MHz band (Leppäkangas et al., 2017). In the numerical design example the bath is synthesized from $\omega_c$5 dissipative resonators with internal quality factor $\omega_c$6, illustrating a route toward strong-coupling quantum simulation of the Ohmic problem (Leppäkangas et al., 2017).

The Ohmic spin-boson model therefore occupies a dual role. It is simultaneously a canonical dissipative two-level-system model with immediate relevance to qubits, transport, and spectroscopy, and a mathematically structured impurity theory with RG flows, exact mappings, integrability, and unresolved questions about the precise infrared nature of the localization transition. The combination of KT phenomenology, bath-induced renormalization, boundary-field-theory structure, and experimental controllability explains why it remains a central reference point across condensed matter, AMO, and quantum-information research (Florens et al., 2011, Vasin et al., 21 Jan 2025).