Spin-Boson Model

The spin-boson model is the canonical model of a two-level system coupled to a bosonic environment. It is the two-state analogue of coordinate-bath models such as the Caldeira–Leggett Model: instead of a continuous coordinate $q$, the system has two localized alternatives represented by Pauli operators.

The model is used for dissipative tunneling, electron transfer, qubit dephasing and relaxation, defects in solids, molecular excitons, superconducting circuits, and conceptual studies of quantum dissipation.

Hamiltonian

A standard spin-boson Hamiltonian is

$$H = H_S + H_B + H_I,$$

with

$$H_S = \frac{\epsilon}{2}\sigma_z - \frac{\Delta}{2}\sigma_x.$$

Here:

• $\epsilon$ is the bias between the two localized states;
• $\Delta$ is the tunneling amplitude;
• $\sigma_z$ distinguishes the localized alternatives;
• $\sigma_x$ mixes them.

The bath is a set of harmonic modes,

$$H_B = \sum_k \hbar\omega_k b_k^\dagger b_k,$$

and a common interaction is longitudinal in the localized basis:

$$H_I = \frac{\sigma_z}{2} \sum_k g_k \left( b_k+b_k^\dagger \right).$$

The bath therefore couples to “which localized state” information. If tunneling is present, this same coupling can also induce relaxation in the energy eigenbasis.

Spectral Density

The environment is summarized by a spectral density. One common convention is

$$J(\omega) = \pi \sum_k g_k^2 \delta(\omega-\omega_k), \qquad \omega>0.$$

Conventions differ by factors of $\pi$, $2$, and $\hbar$. Always compare formulas using the same definition of $J(\omega)$. See Spectral Densities for the general convention map.

A widely studied family is

$$J(\omega) = 2\pi\alpha\, \omega_c^{1-s} \omega^s e^{-\omega/\omega_c}, \qquad \omega>0,$$

where $\omega_c$ is a cutoff, $\alpha$ is a dimensionless coupling strength in the Ohmic case, and $s$ classifies the low-frequency bath:

Regime	Low-frequency behavior	Typical memory
sub-Ohmic	$0\lt s\lt1$	strong low-frequency weight
Ohmic	$s=1$	friction-like damping
super-Ohmic	$s>1$	suppressed low-frequency noise

The low-frequency behavior matters because the spin couples to a slow coordinate-like bath variable. It affects dephasing, tunneling renormalization, and possible localization transitions.

Localized and Energy Bases

The localized basis is natural for the coupling, because $H_I$ is proportional to $\sigma_z$. The energy basis is natural for relaxation. The system Hamiltonian has splitting

$$E = \sqrt{\epsilon^2+\Delta^2}.$$

Define a mixing angle $\theta$ by

$$\cos\theta=\frac{\epsilon}{E}, \qquad \sin\theta=\frac{\Delta}{E}.$$

In the energy basis, with Pauli operators $\tau_z,\tau_x$, the coupling operator becomes

$$\sigma_z = \cos\theta\,\tau_z + \sin\theta\,\tau_x,$$

up to a sign convention for $\tau_x$. Thus the same bath has two roles:

• the $\tau_z$ component produces dephasing in the energy basis;
• the $\tau_x$ component produces transitions between energy eigenstates.

This is why spin-boson physics includes both dephasing and dissipation.

Weak-Coupling Rates

In weak-coupling Markovian theory, the bath noise at the transition frequency controls relaxation, while low-frequency noise controls pure dephasing. Schematically,

$$\Gamma_1 \propto \sin^2\theta\, S_B(E/\hbar),$$

and

$$\Gamma_\phi \propto \cos^2\theta\, S_B(0).$$

The exact prefactors depend on the spectral convention and whether ordered, symmetrized, one-sided, or two-sided spectra are used. At thermal equilibrium, upward and downward transition rates obey Detailed Balance:

$$\frac{\Gamma_\uparrow}{\Gamma_\downarrow} = e^{-\beta E}.$$

This weak-coupling picture is extremely useful, but it is not the whole spin-boson model. Strong coupling and structured baths can invalidate simple rates.

Pure-Dephasing Limit

If $\Delta=0$, then

$$[H_S,\sigma_z]=0.$$

The bath distinguishes the two localized states but does not drive transitions between them. Populations in the $\sigma_z$ basis are constant, while coherences decay.

In one common continuum convention, the coherence takes the form

$$\rho_{+,-}(t) = \rho_{+,-}(0) e^{-i\epsilon t/\hbar} e^{-\Phi(t)},$$

with

$$\Phi(t) = \frac{1}{\pi} \int_0^\infty d\omega\, \frac{J(\omega)}{\omega^2} \left[ 1-\cos(\omega t) \right] \coth \left( \frac{\beta\hbar\omega}{2} \right).$$

This exactly solvable limit is a useful benchmark for dephasing models. The canonical exact-model discussion is Pure Dephasing Model. For the Markovian generator version, see Pure Dephasing Master Equation.

Tunneling Suppression

When $\Delta\ne0$, the bath competes with tunneling. The tunneling term tries to form coherent superpositions of the localized states. The bath coupling to $\sigma_z$ tends to monitor or dress those localized alternatives.

At weak coupling, this produces damping and dephasing of coherent oscillations. At stronger coupling, the bath can renormalize the effective tunneling amplitude downward. In the idealized unbiased Ohmic model at zero temperature, the standard continuum theory has:

• a coherent-to-incoherent crossover near $\alpha=1/2$;
• a localization transition near $\alpha=1$.

These statements belong to the ideal Ohmic spin-boson model with its scaling assumptions. They should not be copied blindly to every qubit, molecule, or finite structured bath.

Bias and Asymmetry

The bias $\epsilon$ favors one localized state over the other. In electron-transfer language it is an energy offset. In qubit language it may be a detuning or static longitudinal field. In a double-well reduction it is an asymmetry between wells.

Bias changes both equilibrium populations and dynamics. It also changes the energy-basis decomposition of the coupling:

• large $|\epsilon|$ makes $\cos\theta$ large, so longitudinal dephasing is more prominent;
• large $\Delta$ near zero bias makes $\sin\theta$ large, so bath-induced transitions are more prominent.

This basis dependence is one of the most common sources of confusion when translating between localized-state and energy-eigenstate descriptions.

Numerical and Non-Markovian Methods

The spin-boson model is simple to write but hard to solve outside perturbative limits. Common methods include:

• noninteracting blip approximation and path-integral methods;
• polaron and variational-polaron transformations;
• reaction-coordinate mappings;
• hierarchical equations of motion;
• tensor-network chain mappings;
• numerical renormalization group for scaling and impurity critical behavior;
• stochastic and influence-functional methods.

For structured or strong-coupling environments, Reaction-Coordinate Mapping and Hierarchical Equations of Motion are common constructive tools.

Relation to Caldeira–Leggett

A particle in a double-well potential coupled to an oscillator bath can often be truncated to its two lowest localized states. The resulting two-state model is a spin-boson model.

In that reduction:

• $q$ becomes approximately proportional to $\sigma_z$ within the two-state subspace;
• tunneling between wells becomes $\Delta\sigma_x$;
• the oscillator bath remains bosonic;
• the spectral density inherits information from the original coordinate-bath coupling.

Thus the spin-boson model is not a separate universe from Caldeira–Leggett physics. It is the two-level reduction of the same dissipative-coordinate idea, with its own special phenomena.

Common Mistakes

Calling every qubit-bath model spin-boson

A qubit coupled to a single cavity mode, a spin bath, a fermionic lead, or a nonlinear detector is not automatically the spin-boson model. The bath structure matters.

Ignoring the basis

Longitudinal in the localized basis need not be longitudinal in the energy basis. Relaxation and dephasing depend on the Hamiltonian diagonalization.

Using symmetrized noise for rates

Transition rates require ordered noise at positive and negative frequencies. Symmetrized spectra can hide detailed-balance asymmetry.

Treating $\alpha$ as universal across conventions

The dimensionless coupling $\alpha$ is convention-dependent unless the spectral-density normalization is fixed.

Overstating localization

The Ohmic localization transition is a result for an idealized continuum scaling model. Finite baths, cutoffs, bias, temperature, and structured environments modify the statement.

Forgetting tunneling renormalization

At strong coupling, the bare $\Delta$ in the Hamiltonian may not be the observed low-energy tunneling scale.

Exercises

Energy Splitting

Diagonalize

$$H_S = \frac{\epsilon}{2}\sigma_z - \frac{\Delta}{2}\sigma_x$$

and find the level splitting.

Solution

The Hamiltonian is a Pauli vector:

$$H_S = \frac12 \left( -\Delta,0,\epsilon \right) \cdot\boldsymbol\sigma.$$

The eigenvalues are

$$E_\pm = \pm \frac12 \sqrt{\epsilon^2+\Delta^2}.$$

Therefore the level splitting is

$$E = E_+ - E_- = \sqrt{\epsilon^2+\Delta^2}.$$

Coupling in the Energy Basis

Using

$$\cos\theta=\frac{\epsilon}{E}, \qquad \sin\theta=\frac{\Delta}{E},$$

explain why the same $\sigma_z$ bath coupling can produce both dephasing and relaxation.

Solution

After diagonalizing $H_S$, the localized operator decomposes as

$$\sigma_z = \cos\theta\,\tau_z + \sin\theta\,\tau_x$$

up to a sign convention. The $\tau_z$ part commutes with the energy Hamiltonian and produces dephasing between energy eigenstates. The $\tau_x$ part does not commute with the energy Hamiltonian and can drive transitions. Thus one physical bath coupling has both effects in the energy basis.

Pure-Dephasing Populations

Show that if $\Delta=0$, the populations in the $\sigma_z$ basis are constants of motion under the full spin-boson Hamiltonian.

Solution

When $\Delta=0$,

$$H_S=\frac{\epsilon}{2}\sigma_z, \qquad H_I=\frac{\sigma_z}{2}B.$$

Both commute with $\sigma_z$, and the bath Hamiltonian acts only on the bath:

$$[H,\sigma_z]=0.$$

Therefore $\sigma_z$ is conserved. Since the projectors onto the two $\sigma_z$ eigenstates are functions of $\sigma_z$, their populations are constants of motion. Coherences can still decay because the two spin states condition different bath evolutions.

Detailed Balance

For a weakly coupled spin-boson model at equilibrium, why should upward and downward rates satisfy

$$\frac{\Gamma_\uparrow}{\Gamma_\downarrow} = e^{-\beta E}?$$

Solution

The bath is thermal and stationary. Its ordered noise spectra satisfy the KMS detailed-balance relation: the spectrum for the bath supplying energy $E$ is suppressed relative to the spectrum for absorbing energy $E$ by $e^{-\beta E}$. Since weak-coupling transition rates are proportional to those ordered spectra at the transition frequency, the same ratio appears for the upward and downward rates.

Cross-Links

• Caldeira–Leggett Model
• Pure Dephasing Model
• Noise Spectra
• Fluctuation–Dissipation Relation
• System–Bath Hamiltonians
• Pure Dephasing Master Equation
• Detailed Balance
• Reaction-Coordinate Mapping
• Hierarchical Equations of Motion
• Approximation Checklist

References

• A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, “Dynamics of the dissipative two-state system,” Reviews of Modern Physics 59, 1-85 (1987).
• A. O. Caldeira and A. J. Leggett, “Quantum tunnelling in a dissipative system,” Annals of Physics 149, 374-456 (1983).
• R. A. Marcus and N. Sutin, “Electron transfers in chemistry and biology,” Biochimica et Biophysica Acta 811, 265-322 (1985).
• R. Silbey and R. A. Harris, “Variational calculation of the dynamics of a two level system interacting with a bath,” Journal of Chemical Physics 80, 2615-2617 (1984).
• U. Weiss, Quantum Dissipative Systems, World Scientific (2012).
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).