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.

A Two-State System Coupled to Oscillators

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:

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

Here $\epsilon$, $\Delta$, and the real couplings $g_k$ have units of energy; $\omega_k$ is an angular frequency. 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

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

With the energy-valued $g_k$, $J$ has units of energy squared times time. Here $\omega_c$ is a cutoff and the displayed power of $\omega_c$ makes $\alpha$ dimensionless for every $s$. The exponent classifies the low-frequency bath:

Regime	Exponent	Low-frequency spectral weight
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

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

For $E>0$, 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

Define $B=\sum_k g_k(b_k+b_k^\dagger)$ and the ordered spectrum $S_B(\omega)=\int_{-\infty}^{\infty}e^{i\omega t}\langle B(t)B(0)\rangle\,dt$. Positive frequency describes the bath absorbing energy. In a weak-coupling, Markov, secular treatment with finite $S_B(0)$,

$$\begin{aligned} \Gamma_\downarrow&=\frac{\sin^2\theta}{4\hbar^2}S_B(E/\hbar),& \Gamma_\uparrow&=\frac{\sin^2\theta}{4\hbar^2}S_B(-E/\hbar),\\ \Gamma_1&=\Gamma_\downarrow+\Gamma_\uparrow,& \Gamma_\phi&=\frac{\cos^2\theta}{2\hbar^2}S_B(0). \end{aligned}$$

The factor $1/4$ in the transition rates comes from $H_I=\sigma_z B/2$. Coherence in the energy basis decays at $\Gamma_2=\Gamma_1/2+\Gamma_\phi$. At thermal equilibrium,

$$S_B(\omega)=2J(\omega)[n_B(\omega)+1],\qquad S_B(-\omega)=2J(\omega)n_B(\omega),\qquad\omega>0,$$

so $\Gamma_\uparrow/\Gamma_\downarrow=e^{-\beta E}$. Symmetrized noise alone cannot distinguish these two rates. At $\epsilon=0$, the coupling is transverse in the energy basis and this first-order longitudinal dephasing rate vanishes, although relaxation remains. Slow or non-Gaussian noise, strong coupling, and unresolved level splittings require a treatment beyond these 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.

For an initially factorized state with an uncoupled thermal oscillator bath, the coherence takes the form

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

with

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

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. For an unbiased Ohmic continuum at zero temperature, in the scaling limit $\Delta/(\hbar\omega_c)\ll1$, the standard theory gives:

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

The coherent-to-incoherent change concerns the form of the dynamics; it is not the localization phase transition. The quoted values approach their stated limits as the scale separation grows. Finite cutoff ratios shift the localization boundary, while finite temperature, bias, and finite baths alter or round these idealized behaviors (Leggett et al., 1987).

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.

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.

References

• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).
• A. O. Caldeira and A. J. Leggett, “Quantum tunnelling in a dissipative system,” Annals of Physics 149, 374–456 (1983).
• 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). doi:10.1103/RevModPhys.59.1.
• 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, 4th ed., World Scientific (2012).