The Spin-Boson Model

Be better to remove them, sash, jamb, lintel, 

Lucretius L-.99-c. 55 BCE) The way things are translated by Rolfe Humphries, Indiana University Press, 1968 

In Sections 2.2 and 2.9 we have discussed the dynamics of the two-level system and of the harmonic oscillator, respectively. These exactly soluble models are often used as prototypes of important classes of physical system. The harmonic oscillator is an exact model for a mode of the radiation field (Chapter 3) and provides good starting points for describing nuclear motions in molecules and in solid environments (Chapter 4). It can also describe the short-time dynamics of liquid environments via the instantaneous normal mode approach (see Section 6.5.4). In fact, many linear response treatments in both classical and quantum dynamics lead to harmonic oscillator models Linear response implies that forces responsible for the return of a system to equilibrium depend linearly on the deviation from equilibrium—a harmonic oscillator property We will see a specific example of this phenomenology in our discussion of dielectric response in Section 16.9. 

The prominence of these quantum dynamical models is also exemplified by the abundance of theoretical pictures based on the spin-boson model—a two (more generally a few) level system coupled to one or many harmonic oscillators. Simple examples are an atom (well characterized at room temperature by its ground and first excited states, that is, a two-level system) interacting with the radiation field (a collection of harmonic modes) or an electron spin interacting with the phonon modes of a surrounding lattice, however this model has found many other applications in a variety of physical and chemical phenomena (and their extensions into the biological world) such as atoms and molecules interacting with the radiation field, polaron formation and dynamics in condensed environments. 

We consider a two-level system coupled to a bath of harmonic oscillators that will be referred to as a boson field. Two variations of this model, which differ from each other by the basis used to describe the two-level system, are frequently encountered. In one, the basis is made of the eigenstates of the two-state Hamiltonian that describes the isolated system. The full Hamiltonian is then written 

Sometimes, you know, we can not see dazzling objects Through an excess of light whoever heard Of doorways, portals, outlooks, in such trouble  

Besides, if eyes are doorways, might it not Be better to remove them, sash, jamb, lintel, 

---

There is a vast field in chemistry where the spin-boson model can serve practical purposes, namely, the exchange reactions of proton transfer in condensed media [Borgis et al. 1989 Suarez and Silbey 1991a Borgis and Hynes 1991 Morillo et al. 1989 Morillo and Cukier 1990]. 

Thus the promoting vibrations reduce the Franck-Condon factor itself, which is not reflected in the spin-boson model (5.55), (5.67). As an illustration, three-dimensional trajectories for various interrelations between symmetric (Ws) and antisymmetric (oja) vibration frequencies, and odo are shown in fig. 33. 

Indeed, consider the spin-boson model, with the Hamiltonian... 

Wilhelm, F. K., Kleff, S., and von Delft, J. (2004). The spin-boson model with a structured environment A comparison of approaches. Chem. Phys., 296 345. 

There is a vast field in chemistry where the spin-boson model can serve practical purposes, namely, proton exchange reactions in condensed media [Borgis and Hynes, 1991 Borgis et al., 1989 Morillo et al., 1989 Morillo and Cukier, 1990 Suarez and Silbey, 1991], The early approaches to this model used a perturbative expansion for weak coupling [Silbey and Harris, 1983], Generally speaking, perturbation theory allows one to consider a TLS coupled to an arbitrary bath via the term ftrz, where / is an operator that acts on the bath variables. The equations of motion in the Heisenberg representation for the operators, daldt = i[H, ], have the form... 

Thus, the promoting vibrations reduce the Franck-Condon factor itself, which is not reflected in the spin-boson model of (5.56) and (5.68). As an illustration, three-dimensional trajectories for various interrelations between symmetric (vibration frequencies and w0 are shown in Figure 5.2 When both vibrations have high frequencies, wa,s w0 the transition proceeds along the MEP (curve 1). In the opposite case of low frequencies, tua s < a>0, the tunneling occurs in the barrier, which is lowered and shortened by the symmetrically coupled vibration qs, so that the position of the antisymmetrically coupled... 

In contrast to the subsystem representation, the adiabatic basis depends on the environmental coordinates. As such, one obtains a physically intuitive description in terms of classical trajectories along Born-Oppenheimer surfaces. A variety of systems have been studied using QCL dynamics in this basis. These include the reaction rate and the kinetic isotope effect of proton transfer in a polar condensed phase solvent and a cluster [29-33], vibrational energy relaxation of a hydrogen bonded complex in a polar liquid [34], photodissociation of F2 [35], dynamical analysis of vibrational frequency shifts in a Xe fluid [36], and the spin-boson model [37,38], which is of particular importance as exact quantum results are available for comparison. 

Simulations using this algorithm [40] and the Trotter-based scheme [38] are able to reproduce the exact quantum results for the spin-boson model, verifying its utility. 

The last term involves derivatives with respect to both mapping and environmental variables. Its contribution is difficult to compute. Calculations on the spin-boson model have shown that even if the last term is neglected, excellent agreement with the exact results for a wide range of system parameters is obtained [53]. 

The main goal in the development of mixed quantum classical methods has as its focus the treatment of large, complex, many-body quantum systems. While applications to models with many realistic elements have been carried out [10,11], here we test the methods and algorithms on the spin-boson model, which is the standard test case in this field. In particular, we focus on the asymmetric spin-boson model and the calculation of off-diagonal density matrix elements, which present difficulties for some simulation schemes. We show that both of the methods discussed here are able to accurately and efficiently simulate this model. 

In this section we present results using the two approaches described in the previous sections the Trotter factorized QCL (TQCL), and iterative linearized density matrix (ILDM) propagation schemes, to study the spin-boson model consisting of a two level system that is bi-linearly coupled to a bath with Mh harmonic modes. This popular model of a quantum system embedded in an environment is described by the following general hamiltonian ... 

As noted earlier, the fundamental equations of the QCL dynamics approach are exact for this model, however, in order to implement these equations in the approach detailed in section 2 the momentum jump approximation of Eq.(14) is made in addition to the Trotter factorization of Eq.(12). Both approximations become more accurate as the size of the time step 5 is reduced. Consequently, the results presented below primarily serve as tests of the validity and utility of the momentum-jump approximation. For a discussion of other simulation schemes for QCL dynamics see Ref. [21] in this volume. The linearized approximate propagator is not exact for the spin-boson model. However when used as a short time approximation for iteration as outlined in section 3 the approach can be made accurate with a sufficient number of iterations [37]. 

In this section we present some applications of the LAND-map approach for computing time correlation functions and time dependent quantum expectation values for realistic model condensed phase systems. These representative applications demonstrate how the methodology can be implemented in general and provide challenging tests of the approach. The first test application is the spin-boson model where exact results are known from numerical path integral calculations [59-62]. The second system we study is a fully atomistic model for excess electronic transport in metal - molten salt solutions. Here the potentials are sufficiently reliable that findings from our calculations can be compared with experimental results. 

Fig. 12.1 The spin-boson model for a two-level molecule coupled to a system of harmonic oscillators.

In the rest of this chapter we focus on the pure vibrational relaxation problem, that is, on the relaxation of an oscillator (usually modeled as harmonic) coupled to its thermal environment. From the theoretical point of view this problem supplements the spin boson model considered in Chapter 12. Indeed, these are the N — 2 and the N = oo limits of an A level system. For a harmonic oscillator there is an added feature, that these states are equally spaced. In this case individual... 

The Golden Rule formula (9.5) for the mean rate constant assumes the Unear response regime of solvent polarization and is completely equivalent in this sense to the result predicted by the spin-boson model, where a two-state electronic system is coupled to a thermal bath of harmonic oscillators with the spectral density of relaxation J(o)) [38,71]. One should keep in mind that the actual coordinates of the solvent are not necessarily harmonic, but if the collective solvent polarization foUows the Unear response, the system can be effectively represented by a set of harmonic oscillators with the spectral density derived from the linear response function [39,182]. Another important point we would like to mention is that the Golden Rule expression is in fact equivalent [183] to the so-called noninteracting blip approximation [71] often used in the context of the spin-boson model. The perturbation theory can be readily applied to... 

This Hamiltonian, which was introduced by Schmickier [12], is equivalent to earher formulations by Levich and Dogo-nadze in terms of wave mechanics [5] it is also related to the spin-boson model for homogeneous electron exchange [13] and to the Anderson—Newns model for specific adsorption [14]. 

---