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The Journal of Chemical Physics

Applying the variational principle to a spin-boson Hamiltonian

Anton Amann

Published Jan 15, 1992• 11 citations
Publisher

Abstract

Ground states of a spin-boson Hamiltonian, describing one two-level system (a spin) coupled to infinitely many harmonic oscillators (bosons), are studied. This spin-boson Hamiltonian is a prototype for the description of a ‘‘small’’ system (e.g., a molecule) coupled to its environment. The respective ground-state vector(s) are approximated by a linear combination of two coherent-state vectors corresponding to the two levels of the spin. Interest concentrates mainly on phase-transition phenomena (generation of superselection rules) in case the parameters of the Hamiltonian (level splitting of the spin, frequencies of the field modes, and coupling constants) exhibit an infrared singularity. The resulting phase diagrams are shown to satisfy reasonably well the rigorous bounds derived by Spohn, and in particular distinguish between the superohmic, ohmic, and subohmic regime in the sense of Leggett. Nevertheless, the approximation method used is simple enough so that everything can be explicitly calculated. Former results by Pfeifer as well as results by Emery and Luther, Zwerger and Harris and Silbey are extended and discussed.

DOI
10.1063/1.462167

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Ground states of a spin-boson Hamiltonian, describing one two-level system (a spin) coupled to infinitely many harmonic oscillators (bosons), are studied. This spin-boson Hamiltonian is a prototype for the description of a ‘‘small’’ system (e.g., a molecule) coupled to its environment. The respective ground-state vector(s) are approximated by a linear combination of two coherent-state vectors corresponding to the two levels of the spin. Interest concentrates mainly on phase-transition phenomena (generation of superselection rules) in case the parameters of the Hamiltonian (level splitting of the spin, frequencies of the field modes, and coupling constants) exhibit an infrared singularity. The resulting phase diagrams are shown to satisfy reasonably well the rigorous bounds derived by Spohn, and in particular distinguish between the superohmic, ohmic, and subohmic regime in the sense of Leggett. Nevertheless, the approximation method used is simple enough so that everything can be explicitly calculated. Former results by Pfeifer as well as results by Emery and Luther, Zwerger and Harris and Silbey are extended and discussed.
The effectiveness of the variational approach \`a la Feynman is proved in the spin-boson model, i.e., the simplest realization of the Caldeira-Leggett model able to reveal the quantum phase transition from delocalized to localized states and the quantum dissipation and decoherence effects induced by a heat bath. After exactly eliminating the bath degrees of freedom, we propose a trial, nonlocal in time, interaction between the spin and itself simulating the coupling of the two-level system with the bosonic bath. It stems from a Hamiltonian where the spin is linearly coupled to a finite number of harmonic oscillators whose frequencies and coupling strengths are variationally determined. We show that a very limited number of these fictitious modes is enough to get a remarkable agreement, up to very low temperatures, with the data obtained by using an approximation-free Monte Carlo approach, predicting (1) in the Ohmic regime, a Berezinski-Thouless-Kosterlitz quantum phase transition exhibiting the typical universal jump at the critical value; and (2) in the sub-Ohmic regime ($s\ensuremath{\le}0.5$), mean-field quantum phase transitions, with logarithmic corrections for s=0.5s=0.5s=0.5.
In a generic quantum mechanical description of a molecule interacting with its thermal environment, the molecule is represented as a few level system (in the simplest description just two, for example, ground and excited states) and the environment is often modeled as a bath of harmonic oscillators. The resulting theoretical framework is known as the spin–boson model, a term that seems to have emerged in the Kondo problem literature (which deals with the behavior of magnetic impurities in metals) during the 1960s, but is now used in a much broader context. Indeed, it has become one of the central models of theoretical physics, with applications in physics, chemistry, and biology that range far beyond the subject of this book. Transitions between molecular electronic states coupled to nuclear vibrations, environmental phonons, and photon modes of the radiation field fall within this class of problems. The present chapter discusses this model and some of its mathematical implications. The reader may note that some of the subjects discussed in Chapter 9 are reiterated here in this more general framework. 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 and provides good starting points for describing nuclear motions in molecules and in solid environments. It can also describe the short-time dynamics of liquid environments via the instantaneous normal mode approach. 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.
A two-level system with a transverse linear infrared singular coupling to a boson gas is treated within a mode-coupling approximation for the dynamics. Static spin susceptibilities and spin polarizations are evaluated consistently with the dynamics by exploiting sum rules. The spin energy is obtained by integration over the excitation spectrum of the boson scattering matrix. In case of a high-frequency boson cutoff large compared to the level splitting, the results demonstrate pronounced precursor phenomena for the known zero-temperature phase transition of the model. In the opposite case the results are in semiquantitative agreement with previous numerical findings for the transition to self-trapping obtained by DeRaedt and DeRaedt.
Symmetrized Trotter approximations to ${e}^{\mathrm{\ensuremath{-}}\ensuremath{\beta}H}$ are employed to compute the low-energy eigenstates of a two-level system linearly coupled to a harmonic oscillator. At high coupling strengths the ground state is nearly doubly degenerate and the projectors become very inefficient. It is demonstrated that the projection technique in combination with the variational principle yields a new efficient method to calculate simultaneously all nearly degenerate eigenvalues and eigenstates. The dynamic properties of the system in the ground state are studied by evaluating the time- and frequency-dependent correlation functions. Depending on the ratio of the tunneling frequency relative to the oscillator frequency, the system behaves differently under increasing coupling strength.
We present some characteristic properties of the ground state of a two-level unit linearly coupled to a bosonic field, exactly deduced without previously solving the eigenvalue problem for the physical system. Such properties are valid in the whole range of the characteristic parameters appearing in the Hamiltonian modelling this system. Our treatment is based on operatorial methods which combine symmetry considerations with some general peculiarities of the lowest energy state. In particular we present rigorous conclusions concerning the expectation value of the two-level unit energy and the mean value of the interaction energy for arbitrary strenght of the coupling constant. The relevance of these rigorous results in connection with an approximate variational determination of the ground state of the system is evidenced.
The dynamics of the spin-boson Hamiltonian is considered in the stochastic approximation. The Hamiltonian describes a two-level system coupled to an environment, and is widely used in physics, chemistry, and the theory of quantum measurement. We demonstrate that the method of stochastic approximation, a general method for consideration of dynamics of an arbitrary system interacting with environment, is powerful enough to reproduce qualitatively the striking results by Leggett et al. [Rev. Mod. Phys. 59, 1 (1987)] found earlier for this model system. The results include an exact expression of the dynamics in terms of the spectral density, and show the appearance of two interesting regimes for the system, i.e., pure oscillating and pure damping ones. Correlators describing the environment are also computed.