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Physical review. B, Condensed matter

Dynamic and thermodynamic properties of a spin-boson system

W. Götze, G. M. Vujiić

Published Nov 15, 1988• 16 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.38.9398

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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.
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 linear and nonlinear dynamical susceptibilities of a two-level system are calculated as it undergoes a transition to a decoherent state. Analogously to the Glover-Tinkham-Ferrell sum rule of superconductivity, spectral weight in the linear susceptibility is continuously transferred from a finite frequency resonance to nearly zero frequency, corresponding to a broken symmetry in the thermodynamic limit. In this respect, the behavior of the present model (the Mermin model) differs from the spin-boson model. The third order nonlinear susceptibility, corresponding to two-photon absorption, has an unexpected nonmonotonic behavior as a function of the environmental coupling, reaching a maximum within the decoherent phase of the model. Both linear and nonlinear susceptibilities may be expressed in a universal form.
We present numerically exact inchworm quantum Monte Carlo results for the real-time dynamics of the spin polarization in the sub-Ohmic spin-boson model at finite temperature. We focus in particular on the localization and coherence behavior of the model, extending our previous study at low temperature [Phys. Rev. Lett. 134, 056502 (2025)]. As the temperature increases, the system becomes less localized and less coherent. The loss of coherence, which is controlled by two independent mechanisms -- a smooth damping-driven crossover and a sharp frequency-driven transition -- exhibits a nontrivial temperature dependence. While both types of coherence loss occur at lower coupling in the high temperature regime, the frequency exhibits a sharper drop at high temperatures and this drop is observed for all values of the sub-Ohmic exponent, in contrast to the zero-temperature case. We discuss the full temperature-dependent dynamical phase diagram of the system and the interplay between coherence and localization across a wide range of physical parameters.
It is known that arrays of trapped ions can be used to efficiently simulate a variety of many-body quantum systems. Here we show how it is possible to build a model representing a spin chain interacting with bosons that is exactly solvable. The exact spectrum of the model at zero temperature and the ground-state properties are studied. We show that a quantum phase transition occurs when the coupling between spins and bosons reaches a critical value, which corresponds to a level crossing in the energy spectrum. Once the critical point is reached, the number of bosonic excitations in the ground state, which can be assumed as an order parameter, starts to be different from zero. The population of the bosonic mode is accompanied by a macroscopic magnetization of the spins. This double effect could represent a useful resource for phase transition detection since a measure of the phonon can give information about the phase of the spin system. A finite-temperature phase diagram is also given in the adiabatic regime.
We propose a density matrix renormalization group approach to tackle a two-state system coupled to a bosonic bath with continuous spectrum. In this approach, the optimized phonon scheme is applied to several hundred phonon modes that are linearly divided among the spectra. Although density matrix renormalization group cannot resolve very small energy scales, the delocalized-localized transition points of the two-state system are extracted by the extrapolation of the flow diagram results. The phase diagram is compared to the numerical renormalization group results and shows a good agreement in both Ohmic and sub-Ohmic cases.
We investigate the dynamics of a 2-level atom (or spin-1/2) coupled to a mass-less bosonic field at positive temperature. We prove that, at small coupling, the combined quantum system approaches thermal equilibrium. Moreover we establish that this approach is exponentially fast in time. We first reduce the question to a spectral problem for the Liouvillean, a self-adjoint operator naturally associated with the system. To compute this operator, we invoke Tomita-Takesaki theory. Once this is done we use complex deformation techniques to study its spectrum. The corresponding zero temperature model is also reviewed and compared.