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Science Technology and Engineering

Application of Double-parameter Variational Approach to the Spin-Boson Model

Hou Zhi-lan

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

Double-parameter variational approach in combination with the spin-boson model to study the macroscopic quantum effect of molecules magnet is emploied, further investigated the ground state energy of spin-boson model and the phase boundary of two-level system from delocalized to localized phases by numerical calculation. The computational results show that the double-parameter variational approach can deduce lower ground state energy than two types of single-parameter variational approaches, which indicates the superiority of the double-parameter variational approach. Calculations also show that different variational approach can result in different ground state energy and phase boundary. The further discussions are focused on the results of the double parameters variational approach, and a brief comparison with two types of single-parameter variational approaches is also presented.

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Double-parameter variational approach in combination with the spin-boson model to study the macroscopic quantum effect of molecules magnet is emploied, further investigated the ground state energy of spin-boson model and the phase boundary of two-level system from delocalized to localized phases by numerical calculation. The computational results show that the double-parameter variational approach can deduce lower ground state energy than two types of single-parameter variational approaches, which indicates the superiority of the double-parameter variational approach. Calculations also show that different variational approach can result in different ground state energy and phase boundary. The further discussions are focused on the results of the double parameters variational approach, and a brief comparison with two types of single-parameter variational approaches is also presented.
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.
We generalize the variational approach proposed by Chen, Zhang, and Wu [Phys. Rev. B 40, 11 326 (1989)] to the ground state of a two-level system coupled to a single boson mode. The resulting displaced-squeezed state is compared to the true ground state (calculated numerically), and to further variational states, by computing energy differences and overlaps. These states are able to describe a bifurcation occurring in the ground state of the quantum system as well as in the equilibrium state of its classical limit, and they are excellent approximations to the true ground state.
We present energy and free-energy analyses on all variational schemes used so far in the spin-boson model at both T=0T=0T=0 and $T\ensuremath{\ne}0$. It is found that all the variational schemes have fail points, at where the variational schemes fail to provide a lower energy (or a lower free energy at $T\ensuremath{\ne}0$) than the displaced-oscillator ground state and therefore the variational ground state becomes unstable, which results in a transition from a variational ground state to a displaced-oscillator ground state when the fail point is reached. Such transitions are always misidentified as crossover from a delocalized to localized phases in variational calculations, leading to an artifact of phonon-induced localization. Physics origin of the fail points and explanations for different transition behaviors with different spectral functions are found by studying the fail points of the variational schemes in the single-mode case.
By using a variation method we study the ground state of a tunneling particle coupled to many bosons in a double-well potential. The boson system is assumed to have dense low-energy excitations, just as taken by Chakravarty and Leggett to discuss the dissipative effect in macroscopic quantum phenomena. We examine reduction of the tunneling matrix and eventual localization of the particle due to boson excitations for arbitrary strengths of tunneling and coupling parameters. Ground state energy is evaluatied. Continuous as well as discontinuous changes from tunneling to localized states are found when the coupling parameter increases. The results are related to the orthogonality of many-particle ground states, as known for many-electron system as Anderson's theorem.
A Bose–Einstein condensate in a double-well potential is considered by using the Feynman path integral theory combined with a variational approach. The system consists of N interacting bosons confined in the double well which is taken as a harmonic potential with a Gaussian barrier. The calculation is carried out within the first cumulant approximation measured with respect to the harmonic action containing variational parameters. Assuming a separable expression for the trial action corresponds to the usual mean field approximation. Performing the variational calculations, we obtain analytical results for the ground state energy and its condensate wavefunction. Using a projection onto the odd contributions, the first excited state energy and condensate wavefunction are determined, too. We find very good agreement with a full numerical solution of the Gross–Pitaevskii equation over the full range of potential parameters.
Abstract We study a one parameter variational wave function to improve the spin density wave ground state of the Hubbard model by inclusion of quantum spin fluctuations. Using a perturbative approach and novel lattice summation techniques we present analytical as well as numerical results for the correlation energies and the staggered magnetizations in one and two dimensions. We find ground state energies which are satisfyingly close to known exact results and are significantly lower than those of existing Gutzwiller and numerical treatments.