Numerical Exact Solution To Spin (Fermi) And Boson Coupling System

In this thesis, numerical exact solutions of spin (fermi) and boson coupling systems is investigated. We propose a general extended coherent state approach to give numerical exact solution to the many-body coupling system by Lanczos exact diagonalization, avoiding truncation of bosons in Fock space. Based on the numerical exact solution with all eigenvalues and eigenfunctions, it facilitates to study the ground state properties in quantum many-body interacting system, e.g., the quantum phase transition (QPT) in two-level atoms coupling with a bosonic cavity system and the dissipative two-state system, and the bipolaron crossover properties in the electron-phonon (e-ph) interacting system. The main topics list in the following three aspects:·The Dicke model, describing the interaction of N two-level atoms with a single bosonic mode, has been solved exactly by the bosonic coherent states approach with the Lanczos exact diagonalization. Attributing to the numer-ical exact solution, the ground state properties in terms of the ground state energy, the expectation value of the photon number, the scaled concurrence (entanglement) and the ground state fidelity as well as its susceptibility are calculated in detail, exhibiting singularities at the critical point of QPT. The pairwise entanglement between arbitrary two atoms is maximum entangled at the critical point, where the fidelity has a drop. With the advantage of the technique, the accessible system size reaches N= 2000 - 4000 and even bigger. Finite-size scaling for several observables are calculated accurately and the scaling exponents obtained are in the same universal class as that in the Lipkin-Meshkov-Glick model, correcting the existing discrepancy in the scaling exponents of previous results limited to the small size of system.·We propose the bosonic coherent approach to solve the dissipative two-state system, so-called the spin-boson model. Based on the discretization of a bosonic bath with arbitrary continuous spectral density of the sub-Ohmic spin-boson model, an accurate solution for finite modes of bosons is obtained. The QPT in the sub-Ohmic spin-boson case can be located by the fidelity, giving the correct phase diagram from delocalized phase to localized phase. The critical exponent for the bath exponent s< 1/2 is correctly given by the fidelity susceptibility, verifying the validity of quantum-to-classical mapping in the sub-Ohmic spin-boson case. It is found that the scaling exponent of the fidelity susceptibility is the same as that of the magnetic susceptibility.·The exact solution of the two-site two-electron Holstein-Hubbard model is obtained numerically by the coherent state approach. The crossover from a two-site dominated to an one-site dominated bipolaron is characterized by the fidelity and the linear entropy, showing the interplay between the electron-phonon interaction and electron-electron Coulomb repulsion. It is found that the quantum entanglement between the electrons and their environment phonons is measured by the linear entropy, which is maximum for the one-site dominated bipolaron.