自旋（费米）玻色耦合系统的数值严格解
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摘要
本文研究了自旋(费米)玻色多体耦合系统,提出一种有效的玻色相干态方法,得到玻色多体作用系统的数值严格解。相干态希尔伯特空间的玻色子数虽然有截断,但相当于Fock态下无穷多个玻色子数展开,通过精确对角化哈密顿量矩阵给出数值精确解。基于本征能量与本征波函数,我们从量子信息角度探讨几个量子多体模型的基态性质,比如N个两能级原子与单模玻色场耦合系统的量子相变,两能级耗散系统中多模玻色场与自旋相互作用的相变性质,电子与声子相互作用系统的双极化子渡越性质。主要内容从以下三个方面展开：
1.提出玻色相干态方法研究N个两能级原子与单模辐射场耦合系统Dicke模型的严格解,结合lanczos精确对角化得到有限尺寸系统的数值精确解。并分析任意两原子的量子纠缠,保真度及其敏感度,发现二级量子相变的奇异行为。在临界点,两原子的纠缠达到最大值,同时保真度有一个跌落现象。由于相干态方法的优越性,原子系统可以计算到N=2000-4000甚至更大,进而计算了基态能量,量子纠缠,保真度的敏感度等可观测量的有限尺寸标度临界指数,分析Dicke模型与Lipkin-Meshkov-Glick模型的临界指数具有相同的普适类。
2.讨论两能级耗散系统中多模谐振腔与自旋耦合的spin-boson模型,用玻色相干态方法处理对数离散化谱函数的哈密顿量,分析亚欧姆耗散0 3.研究电子与声子相互作用系统中两格点两电子的Holstein-Hubbard模型的双极化子渡越性质。对于电子自旋单态与声子耦合哈密顿量,玻色相干态方法在整个耦合区间给出严格解。并计算了电子与其声子环境的纠缠,线性熵度量了单占据格点的双极化子电子与其周围的声子云量子纠缠最大。保真度在中间耦合强度区域有极小值,准确给出了双格点占据主导的双极化子S1与单格点占据主导的双极化子S0的渡越相图。
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.
1.提出玻色相干态方法研究N个两能级原子与单模辐射场耦合系统Dicke模型的严格解,结合lanczos精确对角化得到有限尺寸系统的数值精确解。并分析任意两原子的量子纠缠,保真度及其敏感度,发现二级量子相变的奇异行为。在临界点,两原子的纠缠达到最大值,同时保真度有一个跌落现象。由于相干态方法的优越性,原子系统可以计算到N=2000-4000甚至更大,进而计算了基态能量,量子纠缠,保真度的敏感度等可观测量的有限尺寸标度临界指数,分析Dicke模型与Lipkin-Meshkov-Glick模型的临界指数具有相同的普适类。
2.讨论两能级耗散系统中多模谐振腔与自旋耦合的spin-boson模型,用玻色相干态方法处理对数离散化谱函数的哈密顿量,分析亚欧姆耗散0
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.
引文
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