低维量子多体系统的特性及其量子相变的理论研究

【作者】 张秋兰；

【导师】 万歆； 李有泉；

【作者基本信息】 浙江大学 ， 理论物理， 2008， 博士

【摘要】 本文主要讨论了低维量子体系中的相变和临界现象。这些体系包括一维两分量玻色子体系、一维海森堡链、一维玻色费米混合体系和二维电子气。在低维量子系统中,由量子涨落控制的量子相变是值得研究的问题。论文中应用了解析严格求解和数值对角化等方法对相关问题进行了研究和讨论。论文首先对凝聚态物理中低维量子体系的背景、研究意义和部分研究成果进行了介绍,然后阐述了低维物理学在凝聚态物理研究中的重要性。论文的第二章主要采用Bethe ansatz方法讨论了具有吸引相互作用的一维两分量玻色子体系。利用数值和解析的方法,我们对此体系的基态和低激发态做了详细的分析,并发现其基态是束缚态,粒子之间的结合以两两配对的形式出现,任意打破一对配对,就会导致激发,激发态和其对应的单根有关。对两粒子情况,我们做了特别的研究,得到了其基态是铁磁态且基态与耦合常数无关的结果。第三章,在扭曲边界的一维海森堡链里,利用先前已有的Bethe ansatz根,我们分别用解析和数值的方法对低激发态的持续流谱做了讨论。尽管自旋单态激发和自旋三态激发简并,但是它们的持续流谱已经被明确地证实是完全不同的。非零的自旋流成了有限温度自旋硬度的主要贡献,这些结论给研究一些可积模型中的自旋输运问题提供了参考。目前,有关玻色费米混合体系性质的理论还不完善,仍有很多问题等待探讨和解决。我们在第四章里用Bethe ansatz方法对一维玻色费米混合问题做了详细的讨论,首先选择三种不同的参考态,然后利用量子反散射方法得到三组相应的Bethe ansatz方程。通过对这三组方程的分析,我们给出体系的基态和低激发态的一致结果,并在数值上验证了该结论：体系的基态为全是玻色子的状态,低激发态为玻色子和费米子共存的状态。由于玻色-爱因斯坦凝聚实验(BEC)的实现,光学格点中冷原子问题成了目前凝聚态物理的主要研究方向之一。理论上,冷原子在适当条件下就会经历—个由超流到绝缘的相变。基于此,第五章主要研究了两光学格点中的二玻色子体系,我们用Negativity分析了这个模型的纠缠问题,发现即使在极低温的情况下,纠缠仍可以通过外场来调控。论文对平行场和反平行场的情况做了具体的讨论,当温度升高时,纠缠会增强,反平行磁场下的纠缠比平行磁场下的纠缠大。在一定温度下,我们还观察到化学势对纠缠的影响。在第六章中,论文讨论了有合金杂质的二维电子气系统AlxGa1-xAs/Aly-Ga1-yAs.这个工作的目标是试图解释普林斯顿小组近年来在该材料的整数量子霍尔效应中观测到的反常标度行为。这个工作仍在研究中,论文报告了一些初步的计算结果。特别地,合金杂质势的引入为整数量子霍尔效应的格点模型打开了一个新的自由度。更多还原

【Abstract】 This paper mainly deals with quantum phase transitions and critical phe-nomena in low dimensional systems. These systems include one-dimensional two-component bosons system、one-dimensional Heisenberg model、one-dimensional Bose-Fermi mixture system and two-dimensional electron gas, etc. In low dimen-sional system, quantum phase transition controlled by the quantum fluctuation is worth to study. In this paper, the methods such as analytical exact solu-tion、numerical diagonal method are applied to study the related problems.Firstly this paper gives a brief introduction of low dimensional system in condensate matter physics, such as the background、research significance and partial results, the importance of low dimensional physics in the condensed matter physics is demonstrated.In the second chapter of the paper, the Bethe ansatz method is used to solve one-dimensional two-component bosons with aδ-function potential considering with negative coupling constant. The features of the ground state and low-lying excited states of this model are discussed explicitly by analytical and numerical methods. We conclude that the ground state is a bound state, whose quasi-particles are paired, and when one pair is destroyed, there will be an excited state. In particular we give a thorough discussion of this model to N=2, and show that the ground state is a ferromagnetic state regardless of the coupling constant.Then, in the third chapter, based on the Bethe ansatz method of the one-dimensional Heisenberg model under twisted boundary conditions, we study the spectra of the persistent current carried by the low-lying excited states. Though the energy spectra of the singlet and triplet excitations are degenerate, their transport properties are quite different. The non-vanishing behavior of the per-sistent current is the main contribution to the spin stiffness at finite temperatures, which may provide some useful physical intuitions to the transport properties in integrable models. Since the theory of the Bose-Fermi system is not complete yet, there are lots of issues to be discussed and solved. In the fourth chapter, we study a one-dimensional cold atomic system of Bose-Fermi mixtures based on the Bethe ansatz method. Corresponding to three possible choices of the reference states in the quantum inverse scattering method, three sets of Bethe ansatz equations are derived explicitly. Through the analysis of these equations, the features of the ground state and low-lying excitations are investigated, and we obtain the conclusion that the ground state should be bosonic purely. And the low-lying excitations are mixture of bosons and fermions which can be shown numerically.Since the realization of Bose-Einstein condensation, ultracold atoms in opti-cal lattice becomes a rich field of investigation for theoretical and applied issues of condensed matter physics. Theoretically, ultracold atoms will undergo a quan-tum phase transition from a superfluid state to a Mott insulator state. In the fifth chapter, we investigate the entanglement of bosonic atoms in a two-site optical lattice. Negativity is used to measure the entanglement between two sites. At extremely low temperatures, the entanglement can be controlled by the magnetic field intensity. Increasing the temperature, the entanglement increases and it is larger for the antiparallel fields than for the parallel fields. Given a proper chem-ical potential, we can also demonstrate that anti-parallel fields will enhance the entanglement of two sites.In the sixth chapter, it mainly discusses a two-dimensional electron system realized in an alloy disorder potential AlxGa1-xAs/AlyGa1-yAs heterostructure. The goal is to explain the abnormal scaling behavior in integer quantum hall effect which was observed on this material by Princeton group recently. Since this work is going on, the paper reports some elementary computational results. In particular, the appearance of alloy disorder potential open a new degree of freedom for lattice model of integer quantum hall effect.更多还原