用量子纠缠刻画低维关联量子物态的拓扑性质

发布时间：2018-05-19 05:56

  本文选题：量子纠缠 + SPT态　； 参考：《清华大学》2016年博士论文

【摘要】：低维关联量子物态是当前凝聚态物理的研究热点,其非平庸拓扑信息很大程度上隐藏在基态波函数的量子纠缠特性中。通过将体系一个子部分的约化密度矩阵取对数得到的纠缠哈密顿量,以及其本征值谱、即纠缠谱,则是定量描述纠缠的工具。通过左右划分得到的边缘纠缠谱的低能部分与拓扑态边缘激发能谱一一对应,从而可以反映拓扑信息。本文集中于一维有对称性的短程纠缠态、即对称性保护拓扑态(SPT态),通过研究其共振价键固体(VBS)构型的严格可解点基态波函数的量子纠缠特性,来刻画SPT态的拓扑性质。主要内容如下。在第一部分我们首先介绍了反铁磁整数自旋链的Haldane相,以及作为其严格可解点的AKLT模型。我们说明只有奇数自旋AKLT态才属于非平庸的SPT态,并以该模型为例证明,SPT态边缘纠缠谱简并度中,既有拓扑相关的部分、也有非普适的部分,而非平庸的前者可通过由对称性决定的拓扑退纠缠算符除去。该工作厘清了SPT态边缘纠缠谱的结构,对用纠缠谱刻画拓扑态有指导作用。从第二部分起,我们研究如何通过量子纠缠,从VBS型基态波函数中读取原SPT态与平庸相之间的量子临界点的信息。该临界点不能用朗道的经典理论描述,其低能元激发是原VBS的边缘自由度。我们对VBS波函数引入对称的延展划分:将每l个格点组合成一个子块,以所有奇数子块作为子系统,得到体纠缠哈密顿量HE。我们说明,HE在l选取合适的情况下就描述退禁闭的边缘自由度之间的渗透,从而给出量子临界点的完整能谱。我们证明自旋S-AKLT态的HE是自旋S/2的海森堡模型,对于非平庸的奇数S并取l=偶数,HE就描述Haldane相与平庸相之间的量子临界点,低能有效理论是S U(2)1WZW场论。在第三部分,我们将延展划分的方法运用到具有S O(5)对称性的自旋2-VBS态,并证明HE在l=奇数时就描述与原SPT态相连的量子临界点,低能有效理论是S U(4)1WZW场论。在第四部分,我们用自旋构造了具有S U(N)(N≥3)结构的一维VBS态:在总长度为偶数时,两端边缘互为共轭,破坏镜像对称性;而当总长度为奇数时,两端边缘相同,属于非平庸SPT态,且其对应的量子临界点由取l=奇数的体纠缠哈密顿量描述,低能有效理论是S U(N)1WZW场论。通过延展划分我们不仅可以得到量子临界点低能有效理论,更能写出对应的哈密顿量,从而建立了体-边缘-量子临界点对应,深化了对拓扑态的理解。
[Abstract]:Low-dimensional correlated quantum state is a hot topic in condensed matter physics. Its non-mediocre topological information is largely hidden in the quantum entanglement properties of ground state wave function. The entangled Hamiltonian obtained by taking the reduced density matrix of a subpart of the system, and its eigenvalue spectrum, that is, the entanglement spectrum, is a tool for quantificational description of entanglement. The low-energy part of the edge-entangled spectrum obtained by the left and right partition corresponds to the excited energy spectrum of the topological state, which can reflect the topological information. In this paper, we focus on the one dimensional symmetric short range entangled state, which is the symmetric protected topological state. By studying the quantum entanglement properties of the strictly solvable ground state wave function of its resonant valence bond solid state, we characterize the topological properties of the SPT state. The main contents are as follows. In the first part we first introduce the Haldane phase of the antiferromagnetic integer spin chain and the AKLT model as its strictly solvable point. We show that only the odd-number spin AKLT states are non-mediocre SPT states, and take the model as an example to prove that the degeneracy of the edge entanglement spectrum of the AKLT states is not only topological dependent, but also non-universal. The non-mediocre former can be removed by the topological deentanglement operator determined by symmetry. This work clarifies the structure of the edge entanglement spectrum of SPT states and provides guidance for the characterization of topological states by entanglement spectra. From the second part, we study how to read the information of the quantum critical point between the original SPT state and the mediocre phase from the VBS type ground state wave function by quantum entanglement. This critical point cannot be described by Landau's classical theory, and its low energy element excitation is the edge degree of freedom of the original VBS. We introduce a symmetric extension partition for VBS wave functions: every l lattice points are combined into a subblock, and all odd sub-blocks are used as subsystems to obtain the volume entangled Hamiltonian. We show that the percolation between the edge degrees of freedom of decommissioning is described under the suitable condition of l, and the complete energy spectrum of the quantum critical point is obtained. We prove that the HE of the spin S-AKLT state is the Heisenberg model of spin S / 2. For the non-mediocre odd number S and the even number he, we describe the quantum critical point between the Haldane phase and the mediocre phase. The low energy efficient theory is S U(2)1WZW field theory. In the third part, we apply the method of extended partition to the spin 2-VBS states with S _ O _ (5) symmetry, and prove that he describes the quantum critical points connected with the original SPT states when l = odd. The low energy efficient theory is S U(4)1WZW 's field theory. In the fourth part, we construct a one-dimensional VBS state with S U(N)(N 鈮，

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