The standard Landau-Ginzburg scenario of phase transition is broken down for quantum phase transition. It is difficult to find an order parameter to indicate different phases for quantum fluctuations. Here, we suggest a topological description of the quantum phase transition for the XY model. The ground states are identified as a specialized U(1) principal bundle on the base manifold $S^2$. And then different first Chern numbers of U(1) principal bundle on the base manifold $S^2$ are associated to each phase of quantum fluctuations. The particle-hole picture is used to parameterized the ground states of the XY system. We show that a singularity of the Chern number of the ground states occurs simultaneously with a quantum phase transition. The Chern number is a suitable topological order of the quantum phase transition.
Lee-Yang zeros of modulated XY spin chains with Dzyaloshinskii-Moriya interaction: zero-contour topology and quantum phase-diagram reconstruction
Center of Materials Science and Optoelectronics Engineering, College of Materials Science and Opto-Electronic Technology, University of Chinese Academy of Sciences（中国科学院大学材料科学与光电技术学院物质科学与光电工程中心）；School of Physics, Hangzhou Normal University（杭州师范大学物理学院）
Translation invariance, topology, and protection of criticality in chains of interacting anyons
利用有限尺寸标度变元，可以从有限系统的低能谱中提取相互作用任意子链的临界性质。In Phys. Rev. Lett. 98，160409（2007），Feiguin等人表明，环面上的Fibonacci任意子的反铁磁（AFM）链与三临界伊辛模型处于相同的普适类，并且临界性受到拓扑对称性的保护。在本文中，我们首先回顾了图形形式主义的研究的任意子的光盘，并展示了如何这种形式主义可能会一直延伸到研究系统的表面上的更高的属。然后，我们采用这种图形形式主义研究有限环的相互作用的任意子在光盘和环面，并表明，分析光盘上必然产生的能量谱，这是一个子集，这是获得的环面。对于临界哈密顿量，可以从这个子集中提取出与系统的拓扑对称性有关的局部标度算子的标度维数。相关的考虑也适用于开链。
Phase transitions and thermal entanglement of the distorted Ising-Heisenberg spin chain: topology of multiple-spin exchange interactions in spin ladders
