低维凝聚态系统中的新奇拓扑量子数及相变的理论研究

The study of the novel quantum states and topological orders in quantum condensed matter systems has been at the frontiers of the research of theoretical Physics. Meanwhile, the novel topological quantum states and the phase transitions emerging from the low-dimensional quantum many-body systems, i.e., quantum spin Hall effect, Z2 topological insulator, and so on, have become important research topics. In this project, we investigate the ground-state quantum geometric tensor, Berry phase, topological order and the quantum phase transitions in 1D and 2D exactly solved and with interactions fermionic and spin systems. Based on our previous works, we propose a topological Euler number to characterize nontrivial topological phases of gapped fermionic systems, which originates from the topological properties on the Riemannian structure established by the ground-state quantum geometric tensor. In this project, we plan to carry on further studies in the following aspects: (1) Quantized Berry phase, topological Z2 number and the topological orders in 1D fermionic and spin systems; (2) Quantum geometric tensor, topological Euler number and the topological orders in 2D fermionic and spin systems; (3) Non-Abelian quantum geometric tensor and its applications. From these studies, we will reveal some new geometric nature and topological quantum numbers of the ground state in low-dimensional quantum systems, establish the physical mechanism, reveal new physical properties and promote the related experiments research.

凝聚态系统中的新奇量子态与拓扑序的研究处于当前理论物理研究的前沿。同时，量子自旋Hall效应、Z2拓扑绝缘体等低维量子体系中涌现的拓扑量子态及其相变成为了重要的研究课题。本项目将从量子态空间几何的角度，研究一维、二维严格可解及存在相互作用的量子自旋及费米系统的基态量子几何张量、Berry相位、拓扑序及其之间的量子相变。在前期工作的基础上，我们提出一种新的拓扑量子数－拓扑Euler数以刻画具有体能隙费米系统基态的非平凡拓扑相，它来源于基态量子几何张量所诱导的Riemannian结构的拓扑性质。本项目拟在以下几个方面进行更深入的研究：（1）量子化Berry相位、拓扑Z2量子数与一维费米及自旋系统的拓扑序；（2）量子几何张量、拓扑Euler数与二维费米及自旋系统的拓扑序；（3）非阿贝尔量子几何张量及其应用。本项目将揭示低维系统基态的一些新的几何与拓扑性质，确定它们的物理机制并促进相关实验的研究。

本项目基于量子多体系统的内蕴几何性质，着重研究了低维量子系统的能带以及基态的Berry曲率、Riemannian度规，以及更一般的量子几何张量，并由此得到了能带的拓扑Euler数指标，用以特征具有体能隙费米系统的非平凡拓扑相。具体来说，我们在以下几个方面取得了研究进展：（1）阐明了能带的拓扑Euler数来源于Gauss-Bonnet定理对第一布里渊区上闭合Bloch态流形的拓扑刻画；得到了计算能带Euler数的普遍公式，揭示了Euler数与能带绝缘体的第一类Chern数之间的关系。（2）研究了一类具有多格点自旋耦合交换作用的一维自旋模型，我们指出此系统的铁磁-顺磁量子相变可以用能带的拓扑Euler数来标志。同时，由于该一维系统的哈密顿量属于BDI对称类，因此不能使用能带的Chern数来刻画其拓扑序。（3）在一维自旋模型中，研究了能带的量子几何张量、Euler数与自旋关联函数之间的内在联系。（4）在一类具有时间反演对称性破缺的两维格点费米模型中，利用能带的拓扑Euler数与Chern数研究了该系统的拓扑相变。