Geometrical Phases in Condensed Matter Theory : Qutnaum Phase Transition and Classification by topological orders

有一类基本的多体量子态不具有相关的对称性破缺。它包括量子霍尔态、各向异性超导体、石墨烯、近藤绝缘体、带挫折的量子自旋等，在这些低维系统中，强的量子涨落阻碍了表征态的经典有序的发展。这些是量子液体，量子效应本身就是它们的特征。在态为带隙态的假设下，通过定义拓扑局域序参量刻画了无对称破缺的量子基态。它由量子化的Berry相位定义，其中一些反幺正对称性，如时间反转或粒子空穴保证了Berry相位的量子化。这个拓扑序参量对于描述系统对称性保持不变的量子相变是非常有用的。基于体-边对应，我们还利用纠缠熵对带隙量子液体的量子相变进行了表征。我们已经成功地发展了我们的原始理论，并通过应用到许多具体的系统中来证明它们的有效性。

There is a fundamental class of many-body quantum states that do not possess relevant symmetry breaking. It includes quantum Hall states, anisotropic superconductors, graphene, Kondo insulators, quantum spins with frustrations and so on. In these low dimensional systems, strong quantum fluctuations prevent from developing classical order characterizing the states. These are quantum liquids where the quantum effects themselves are their characteristic features. Assuming the state is gapped, we characterized the quantum ground state without symmetry breaking by defining a topological local order parameter. It is defined by a quantized Berry phase where some anti-unitary symmetry such as time reversal or particle-hole guarantee the quantization of the Berry phase. This topological order parameter is quite useful to describe quantum phase transitions where the symmetry of the system remains the same. We also characterized the state by the uses of an entanglement entropy for the these quantum phase transition of the gapped quantum liquids based on the bulk-edge correspondence. We have successfully developed our original theory and show their validity by application to many concrete systems.