Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
复制标题
DOI：
10.1103/physrevb.82.155138
复制
发表时间：
2010-10-26
影响因子：
3.7
通讯作者：
Wen, Xiao-Gang
中科院分区：
文献类型：
--
作者：
Chen, Xie;Gu, Zheng-Cheng;Wen, Xiao-Gang
处于同一相位的两个缺口量子基态通过绝热演化连接起来，从而产生在状态之间映射的局部酉变换。另一方面，在局部酉变换下，间隙基态保持在同一相内。因此，局部酉变换定义了一个等价关系，等价类是定义间隙量子系统不同相位的普适性类。由于局部酉变换可以去除局部纠缠，所以上述等价/普适性类对应的是拓扑秩序的本质——远程纠缠模式。局部酉变换还允许我们定义波函数重整化方案，在该方案下，波函数可以在相同的等价/普适性类中转化为更简单的波函数。利用这种设置，我们找到了局部酉变换具有有限维的可能不动点波函数的条件。这些条件的解使我们可以对这类拓扑序进行分类，从而推广了拓扑序的弦网分类。我们还描述了一种由局部酉变换引起的波函数重整化算法。该算法允许我们计算不在不动点处的张量积波函数的流动。这将允许我们在一般张量积状态下计算拓扑阶数以及对称破缺阶数。
Two gapped quantum ground states in the same phase are connected by an adiabatic evolution which gives rise to a local unitary transformation that maps between the states. On the other hand, gapped ground states remain within the same phase under local unitary transformations. Therefore, local unitary transformations define an equivalence relation and the equivalence classes are the universality classes that define the different phases for gapped quantum systems. Since local unitary transformations can remove local entanglement, the above equivalence/universality classes correspond to pattern of long-range entanglement, which is the essence of topological order. The local unitary transformation also allows us to define a wave function renormalization scheme, under which a wave function can flow to a simpler one within the same equivalence/universality class. Using such a setup, we find conditions on the possible fixed-point wave functions where the local unitary transformations have finite dimensions. The solutions of the conditions allow us to classify this type of topological orders, which generalize the string-net classification of topological orders. We also describe an algorithm of wave function renormalization induced by local unitary transformations. The algorithm allows us to calculate the flow of tensor-product wave functions which are not at the fixed points. This will allow us to calculate topological orders as well as symmetry-breaking orders in a generic tensor-product state.