标题： 用神经网络学习多个不相交区间的Renyi熵 显示英文标题

标题： Machine learning the Renyi entropy of multiple disjoint intervals with neural networks

摘要： Renyi熵通过改进的交换操作用两种方法计算：一种是通过哈密顿量的直接对角化，另一种是通过使用神经网络的最先进的机器学习方法。 我们使用一维横场伊辛模型来演示改进的交换操作策略。 特别是，我们研究了具有两个、三个和四个不相交区间的第二Renyi熵。 我们发现这两种方法的结果在误差范围内非常吻合，这表明机器学习方法适用于计算多个不相交区间的Renyi熵。 此外，随着磁场的增加，Renyi熵也会增加，直到系统到达相变的临界点。 然而，当磁场超过临界值时，Renyi熵会下降，因为系统进入了顺磁相。 总体而言，这些结果与理论预测非常吻合，并展示了神经网络机器学习方法的高准确性。 显示英文摘要

摘要： Renyi entropy with multiple disjoint intervals are computed from the improved swapping operations by two methods: one is from the direct diagonalization of the Hamiltonian and the other one is from the state-of-the-art machine learning method with neural networks. We use the paradigmatic transverse-field Ising model in one-dimension to demonstrate the strategy of the improved swapping operation. In particular, we study the second Renyi entropy with two, three and four disjoint intervals. We find that the results from the above two methods match each other very well within errors, which indicates that the machine learning method is applicable for calculating the Renyi entropy with multiple disjoint intervals. Moreover, as the magnetic field increases, the Renyi entropy grows as well until the system arrives at the critical point of the phase transition. However, as the magnetic field exceeds the critical value, the Renyi entropy will decrease since the system enters the paramagnetic phase. Overall, these results match the theoretical predictions very well and demonstrate the high accuracy of the machine learning methods with neural networks.