【摘要】 通过序参量来研究量子相变是比较传统的做法,而从机器学习的角度研究相变是一块全新的领域.本文提出了先采用无监督学习算法中的高斯混合模型对J₁-J₂反铁磁海森伯自旋链系统的态矢量进行分类,再使用监督学习算法中的卷积神经网络鉴别无监督学习算法给出的分类点是否是相变点的方法,并使用交叉验证的方法对学习效果进行验证.结果表明,上述机器学习方法可以从基态精确找到J₁–J₂反铁磁海森伯自旋链系统的一阶相变点、无法找到无穷阶相变点,从第一激发态不仅能找到一阶相变点,还能找到无穷阶相变点.更多还原

【Abstract】 Studying quantum phase transitions through order parameters is a traditional method, but studying phase transitions by machine learning is a brand new field. The ability of machine learning to classify, identify, or interpret massive data sets may provide physicists with similar analyses of the exponentially large data sets embodied in the Hilbert space of quantum many-body system. In this work,we propose a method of using unsupervised learning algorithm of the Gaussian mixture model to classify the state vectors of the J₁-J₂ antiferromagnetic Heisenberg spin chain system, then the supervised learning algorithm of the convolutional neural network is used to identify the classification point given by the unsupervised learning algorithm,and the cross-validation method is adopted to verify the learning effect. Using this method, we study the J₁-J₂ Heisenberg spin chain system with chain length N = 8, 10, 12, 16 and obtain the same conclusion. The first order phase transition point of J₁-J₂ antiferromagnetic Heisenberg spin chain system can be accurately found from the ground state vector, but the infinite order phase transition point cannot be found from the ground state vector. The first order and the infinite order phase transition point can be found from the first excited state vector,which indirectly shows that the first excited state may contain more information than the ground state of J₁-J₂ antiferromagnetic Heisenberg spin chain system. The visualization of the state vector shows the reliability of the machine learning algorithm, which can extract the feature information from the state vector.The result reveals that the machine learning techniques can directly find some possible phase transition points from a large set of state vectorwithout prior knowledge of the energy or locality conditions of the Hamiltonian,which may assists us in studying unknown systems. Supervised learning can verify the phase transition points given by unsupervised learning, thereby indicating that we can discover some useful information about unknown systems only through machine learning techniques. Machine learning techniques can be a basic research tool in strong quantum-correlated systems, and it can be adapted to more complex systems, which can help us dig up hidden information.更多还原