量子多体问题的随机抽样神经网络
Random Sampling Neural Network for Quantum Many-Body Problems
摘要：
量子多体系统的本征值问题是凝聚态物理中的一个基本且具有挑战性的课题，因为希尔伯特空间的维数（以及因此所需的计算内存和时间）随着系统尺寸的增加而呈指数增长。一些数值方法已经开发了一些特定的系统，但可能不适用于其他。在这里，我们提出了一个通用的数值方法，随机采样神经网络（RSNN），利用模式识别技术的随机采样矩阵元素的相互作用的多体系统通过自监督学习方法。用几种精确可解的一维模型，包括带横场的伊辛模型、哈伯德模型和自旋- $1/2$ $XXZ$模型，检验了RSNN的适用性.在强关联区或量子相变点附近，即使在弱相互作用区训练相应的RSNN模型，也能获得相当高的能谱、磁化强度和临界指数等精度。所需的计算时间与系统大小成线性关系。我们的研究结果表明，它是可能的，结合现有的数值方法的训练过程和RSNN探索量子多体问题在一个更广泛的参数制度，即使是强关联系统。
英文摘要：
The eigenvalue problem of quantum many-body systems is a fundamental and challenging subject in condensed matter physics, since the dimension of the Hilbert space (and hence the required computational memory and time) grows exponentially as the system size increases. A few numerical methods have been developed for some specific systems, but may not be applicable in others. Here we propose a general numerical method, Random Sampling Neural Networks (RSNN), to utilize the pattern recognition technique for the random sampling matrix elements of an interacting many-body system via a self-supervised learning approach. Several exactly solvable 1D models, including Ising model with transverse field, Fermi-Hubbard model, and spin-$1/2$ $XXZ$ model, are used to test the applicability of RSNN. Pretty high accuracy of energy spectrum, magnetization and critical exponents etc. can be obtained within the strongly correlated regime or near the quantum phase transition point, even the corresponding RSNN models are trained in the weakly interacting regime. The required computation time scales linearly to the system size. Our results demonstrate that it is possible to combine the existing numerical methods for the training process and RSNN to explore quantum many-body problems in a much wider parameter regime, even for strongly correlated systems.
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