We show that Restricted Boltzmann Machines (RBMs) provide a flexible generative framework for modeling spin configurations in disordered yet strongly correlated phases of frustrated magnets. As a benchmark, we first demonstrate that an RBM can learn the zero-temperature ground-state manifold of the one-dimensional ANNNI model at its multiphase point, accurately reproducing its characteristic oscillatory and exponentially decaying correlations. We then apply RBMs to kagome spin ice and show that they successfully learn the local ice rules and short-range correlations of the extensively degenerate ice-I manifold. Correlation functions computed from RBM-generated configurations closely match those from direct Monte Carlo simulations. For the partially ordered ice-II phase -- featuring long-range charge order and broken time-reversal symmetry -- accurate modeling requires RBMs with uniform-sign bias fields, mirroring the underlying symmetry breaking. These results highlight the utility of RBMs as generative models for learning constrained and highly frustrated magnetic states.

翻译：我们证明，受限玻尔兹曼机（RBMs）为建模受挫磁体无序但强关联相中的自旋构型提供了一个灵活的生成框架。作为基准测试，我们首先证明RBM能够学习一维ANNNI模型在其多相点处的零温基态流形，精确复现了其特征性的振荡和指数衰减关联。随后，我们将RBMs应用于Kagome自旋冰，并表明它们成功学习了广泛简并的ice-I流形的局域冰规则和短程关联。由RBM生成构型计算得到的关联函数与直接蒙特卡洛模拟的结果高度吻合。对于具有长程电荷序和破缺时间反演对称性的部分有序ice-II相，精确建模需要采用具有均匀符号偏置场的RBMs，这反映了底层的对称性破缺。这些结果凸显了RBMs作为生成模型在学习受约束和高度受挫磁态方面的实用性。