Monte Carlo methods have led to profound insights into the strong-coupling behaviour of lattice gauge theories and produced remarkable results such as first-principles computations of hadron masses. Despite tremendous progress over the last four decades, fundamental challenges such as the sign problem and the inability to simulate real-time dynamics remain. Neural network quantum states have emerged as an alternative method that seeks to overcome these challenges. In this work, we use gauge-invariant neural network quantum states to accurately compute the ground state of $\mathbb{Z}_N$ lattice gauge theories in $2+1$ dimensions. Using transfer learning, we study the distinct topological phases and the confinement phase transition of these theories. For $\mathbb{Z}_2$, we identify a continuous transition and compute critical exponents, finding excellent agreement with existing numerics for the expected Ising universality class. In the $\mathbb{Z}_3$ case, we observe a weakly first-order transition and identify the critical coupling. Our findings suggest that neural network quantum states are a promising method for precise studies of lattice gauge theory.

翻译：蒙特卡洛方法为格点规范理论的强耦合行为提供了深刻见解，并产生了诸如强子质量第一性原理计算等显著成果。尽管过去四十年取得了巨大进展，但符号问题和无法模拟实时动力学等根本性挑战依然存在。神经网络量子态作为一种替代方法应运而生，旨在克服这些挑战。本工作中，我们采用规范不变的神经网络量子态，精确计算了$2+1$维$\mathbb{Z}_N$格点规范理论的基态。通过迁移学习，我们研究了这些理论的拓扑相及禁闭相变。对于$\mathbb{Z}_2$理论，我们识别出连续相变并计算了临界指数，结果与预期伊辛普适类的现有数值模拟高度吻合。在$\mathbb{Z}_3$情形中，我们观察到弱一级相变并确定了临界耦合值。我们的研究结果表明，神经网络量子态是精确研究格点规范理论的一种极具前景的方法。