We introduce the group-equivariant autoencoder (GE-autoencoder) -- a deep neural network (DNN) method that locates phase boundaries by determining which symmetries of the Hamiltonian have spontaneously broken at each temperature. We use group theory to deduce which symmetries of the system remain intact in all phases, and then use this information to constrain the parameters of the GE-autoencoder such that the encoder learns an order parameter invariant to these ``never-broken'' symmetries. This procedure produces a dramatic reduction in the number of free parameters such that the GE-autoencoder size is independent of the system size. We include symmetry regularization terms in the loss function of the GE-autoencoder so that the learned order parameter is also equivariant to the remaining symmetries of the system. By examining the group representation by which the learned order parameter transforms, we are then able to extract information about the associated spontaneous symmetry breaking. We test the GE-autoencoder on the 2D classical ferromagnetic and antiferromagnetic Ising models, finding that the GE-autoencoder (1) accurately determines which symmetries have spontaneously broken at each temperature; (2) estimates the critical temperature in the thermodynamic limit with greater accuracy, robustness, and time-efficiency than a symmetry-agnostic baseline autoencoder; and (3) detects the presence of an external symmetry-breaking magnetic field with greater sensitivity than the baseline method. Finally, we describe various key implementation details, including a new method for extracting the critical temperature estimate from trained autoencoders and calculations of the DNN initialization and learning rate settings required for fair model comparisons.

翻译：我们提出了群等变自编码器（GE-autoencoder）——一种深度神经网络（DNN）方法，通过确定哈密顿量的各对称性在每个温度下是否自发破缺来定位相边界。我们利用群论推导系统中在所有相中保持完整的对称性，并据此约束GE-autoencoder的参数，使得编码器学习到一个对这些"永不破缺"对称性保持不变性的序参量。这一过程大幅减少了自由参数的数量，使得GE-autoencoder的规模与系统规模无关。我们在GE-autoencoder的损失函数中引入对称性正则化项，使所学习的序参量也对系统的其余对称性具有等变性。通过分析所学习序参量变换的群表示，我们能够提取相关的自发对称性破缺信息。我们在二维经典铁磁和反铁磁伊辛模型上测试了GE-autoencoder，发现该模型能够：（1）精确确定每个温度下哪些对称性已自发破缺；（2）在热力学极限下以更高精度、鲁棒性和时效性估算临界温度，优于不利用对称性的基准自编码器；（3）以高于基准方法的灵敏度检测外部对称性破缺磁场的存在。最后，我们描述了若干关键实现细节，包括从训练好的自编码器中提取临界温度估计值的新方法，以及为公平模型比较所需的DNN初始化和学习率设置计算。