论文标题
一维$ J_1-J_2 $ HEISENBERG型号的限制性玻尔兹曼机器的精度
Accuracy of Restricted Boltzmann Machines for the one-dimensional $J_1-J_2$ Heisenberg model
论文作者
论文摘要
最近已提出神经网络作为量子多体系统的变异波函数[G. Carleo和M. Troyer，Science 355，602（2017）]。在这项工作中，我们专注于一种特定的体系结构，称为限制性玻尔兹曼机器（RBM），并分析了其在一个空间维度中的Spin-1/2 $ J_1-J_2 $ Antiferromagnetic Heisenberg模型的准确性。该模型的基础状态具有非平凡的标志结构，尤其是对于$ J_2/j_1> 0.5 $，迫使我们与复杂价值的RBM一起工作。讨论了两个变异的Ansätze：一个通过完全复杂的RBM定义，其中两个不同的实现网络用于近似模量和波函数的相位。在这两种情况下，都通过考虑RBM的线性组合来实现翻译不变性，还可以访问固定动量$ k $的最低能量激发。与精确结果相比，我们对小簇进行了系统的研究，以评估这些波功能的准确性，从而提供了完全复杂的RBM的至高无上的证据。我们的计算表明，这种Ansätze非常灵活，并描述了无间隙和间隙的接地状态，还以$ J_2/j_1> 0.5 $的价格捕获了不相称的自旋旋转相关性和低能频谱。还将RBM结果与用Gutzwiller-Pro注射的费米度态获得的RBM结果进行了比较，该结果通常用于描述量子自旋模型[F。 Ferrari，A。Parola，S。Sorella和F. Becca，物理学。 Rev. B 97，235103（2018）]。与后者类别的变分状态相反，RBM的完全连接结构阻碍了波函数从小群集到大簇的可传递性，这意味着随着系统大小的计算成本增加。
Neural networks have been recently proposed as variational wave functions for quantum many-body systems [G. Carleo and M. Troyer, Science 355, 602 (2017)]. In this work, we focus on a specific architecture, known as Restricted Boltzmann Machine (RBM), and analyse its accuracy for the spin-1/2 $J_1-J_2$ antiferromagnetic Heisenberg model in one spatial dimension. The ground state of this model has a non-trivial sign structure, especially for $J_2/J_1>0.5$, forcing us to work with complex-valued RBMs. Two variational Ansätze are discussed: one defined through a fully complex RBM, and one in which two different real-valued networks are used to approximate modulus and phase of the wave function. In both cases, translational invariance is imposed by considering linear combinations of RBMs, giving access also to the lowest-energy excitations at fixed momentum $k$. We perform a systematic study on small clusters to evaluate the accuracy of these wave functions in comparison to exact results, providing evidence for the supremacy of the fully complex RBM. Our calculations show that this kind of Ansätze is very flexible and describes both gapless and gapped ground states, also capturing the incommensurate spin-spin correlations and low-energy spectrum for $J_2/J_1>0.5$. The RBM results are also compared to the ones obtained with Gutzwiller-projected fermionic states, often employed to describe quantum spin models [F. Ferrari, A. Parola, S. Sorella and F. Becca, Phys. Rev. B 97, 235103 (2018)]. Contrary to the latter class of variational states, the fully-connected structure of RBMs hampers the transferability of the wave function from small to large clusters, implying an increase of the computational cost with the system size.
