张量网络状态高阶矩和累积量的有效计算
Efficient evaluation of high-order moments and cumulants in tensor network states
摘要：
我们提出了一个数值方案，有效地提取高阶矩和累积量的各种运营商的自旋系统表示为张量积状态，有限和无限的系统，并提出了几个应用这些数量。例如，一个状态的能量的第二累积量，$\langle ΔH^2 \rangle$，给出了一个简单的方法来检查数值基态近似算法的收敛性。此外，我们还讨论了矩和累积量在相变研究中的应用。特别感兴趣的是我们的方法来计算所谓的宾德累积量，我们用它来检测临界点和研究的相关长度的临界指数只有小的有限的数值计算的应用。我们应用这些方法来研究一族一维模型（横向伊辛模型、自旋1伊辛模型和晶场中自旋1伊辛模型）以及正方形晶格上的二维伊辛模型的行为。我们的研究结果表明，在一维，基于累积量的方法可以产生精确的估计的临界点在一个较低的计算成本，并显示二维系统以及承诺。
英文摘要：
We present a numerical scheme for efficiently extracting the higher-order moments and cumulants of various operators on spin systems represented as tensor product states, for both finite and infinite systems, and present several applications for such quantities. For example, the second cumulant of the energy of a state, $\langle ΔH^2 \rangle$, gives a straightforward method to check the convergence of numerical ground-state approximation algorithms. Additionally, we discuss the use of moments and cumulants in the study of phase transitions. Of particular interest is the application of our method to calculate the so-called Binder's cumulant, which we use to detect critical points and study the critical exponent of the correlation length with only small finite numerical calculations. We apply these methods to study the behavior of a family of one-dimensional models (the transverse Ising model, the spin-1 Ising model, and the spin-1 Ising model in a crystal field), as well as the two-dimensional Ising model on a square lattice. Our results show that in one dimension, cumulant-based methods can produce precise estimates of the critical points at a low computational cost, and show promise for two-dimensional systems as well.
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