论文标题
基于典型性估计的有限尺寸缩放尺寸
Finite-size scaling of typicality-based estimates
论文作者
论文摘要
根据典型性的概念，相对于从高维希尔伯特空间随机绘制的单个纯状态，可以通过期望值准确地近似集合平均值。这种随机矢量近似或痕量估计器为具有较大希尔伯特空间尺寸的系统的热力学数量提供了强大的方法，通常无法准确地，分析或数值对其进行处理。在这里，我们从两个角度讨论了此类痕量估计器的准确性的有限尺寸缩放。首先，我们研究了随机矢量预期值的完全概率分布，其次是标准偏差的全温度依赖性。在数值示例的帮助下，我们发现了明显的高斯概率分布，并且与系统大小的预期偏差的预期减小，至少高于某些系统特异性温度。在下方，尤其是对于比激发差距小的温度，没有简单的规则。
According to the concept of typicality, an ensemble average can be accurately approximated by an expectation value with respect to a single pure state drawn at random from a high-dimensional Hilbert space. This random-vector approximation, or trace estimator, provides a powerful approach to, e.g., thermodynamic quantities for systems with large Hilbert-space sizes, which usually cannot be treated exactly, analytically or numerically. Here, we discuss the finite-size scaling of the accuracy of such trace estimators from two perspectives. First, we study the full probability distribution of random-vector expectation values and, second, the full temperature dependence of the standard deviation. With the help of numerical examples, we find pronounced Gaussian probability distributions and the expected decrease of the standard deviation with system size, at least above certain system-specific temperatures. Below and in particular for temperatures smaller than the excitation gap, simple rules are not available.
