Dipartimento di Fisica, Università degli Studi di Milano, via Celoria 16, 20133 Milano, Italy.
J Chem Phys. 2009 Oct 21;131(15):154108. doi: 10.1063/1.3247833.
Generally "exact" quantum Monte Carlo computations for the ground state of many bosons make use of importance sampling. The importance sampling is based either on a guiding function or on an initial variational wave function. Here we investigate the need of importance sampling in the case of path integral ground state (PIGS) Monte Carlo. PIGS is based on a discrete imaginary time evolution of an initial wave function with a nonzero overlap with the ground state, which gives rise to a discrete path which is sampled via a Metropolis-like algorithm. In principle the exact ground state is reached in the limit of an infinite imaginary time evolution, but actual computations are based on finite time evolutions and the question is whether such computations give unbiased exact results. We have studied bulk liquid and solid (4)He with PIGS by considering as initial wave function a constant, i.e., the ground state of an ideal Bose gas. This implies that the evolution toward the ground state is driven only by the imaginary time propagator, i.e., there is no importance sampling. For both phases we obtain results converging to those obtained by considering the best available variational wave function (the shadow wave function) as initial wave function. Moreover we obtain the same results even by considering wave functions with the wrong correlations, for instance, a wave function of a strongly localized Einstein crystal for the liquid phase. This convergence is true not only for diagonal properties such as the energy, the radial distribution function, and the static structure factor, but also for off-diagonal ones, such as the one-body density matrix. This robustness of PIGS can be traced back to the fact that the chosen initial wave function acts only at the beginning of the path without affecting the imaginary time propagator. From this analysis we conclude that zero temperature PIGS calculations can be as unbiased as those of finite temperature path integral Monte Carlo. On the other hand, a judicious choice of the initial wave function greatly improves the rate of convergence to the exact results.
一般来说，许多玻色子基态的“精确”量子蒙特卡罗计算都利用了重要性抽样。重要性抽样要么基于引导函数，要么基于初始变分波函数。在这里，我们研究了路径积分基态（PIGS）蒙特卡罗中重要性抽样的必要性。PIGS 基于初始波函数在非零与基态重叠的虚时间上的离散演化，这导致了一条离散路径，通过类似于 Metropolis 的算法进行采样。原则上，在无限虚时间演化的极限下可以达到精确的基态，但实际计算是基于有限的时间演化，问题是这样的计算是否能给出无偏的精确结果。我们通过将初始波函数视为常数（即理想玻色气体的基态）来研究体相液体和固体（4）He 的 PIGS。这意味着向基态的演化仅由虚时间传播子驱动，即没有重要性抽样。对于这两个相，我们得到的结果都收敛于以最佳可用变分波函数（影子波函数）为初始波函数得到的结果。此外，即使考虑具有错误相关性的波函数，例如液体相的强局域爱因斯坦晶体的波函数，我们也得到相同的结果。这种收敛不仅适用于对角性质，如能量、径向分布函数和静态结构因子，也适用于非对角性质，如单粒子密度矩阵。PIGS 的这种稳健性可以追溯到所选择的初始波函数仅在路径的开始起作用而不影响虚时间传播子的事实。从这个分析中，我们得出结论，零温 PIGS 计算可以与有限温路径积分蒙特卡罗计算一样无偏。另一方面，初始波函数的明智选择可以大大提高收敛到精确结果的速度。
