费米子的路径积分分子动力学：用Bogoliubov不等式解决符号问题
Path Integral Molecular Dynamics for Fermions: Alleviating the Sign Problem with the Bogoliubov Inequality
摘要：
提出了一种对费米子进行路径积分分子动力学（PIMD）模拟的方法，并解决了其符号问题。PIMD模拟被广泛用于研究处于热平衡的多体量子系统。然而，他们假设粒子是可区分的，忽略了玻色子和费米子交换效应。相互作用的费米子在许多化学和物理系统中起着关键作用，例如量子点中的电子和超冷囚禁原子。由于费米子配分函数的被积函数不是正定的，因此不可能用PIMD直接采样。我们表明，PIMD模拟费米子是可行的，采用我们最近开发的方法玻色子PIMD和重新加权的结果，以获得费米子的期望值。测试的方法对路径积分蒙特卡罗（PIMC）模拟多达7个电子在一个二维量子点的相互作用强度的范围。然而，与PIMC一样，该方法在低温下也存在符号问题。我们提出了一个简单的方法来减轻它模拟一个辅助系统具有较大的平均符号，并获得一个上界的原系统的能量使用Bogoliubov不等式。这使得费米子可以在比PIMD更低的温度下进行研究，就像三电子量子点的情况一样。我们的研究结果扩展了费米子的PIMD模拟的边界，并有望刺激解决符号问题的新方法的发展。
英文摘要：
We present a method for performing path integral molecular dynamics (PIMD) simulations for fermions and address its sign problem. PIMD simulations are widely used for studying many-body quantum systems at thermal equilibrium. However, they assume that the particles are distinguishable and neglect bosonic and fermionic exchange effects. Interacting fermions play a key role in many chemical and physical systems, such as electrons in quantum dots and ultracold trapped atoms. A direct sampling of the fermionic partition function is impossible using PIMD since its integrand is not positive definite. We show that PIMD simulations for fermions are feasible by employing our recently developed method for bosonic PIMD and reweighting the results to obtain fermionic expectation values. The approach is tested against path integral Monte Carlo (PIMC) simulations for up to 7 electrons in a two-dimensional quantum dot for a range of interaction strengths. However, like PIMC, the method suffers from the sign problem at low temperatures. We propose a simple approach for alleviating it by simulating an auxiliary system with a larger average sign and obtaining an upper bound to the energy of the original system using the Bogoliubov inequality. This allows fermions to be studied at temperatures lower than would otherwise have been feasible using PIMD, as demonstrated in the case of a three-electron quantum dot. Our results extend the boundaries of PIMD simulations of fermions and will hopefully stimulate the development of new approaches for tackling the sign problem.
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